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Types of statistical analysis, importance of statistical analysis, benefits of statistical analysis, statistical analysis process, statistical analysis methods, statistical analysis software, statistical analysis examples, career in statistical analysis, choose the right program, become proficient in statistics today, what is statistical analysis types, methods and examples.

What Is Statistical Analysis?

Statistical analysis is the process of collecting and analyzing data in order to discern patterns and trends. It is a method for removing bias from evaluating data by employing numerical analysis. This technique is useful for collecting the interpretations of research, developing statistical models, and planning surveys and studies.

Statistical analysis is a scientific tool in AI and ML that helps collect and analyze large amounts of data to identify common patterns and trends to convert them into meaningful information. In simple words, statistical analysis is a data analysis tool that helps draw meaningful conclusions from raw and unstructured data. 

The conclusions are drawn using statistical analysis facilitating decision-making and helping businesses make future predictions on the basis of past trends. It can be defined as a science of collecting and analyzing data to identify trends and patterns and presenting them. Statistical analysis involves working with numbers and is used by businesses and other institutions to make use of data to derive meaningful information. 

Given below are the 6 types of statistical analysis:

Descriptive Analysis

Descriptive statistical analysis involves collecting, interpreting, analyzing, and summarizing data to present them in the form of charts, graphs, and tables. Rather than drawing conclusions, it simply makes the complex data easy to read and understand.

Inferential Analysis

The inferential statistical analysis focuses on drawing meaningful conclusions on the basis of the data analyzed. It studies the relationship between different variables or makes predictions for the whole population.

Predictive Analysis

Predictive statistical analysis is a type of statistical analysis that analyzes data to derive past trends and predict future events on the basis of them. It uses machine learning algorithms, data mining , data modelling , and artificial intelligence to conduct the statistical analysis of data.

Prescriptive Analysis

The prescriptive analysis conducts the analysis of data and prescribes the best course of action based on the results. It is a type of statistical analysis that helps you make an informed decision. 

Exploratory Data Analysis

Exploratory analysis is similar to inferential analysis, but the difference is that it involves exploring the unknown data associations. It analyzes the potential relationships within the data. 

Causal Analysis

The causal statistical analysis focuses on determining the cause and effect relationship between different variables within the raw data. In simple words, it determines why something happens and its effect on other variables. This methodology can be used by businesses to determine the reason for failure. 

Statistical analysis eliminates unnecessary information and catalogs important data in an uncomplicated manner, making the monumental work of organizing inputs appear so serene. Once the data has been collected, statistical analysis may be utilized for a variety of purposes. Some of them are listed below:

  • The statistical analysis aids in summarizing enormous amounts of data into clearly digestible chunks.
  • The statistical analysis aids in the effective design of laboratory, field, and survey investigations.
  • Statistical analysis may help with solid and efficient planning in any subject of study.
  • Statistical analysis aid in establishing broad generalizations and forecasting how much of something will occur under particular conditions.
  • Statistical methods, which are effective tools for interpreting numerical data, are applied in practically every field of study. Statistical approaches have been created and are increasingly applied in physical and biological sciences, such as genetics.
  • Statistical approaches are used in the job of a businessman, a manufacturer, and a researcher. Statistics departments can be found in banks, insurance businesses, and government agencies.
  • A modern administrator, whether in the public or commercial sector, relies on statistical data to make correct decisions.
  • Politicians can utilize statistics to support and validate their claims while also explaining the issues they address.

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Statistical analysis can be called a boon to mankind and has many benefits for both individuals and organizations. Given below are some of the reasons why you should consider investing in statistical analysis:

  • It can help you determine the monthly, quarterly, yearly figures of sales profits, and costs making it easier to make your decisions.
  • It can help you make informed and correct decisions.
  • It can help you identify the problem or cause of the failure and make corrections. For example, it can identify the reason for an increase in total costs and help you cut the wasteful expenses.
  • It can help you conduct market analysis and make an effective marketing and sales strategy.
  • It helps improve the efficiency of different processes.

Given below are the 5 steps to conduct a statistical analysis that you should follow:

  • Step 1: Identify and describe the nature of the data that you are supposed to analyze.
  • Step 2: The next step is to establish a relation between the data analyzed and the sample population to which the data belongs. 
  • Step 3: The third step is to create a model that clearly presents and summarizes the relationship between the population and the data.
  • Step 4: Prove if the model is valid or not.
  • Step 5: Use predictive analysis to predict future trends and events likely to happen. 

Although there are various methods used to perform data analysis, given below are the 5 most used and popular methods of statistical analysis:

Mean or average mean is one of the most popular methods of statistical analysis. Mean determines the overall trend of the data and is very simple to calculate. Mean is calculated by summing the numbers in the data set together and then dividing it by the number of data points. Despite the ease of calculation and its benefits, it is not advisable to resort to mean as the only statistical indicator as it can result in inaccurate decision making. 

Standard Deviation

Standard deviation is another very widely used statistical tool or method. It analyzes the deviation of different data points from the mean of the entire data set. It determines how data of the data set is spread around the mean. You can use it to decide whether the research outcomes can be generalized or not. 

Regression is a statistical tool that helps determine the cause and effect relationship between the variables. It determines the relationship between a dependent and an independent variable. It is generally used to predict future trends and events.

Hypothesis Testing

Hypothesis testing can be used to test the validity or trueness of a conclusion or argument against a data set. The hypothesis is an assumption made at the beginning of the research and can hold or be false based on the analysis results. 

Sample Size Determination

Sample size determination or data sampling is a technique used to derive a sample from the entire population, which is representative of the population. This method is used when the size of the population is very large. You can choose from among the various data sampling techniques such as snowball sampling, convenience sampling, and random sampling. 

Everyone can't perform very complex statistical calculations with accuracy making statistical analysis a time-consuming and costly process. Statistical software has become a very important tool for companies to perform their data analysis. The software uses Artificial Intelligence and Machine Learning to perform complex calculations, identify trends and patterns, and create charts, graphs, and tables accurately within minutes. 

Look at the standard deviation sample calculation given below to understand more about statistical analysis.

The weights of 5 pizza bases in cms are as follows:

Calculation of Mean = (9+2+5+4+12)/5 = 32/5 = 6.4

Calculation of mean of squared mean deviation = (6.76+19.36+1.96+5.76+31.36)/5 = 13.04

Sample Variance = 13.04

Standard deviation = √13.04 = 3.611

A Statistical Analyst's career path is determined by the industry in which they work. Anyone interested in becoming a Data Analyst may usually enter the profession and qualify for entry-level Data Analyst positions right out of high school or a certificate program — potentially with a Bachelor's degree in statistics, computer science, or mathematics. Some people go into data analysis from a similar sector such as business, economics, or even the social sciences, usually by updating their skills mid-career with a statistical analytics course.

Statistical Analyst is also a great way to get started in the normally more complex area of data science. A Data Scientist is generally a more senior role than a Data Analyst since it is more strategic in nature and necessitates a more highly developed set of technical abilities, such as knowledge of multiple statistical tools, programming languages, and predictive analytics models.

Aspiring Data Scientists and Statistical Analysts generally begin their careers by learning a programming language such as R or SQL. Following that, they must learn how to create databases, do basic analysis, and make visuals using applications such as Tableau. However, not every Statistical Analyst will need to know how to do all of these things, but if you want to advance in your profession, you should be able to do them all.

Based on your industry and the sort of work you do, you may opt to study Python or R, become an expert at data cleaning, or focus on developing complicated statistical models.

You could also learn a little bit of everything, which might help you take on a leadership role and advance to the position of Senior Data Analyst. A Senior Statistical Analyst with vast and deep knowledge might take on a leadership role leading a team of other Statistical Analysts. Statistical Analysts with extra skill training may be able to advance to Data Scientists or other more senior data analytics positions.

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Hope this article assisted you in understanding the importance of statistical analysis in every sphere of life. Artificial Intelligence (AI) can help you perform statistical analysis and data analysis very effectively and efficiently. 

If you are a science wizard and fascinated by the role of AI in statistical analysis, check out this amazing Caltech Post Graduate Program in AI & ML course in collaboration with Caltech. With a comprehensive syllabus and real-life projects, this course is one of the most popular courses and will help you with all that you need to know about Artificial Intelligence. 

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What Is Statistical Analysis: Types, Methods, Steps & Examples

Statistical Analysis

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Statistical analysis is the process of analyzing data in an effort to recognize patterns, relationships, and trends. It involves collecting, arranging and interpreting numerical data and using statistical techniques to draw conclusions.

Statistical analysis in research is a powerful tool used in various fields to make sense of quantitative data. Numbers speak for themselves and help you make assumptions on what may or may not happen if you take a certain course of action.

For example, let's say that you run an ecommerce business that sells coffee. By analyzing the amount of sales and the quantity of coffee produced, you can guess how much more coffee you should manufacture in order to increase sales.

In this blog by dissertation services , we will explore the basics of statistical analysis, including its types, methods, and steps on how to analyze statistical data. We will also provide examples to help you understand how statistical analysis methods are applied in different contexts.

What Is Statistical Analysis: Definition

Statistical analysis is a set of techniques used to analyze data and draw inferences about the population being studied. It involves organizing data, summarizing key patterns , and calculating the probability that observations could have occurred randomly. Statistics help to test hypotheses and determine the link between independent and dependent variables .

It is widely used to optimize processes, products, and services in various fields, including:

  • Social sciences, etc.

The ultimate goal of statistical analysis is to extract meaningful insights from data and make predictions about causal relationships. It can also allow researchers to make generalizations about entire populations.

Types of Statistical Analysis

In general, there are 7 different types of statistical analysis, with descriptive, inferential and predictive ones being the most commonly used.

  • Summarizes data in tables, charts, or graphs to help you find patterns.
  • Includes calculating averages, percentages, mean, median and standard deviation.
  • Draws inferences from a sample and estimates characteristics of a population, generalizing insights from a smaller group to a larger one.
  • Includes hypothesis testing and confidence intervals.
  • Uses data to oversee future trends and patterns.
  • Relies on regression analysis and machine learning techniques.
  • Uses data to make informed decisions and suggest actions.
  • Comprises optimization models and network analysis.
  • Investigates data and discovers relationships between variables.
  • Requires cluster analysis, principal component analysis, and factor analysis.
  • Examines the effect of one or more independent variables on a dependent variable.
  • Implies experiments, surveys, and interviews.
  • Studies how different variables interact and affect each other.
  • Includes mathematical models and simulations.

What Are Statistics Used for?

People apply statistics for a variety of purposes across numerous fields, including research, business and even everyday life. Researchers most frequently opt for statistical methods in research in such cases:

  • To scrutinize a dataset in experimental and non-experimental research designs and describe the core features
  • To test the validity of a claim and determine whether occurring outcomes are due to an actual effect
  • To model a causal connection between variables and foresee potential links
  • To monitor and improve the quality of products or services by spotting trends
  • To assess and manage potential risks.

As you can see, we can avail from statistical analysis tools in literally any area of our life to interpret our surroundings and observe tendencies. Any assumptions that we make after studying a sample can either make or break our research efforts. And a meticulous statistical analysis will ensure that you are making the best guess.

Statistical Analysis Methods

There is no shortage of statistical methods and techniques that can be exercised to make assumptions. When done right, these methods will streamline your research and enlighten you with meaningful insights into the correlation between various factors or processes.

As a student or researcher, you will most likely deal with the following statistical methods of data analysis in your studies:

  • Mean: average value of a dataset.
  • Standard deviation: measure of variability in data.
  • Regression: predicting one variable based on another.
  • Hypothesis testing: statistical testing of a hypothesis.
  • Sample size: number of individuals to be observed.

Let's discuss each of these statistical analysis techniques in more detail.

Imagine that you need to figure out the standard value in a set of numbers. Mean is a common type of statistical research methods that gives a measure of the average value.

The mean value is calculated by summing up all data points and then dividing it by the number of individuals. It's a useful method for exploratory analysis as it shows how much of the data fall close to the average.

You want to calculate the average age of 500 people working in your enterprise. You would add up the ages of all 500 people and divide by 500 to calculate the mean age: (25+31+27+28+34...)/500=27.

Standard Deviation

Sometimes, you will need to figure out how your data is distributed. That's where a standard deviation comes in! The standard deviation is a statistical method that gives a clue of how far your data is located from the average value (mean).

A higher standard deviation indicates that the data is more spread out from the mean, while a lower standard deviation indicates that the data is more tightly clustered around the mean.

Let's take the same example as above and calculate how much the ages fluctuate from the average value, which is 27. You would subtract each age from the mean and then square the result. Then you add up all results and divide them by 500 (the number of individuals). You would end up with the standard deviation of your data set.

Regression is one of the most powerful types of statistical methods, as it allows you to make accurate predictions based on existing data. It showcases the link between two or more variables and allows you to estimate any unknown values. By using regression, you can measure how one factor impacts another one and forecast future values of the dependent variable.

You want to predict the price of a house based on its size. You would retrieve details on the size and price of several houses in a given district. You would then use regression analysis to determine if the size affects pricing. After recognizing a positive correlation between variables, you could then develop an equation that will allow to prognose the price of a house based on its size.

Hypothesis Testing

Hypothesis testing is another statistical analysis tool which allows you to ascertain if your assumptions hold true or not. By conducting tests, you can prove or disprove your hypothesis.

You are testing a new drug and would like to know if it has any effect on lowering cholesterol level. You can use hypothesis testing to compare the results of your treatment group and control group . Significant difference between results would imply that the drug can decrease cholesterol levels.

Sample Size

In order to draw reliable conclusions from your data analysis, you need to have a sample size large enough to provide you with accurate results. The size of the sample can greatly influence the reliability of your analysis, so it's important to decide on the right number of individuals.

You want to conduct a survey about customer satisfaction in your business. The sample size should be broad enough to offer you representative results. You would need to question as many clients as possible to obtain insightful information.

These are just a few examples of statistical analysis and its methods. By using them wisely, you will be able to make accurate verdicts.

Statistical Analysis Process

Now that you are familiar with the most essential methods and tools for statistical analysis, you are ready to get started with the process itself. Below we will explain how to perform statistical analysis in the right order. Stick to our detailed steps to run a foolproof study like a professional statistical analyst.

1. Prepare Your Hypotheses

Before you start digging into the numbers, it's important to formulate a hypothesis . 

Generally, there are two types of hypotheses that you will need to divide – a null hypothesis and an alternative hypothesis. The null assumption implies that the studied phenomenon is not true, while the alternative one suggests that it’s actually true.

First, detect a research question or problem that you want to investigate. Then, you should build 2 opposite statements that outline the relationship between variables in your study.

For example if you want to check how some specific exercise influences a person's resting heart rate, your hypotheses might look like this:

Null hypothesis: The exercise has no effect on resting heart rate. Alternative hypothesis:  The exercise reduces resting heart rate.

2. Collect Data

Your next step in conducting statistical data analysis is to make sure that you are working with the right data. After all, you don't want to realize that the information you obtained doesn't fit your research design .

To choose appropriate data for your study, keep a few key points in mind. First, you'll want to identify a trustworthy data source. This could be data from the primary source – a survey, poll or experiment you've conducted, or the secondary source – from existing databases, research articles, or other scholarly publications. If you are running an authentic research, most likely you will need to organize your own experimental study or survey.

You should also have enough data to work with. Decide on an adequate sample size or a sufficient time period. This will help make your data analysis applicable to broader populations.

As you're gathering data , don't forget to check its format and accessibility. You'll want the data to be in a usable form, so you might need to convert or aggregate it as needed.

Sampling Techniques for Data Analysis

Now, let's ensure that you are acquainted with the sampling methods. In general, they fall into 2 main categories: probability and non-probability sampling.

If you are performing a survey to investigate the shopping behaviors of people living in the USA, you can use simple random sampling. This means that you will randomly select individuals from a larger population.

3. Arrange and Clean Your Data

The information you retrieve from a sample may be inconsistent and contain errors. Before doing further manipulations, you will need to preprocess data. This is a crucial step in the process of statistical analysis as it allows us to prepare information for the next step.

Arrange your data in a logical fashion and see if you can detect any discrepancies. At this stage, you will need to look for potential missing values or duplicate entries. Here are some typical issues researchers deal with when digesting their data for a statistical study:

  • Handling missing values Sometimes, certain entries might be absent. To fix this, you can either remove the entries with missing values or fill in the blanks based on already available data.
  • Transforming variables In some cases, you might need to change the way a variable is measured or presented to make it more suitable for your data analysis. This can involve adjusting the scale of the variable or making its distribution more "normal."
  • Resampling data Resampling is a technique used to alter data organization, like taking a smaller sample from a larger dataset or rearranging data points to create a new sample. This way, you will be able to enhance the accuracy of your analysis or test different scenarios.

Once your data is shovel-ready, you are ready to select statistical tools for data analysis and scrutinize the information.

4. Perform Data Analysis

Finally, we got to the most important stage – conducting data analysis. You will be surprised by the abundance of statistical methods. Your choice should largely depend on the type and scope of your research proposal or project. Keep in mind that there is no one-size-fits-all approach and your preference should be tailored to your particular research objective.

In some cases, descriptive statistics may be sufficient to answer the research question or hypothesis. For example, if you want to describe the characteristics of a population, such as the average income or education level, then descriptive statistics alone may be appropriate.

In other cases, you may need to use both descriptive and inferential statistics. For example, if you want to compare the means of 2 or more groups, such as the average income of men and women, then you would need to develop predictive models using inferential statistics or run hypothesis tests.

We will go through all scenarios so you can pick the right statistical methods for your specific instance.

Summing Up Data With Descriptive Statistics

To perform efficient statistical analysis, you need to see how numbers create a bigger picture. Some patterns aren't apparent from the first glance and may be hidden deep in raw data.

That's why your data should be presented in a clear manner. Descriptive statistics is the best way to handle this task.

Using Graphs

Your departure point is categorizing your information. Divide data into logical groups and think how to further visualize it. There are various graphical methods to uncover patterns:

  • Bar charts: present relative frequencies of different groups
  • Line charts: demonstrate how different variables change over time
  • Scatter plots: show the connection between two variables
  • Histograms: enable to detect the shape of data distribution
  • Pie charts: provide visual representation of relative frequencies
  • Box plots: help to identify significant outliers.
Imagine that you are analyzing the relationship between a person's age and their income. You have collected data on the age and income of 50 individuals, and you want to confirm if there is any relationship. You decide to use a scatter plot with age on x-axis and income on y-axis. When you look at the scatter plot, you might notice that there is a general trend of income increasing with age. This might indicate that older individuals tend to have higher incomes. However, there may be some variation in data, with some people having higher or lower incomes than you expected.

Calculating Averages

Based on how your data is distributed, you will need to calculate your averages, otherwise known as measures of central tendency. There are 3 methods allowing to analyze statistical data:

  • Mean: useful when data is normally distributed.
  • Median: a better measure in data sets with extreme outliers.
  • Mode: handy when looking for the most common value in a data set.

Assessing Variability

In addition to measures of central tendency, statistical analysts often want to assess the spread or variability of their data. There are several measures of variability popular in statistical analysis:

  • Range: Difference between the maximum and minimum values.
  • Interquartile range (IQR): Difference between the 75th percentile and 25th percentile.
  • Standard deviation: Measure of how widely values are dispersed from the mean.
  • Variance: Measure of how far a set of numbers is spread out.

While range is the simplest one, it can be influenced by extreme values. The variance and standard deviation require additional calculations, but they are more robust in terms of showing the distance of each data point from the mean.

Testing Hypotheses with Inferential Statistics

After conducting descriptive statistics, researchers can use inferential statistics to build assumptions about a larger population.

One common method of inferential statistics is hypothesis testing. This involves determining the probability that the null hypothesis is correct. If the probability is low, the null hypothesis can be denied and the alternative hypothesis is accepted. When testing hypotheses, it is important to pick the appropriate statistical test (test statistic or p value) and consider factors such as sample size, statistical significance, and effect size.

Researchers test whether a new medication is effective at treating a medical condition by randomly assigning patients to a treatment group and a control group. They measure the outcome of interest and use a t-test to determine whether the medication is effective. As a result of calculation, researchers reveal that their t-value is less than the critical value. This indicates that the difference between the treatment and control groups is not statistically significant and the null hypothesis cannot be denied. As a result, researchers can conclude that the new medication is not effective at treating this medical condition.

Another method of inferential statistics is confidence intervals, which estimate the range of values that the true population parameter is likely to fall within.

If certain conditions for variables are satisfied, you can draw statistical inference using regression analysis. This technique helps researchers devise a scheme of how variables are interconnected in a study. There are different types of regression depending on the variables you're working with:

  • Linear regression: used for predicting the value of a continuous variable.
  • Logistic regression: chosen if scientists work with categorical data.
  • Multiple regression: used to determine the relationship between several independent variables and a single outcome variable.

As you can see, there are various approaches in statistical analytics. Depending on the kind of data you are processing, you have to choose the right type of statistical analysis.

5. Interpret the Outcomes

After conducting the statistical analysis, it is important to interpret the results. This includes determining whether a hypothesis was accepted or rejected. If the hypothesis is accepted, it means that the data supports the original claim. You should further assess if data followed any patterns, and if so, what those patterns mean.

It is also important to consider any errors that could have occurred during the analysis, such as measurement error or sampling bias. These errors can affect your results and can lead to incorrect interpretations if not accounted for.

Make sure you communicate the results effectively to others. This may involve creating reports, or giving a presentation to other members of your research team. The choice of format for presenting the results will depend on the intended audience and the goals of your statistical analysis. You may also need to check the guidelines of any specific paper format you are working with. For example, if you are writing in APA style , you might need to learn more about reporting statistics in APA . 

After conducting a regression analysis, you found that there is a statistically significant positive relationship between the number of hours spent studying and the exam scores. Specifically, for every additional hour of studying, the exam score increased by an average of 5 points (β = 5.0, p < 0.001). Based on these results, you can conclude that the more time students spend studying, the higher their exam scores tend to be. However, it's important to note that there may be other factors that could also be influencing the exam scores, such as prior knowledge or natural ability. Therefore, you should account for these confounding variables when interpreting the results.

Benefits of Statistical Analysis

Statistics in research is a solid instrument for understanding numerical data in a quantitative study . Here are some of the key benefits of statistical analysis:

  • Identifying patterns and relationships
  • Testing hypotheses
  • Making assumptions and forecasts
  • Measuring uncertainty
  • Comparing data.

Statistics Drawbacks

Statistical analysis can be powerful and useful, but it also has some limitations. Some of the key cons of statistics include:

  • Reliance on data accuracy and quality
  • Inability to provide complete explanations for results
  • Chance of incorrect interpretation or application of results
  • Need for specialized knowledge or software
  • Complexity of analysis.

Bottom Line on Statistical Analysis

Statistical analysis is an essential tool for any researcher, scientist, or student who are coping with quantitative data. However, accuracy of data is paramount in any statistical analysis – if the data fails, then the results can be misleading. Therefore, you should be aware of how to do statistics and account for potential errors to obtain dependable results.

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FAQ About Statistics

1. what is a statistical method.

A statistical method is a set of techniques used to analyze data and draw conclusions about a population. Statistical methods involve using mathematical formulas, models, or algorithms to summarize data and  investigate causal relationships. They are also utilized to estimate population parameters and make predictions.

2. What is the importance of statistical analysis?

Statistical analysis is important because it allows us to make sense of data and draw conclusions that are supported by evidence, rather than relying solely on intuition. It helps us to understand the relationships between variables, test hypotheses and make predictions, which can further drive progress in various fields of study. Additionally, statistical analysis can provide a means of objectively evaluating the effectiveness of interventions, policies, or programs.

3. How can I ensure the validity of my statistical analysis results?

To ensure the validity of statistical analysis results, it's essential to use techniques that are appropriate for your research question and data type. Most statistical methods assume certain conditions about the data. Verify whether the assumptions are met before applying any method. Outliers can also significantly affect the results of statistical analysis. Remove them if they are due to data entry errors, or analyze them separately if they are legitimate data points.

4. What is the difference between statistical analysis and data analysis?

Statistical analysis is a type of data analysis that uses statistical methods, while data analysis is a broader process of examining data using various techniques. Statistical analysis is just one tool used in data analysis.

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How To Write a Statistical Research Paper: Tips, Topics, Outline

Statistical Research Paper

Working on a research paper can be a bit challenging. Some people even opt for paying online writing companies to do the job for them. While this might seem like a better solution, it can cost you a lot of money. A cheaper option is to search online for the critical parts of your essay. Your data should come from reliable sources for your research paper to be authentic. You will also need to introduce your work to your readers. It should be straightforward and relevant to the topic.  With this in mind, here is a guideline to help you succeed in your research writing. But before that, let’s see what the outline should look like.

The Outline

Table of Contents

How to write a statistical analysis paper is a puzzle many people find difficult to crack. It’s not such a challenging task as you might think, especially if you learn some helpful tips to make the writing process easier. It’s just like working on any other essay. You only need to get the format and structure right and study the process. Here is what the general outline should look like:

  • introduction;
  • problem statement;
  • objectives;
  • methodology;
  • data examination;
  • discussion;
  • conclusion and recommendations.

Let us now see some tips that can help you become a better statistical researcher.

  • Top 99+ Trending Statistics Research Topics for Students

Tips for Writing Statistics Research Paper

If you are wondering how people write their papers, you are in the right place. We’ll take a look at a few pointers that can help you come up with amazing work.

Choose A Topic

Basically, this is the most important stage of your essay. Whether you want to pay for it or not, you need a simple and accessible topic to write about. Usually, the paid research papers have a well-formed and clear topic. It helps your paper to stand out. Start off by explaining to your audience what your papers are all about. Also, check whether there is enough data to support your idea. The weaker the topic is, the harder your work will be. Is the potential theme within the realm of statistics? Can the question at hand be solved with the help of the available data? These are some of the questions someone should answer first. In the end, the topic you opt for should provide sufficient space for independent information collection and analysis.

Collect Data

This stage relies heavily on the quantity of data sources and the method used to collect them. Keep in mind that you must stick to the chosen methodology throughout your essay. It is also important to explain why you opted for the data collection method used. Plus, be cautious when collecting information. One simple mistake can compromise the entire work. You can source your data from reliable sources like google, read published articles, or experiment with your own findings. However, if your instructor provides certain recommendations, follow them instead. Don’t twist the information to fit your interest to avoid losing originality. And in case no recommendations are given, ask your instructor to provide some.

Write Body Paragraphs

Use the information garnered to create the main body of your essay. After identifying an applicable area of interest, use the data to build your paragraphs. You can start off by making a rough draft of your findings and then use it as a guide for your main essay. The next step is to construe numerical figures and make conclusions. This stage requires your proficiency in interpreting statistics. Integrate your math engagement strategies to break down those figures and pinpoint only the most meaningful parts of them. Also, include some common counterpoints and support the information with specific examples.

Create Your Essay

Now that you have all the appropriate materials at hand, this section will be easy. Simply note down all the information gathered, citing your sources as well. Make sure not to copy and paste directly to avoid plagiarism. Your content should be unique and easy to read, too. We recommend proofreading and polishing your work before making it public. In addition, be on the lookout for any grammatical, spelling, or punctuation mistakes.

This section is a summary of all your findings. Explain the importance of what you are doing. You can also include suggestions for future work. Make sure to restate what you mentioned in the introduction and touch a little bit on the method used to collect and analyze your data. In short, sum up everything you’ve written in your essay.

How to Find Statistical Topics for your Paper

Statistics is a discipline that involves collecting, analyzing, organizing, presenting, and interpreting data. If you are looking for the right topic for your work, here are a few things to consider.

●   Start by finding out what topics have already been worked on and pick the remaining areas.

●   Consider recent developments in your field of study that may inspire a new topic.

●   Think about any specific questions or problems that you have come across on your own that could be explored further.

●   Ask your advisor or mentor for suggestions.

●   Review conference proceedings, journal articles, and other publications.

●   Try using a brainstorming technique. For instance, list out related keywords and combine them in different ways to generate new ideas.

Try out some of these tips. Be sure to find something that will work for you.

Working on a statistics paper can be quite challenging to work on. But with the right information sources, everything becomes easy. This guide will help you reveal the secret of preparing such essays. Also, don’t forget to do more reading to broaden your knowledge. You can find statistics research paper examples and refer to them for ideas. Nonetheless, if you’re still not confident enough, you can always hire a trustworthy writing company to get the job done.

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Step-by-Step Guide to Statistical Analysis

It would not be wrong to say that statistics are utilised in almost every aspect of society. You might have also heard the phrase, “you can prove anything with statistics.” Or “facts are stubborn things, but statistics are pliable, which implies the results drawn from statistics can never be trusted.

But what if certain conditions are applied, and you analyse these statistics before getting somewhere? Well, that sounds totally reliable and straight from the horse’s mouth. That is what statistical analysis is.

It is the branch of science responsible for rendering various analytical techniques and tools to deal with big data. In other words, it is the science of identifying, organising, assessing and interpreting data to make interferences about a particular populace.Every statistical dissection follows a specific pattern, which we call the Statistical Analysis Process.

It precisely concerns data collection , interpretation, and presentation. Statistical analyses can be carried out when handling a huge extent of data to solve complex issues. Above all, this process delivers importance to insignificant numbers and data that often fills in the missing gaps in research.

This guide will talk about the statistical data analysis types, the process in detail, and its significance in today’s statistically evolved era.

Types of Statistical Data Analysis

Though there are many types of statistical data analysis, these two are the most common ones:

Descriptive Statistics

Inferential statistics.

Let us discuss each in detail.

It quantitatively summarises the information in a significant way so that whoever is looking at it might detect relevant patterns instantly. Descriptive statistics are divided into measures of variability and measures of central tendency. Measures of variability consist of standard deviation, minimum and maximum variables, skewness, kurtosis, and variance , while measures of central tendency include the mean, median , and mode .

  • Descriptive statistics sum up the characteristics of a data set
  • It consists of two basic categories of measures: measures of variability and measures of central tendency
  • Measures of variability describe the dispersion of data in the data set
  • Measures of central tendency define the centre of a data set

With inferential statistics , you can be in a position to draw conclusions extending beyond the immediate data alone. We use this technique to infer from the sample data what the population might think or make judgments of the probability of whether an observed difference between groups is dependable or undependable. Undependable means it has happened by chance.

  • Inferential Statistics is used to estimate the likelihood that the collected data occurred by chance or otherwise
  • It helps conclude a larger population from which you took samples
  • It depends upon the type of measurement scale along with the distribution of data

Other Types Include:

Predictive Analysis: making predictions of future events based on current facts and figures

Prescriptive Analysis: examining data to find out the required actions for a particular situation

Exploratory Data Analysis (EDA): previewing of data and assisting in getting key insights into it

Casual Analysis: determining the reasons behind why things appear in a certain way

Mechanistic Analysis: explaining how and why things happen rather than how they will take place subsequently

Statistical Data Analysis: The Process

The statistical data analysis involves five steps:

  • Designing the Study
  • Gathering Data
  • Describing the Data
  • Testing Hypotheses
  • Interpreting the Data

Step 1: Designing the Study

The first and most crucial step in a scientific inquiry is stating a research question and looking for hypotheses to support it.

Examples of research questions are:

  • Can digital marketing increase a company’s revenue exponentially?
  • Can the newly developed COVID-19 vaccines prevent the spreading of the virus?

As students and researchers, you must also be aware of the background situation. Answer the following questions.

What information is there that has already been presented by other researchers?

How can you make your study stand apart from the rest?

What are effective ways to get your findings?

Once you have managed to get answers to all these questions, you are good to move ahead to another important part, which is finding the targeted population .

What population should be under consideration?

What is the data you will need from this population?

But before you start looking for ways to gather all this information, you need to make a hypothesis, or in this case, an educated guess. Hypotheses are statements such as the following:

  • Digital marketing can increase the company’s revenue exponentially.
  • The new COVID-19 vaccine can prevent the spreading of the virus.

Remember to find the relationship between variables within a population when writing a statistical hypothesis. Every prediction you make can be either null or an alternative hypothesis.

While the former suggests no effect or relationship between two or more variables, the latter states the research prediction of a relationship or effect.

How to Plan your Research Design?

After deducing hypotheses for your research, the next step is planning your research design. It is basically coming up with the overall strategy for data analysis.

There are three ways to design your research:

1. Descriptive Design:

In a descriptive design, you can assess the characteristics of a population by using statistical tests and then construe inferences from sample data.

2. Correlational Design:

As the name suggests, with this design, you can study the relationships between different variables .

3. Experimental Design:

Using statistical tests of regression and comparison, you can evaluate a cause-and-effect relationship.

Step 2: Collecting Data

Collecting data from a population is a challenging task. It not only can get expensive but also take years to come to a proper conclusion. This is why researchers are instead encouraged to collect data from a sample.

Sampling methods in a statistical study refer to how we choose members from the population under consideration or study. If you select a sample for your study randomly, the chances are that it would be biased and probably not the ideal data for representing the population.

This means there are reliable and non-reliable ways to select a sample.

Reliable Methods of Sampling

Simple Random Sampling: a method where each member and set of members have an equal chance of being selected for the sample

Stratified Random Sampling: population here is first split into groups then members are selected from each group

Clutter Random Sampling: the population is divided into groups, and members are randomly chosen from some groups.

Systematic Random Sampling: members are selected in order. The starting point is chosen by chance, and every nth member is set for the sample.

Non-Reliable Methods of Sampling

Voluntary Response Sampling: choosing a sample by sending out a request for members of a population to join. Some might join, and others might not respond

Convenient Sampling: selecting a sample readily available by chance

Here are a few important terms you need to know for conducting samples in statistics:

Population standard deviation: estimated population parameter on the basis of the previous study

Statistical Power: the chances of your study detecting an effect of a certain size

Expected Effect Size: it is an indication of how large the expected findings of your research be

Significance Level (alpha): it is the risk of rejecting a true null hypothesis

Step 3: Describing the Data

Once you are done finalising your samples, you are good to go with their inspection by calculating descriptive statistics , which we discussed above.

There are different ways to inspect your data.

  • By using a scatter plot to visualise the relationship between two or more variables
  • A bar chart displaying data from key variables to view how the responses have been distributed
  • Via frequency distribution where data from each variable can be organised

When you visualise data in the form of charts, bars, and tables, it becomes much easier to assess whether your data follow a normal distribution or skewed distribution. You can also get insights into where the outliers are and how to get them fixed.

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How is a Skewed Distribution Different from a Normal One?

A normal distribution is where the set of information or data is distributed symmetrically around a centre. This is where most values lie, with the values getting smaller at the tail ends.

On the other hand, if one of the tails is longer or smaller than the other, the distribution would be skewed . They are often called asymmetrical distributions, as you cannot find any sort of symmetry in them.

The skewed distribution can be of two ways: left-skewed distribution and right-skewed distribution . When the left tail is longer than the right one, it is left-stewed distribution, while the right tail is longer in a right-strewed distribution.

Now, let us discuss the calculation of measures of central tendency. You might have heard about this one already.

What do Measures of Central Tendency Do?

Well, it precisely describes where most of the values lie in a data set. Having said that, the three most heard and used measures of central tendency are:

When considered from low to high, this is the value in the exact centre.

Mode is the most wanted or popular response in the data set.

You calculate the mean by simply adding all the values and dividing by the total number.Coming to how you can calculate the , which is equally important.

Measures of variability

Measures of variability give you an idea of how to spread out or dispersed values in a data set.

The four most common ones you must know about are:

Standard Deviation

The average distance between different values in your data set and the mean

Variance is the square of the standard deviation.

The range is the highest value subtracted from the data set's minimum value.

Interquartile Range

Interquartile range is the highest value minus lowest of the data set

Step 4: Testing your Hypotheses

Two terms you need to know in order to learn about testing a hypothesis:

Statistic-a number describing a sample

Parameter-a number describing a population

So, what exactly are hypotheses testing?

It is where an analyst or researcher tests all the assumptions made earlier regarding a population parameter. The methodology opted for by the researcher solely depends on the nature of the data utilised and the reason for its analysis.

The only objective is to evaluate the plausibility of hypotheses with the help of sample data. The data here can either come from a larger population or a sample to represent the whole population .

How it Works?

These four steps will help you understand what exactly happens in hypotheses testing.

  • The first thing you need to do is state the two hypotheses made at the beginning.
  • The second is formulating an analysis plan that depicts how the data can be assessed.
  • Next is physically analysing the sample data about the plan.
  • The last and final step is going through the results and assessing whether you need to reject the null hypothesis or move forward with it.

Questions might arise on knowing if the null hypothesis is plausible, and this is where statistical tests come into play.

Statistical tests let you determine where your sample data could lie on an expected distribution if the null hypotheses were plausible. Usually, you get two types of outputs from statistical tests:

  • A test statistic : this shows how much your data differs from the null hypothesis
  • A p-value: this value assesses the likelihood of getting your results if the null hypothesis is true

Step 5: Interpreting the Data

You have made it to the final step of statistical analysis , where all the data you found useful till now will be interpreted. In order to check the usability of data, researchers compare the p-value to a set significant level, which is 0.05, so that they can know if the results are statistically important or not. That is why this process in hypothesis testing is called statistical significance .

Remember that the results you get here are unlikely to have arisen because of probability. There are lower chances of such findings if the null hypothesis is plausible.

By the end of this process, you must have answers to the following questions:

  • Does the interpreted data answer your original question? If yes, how?
  • Can you defend against objections with this data?
  • Are there limitations to your conclusions?

If the final results cannot help you find clear answers to these questions, you might have to go back, assess and repeat some of the steps again. After all, you want to draw the most accurate conclusions from your data.

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  • Indian J Anaesth
  • v.60(9); 2016 Sep

Basic statistical tools in research and data analysis

Zulfiqar ali.

Department of Anaesthesiology, Division of Neuroanaesthesiology, Sheri Kashmir Institute of Medical Sciences, Soura, Srinagar, Jammu and Kashmir, India

S Bala Bhaskar

1 Department of Anaesthesiology and Critical Care, Vijayanagar Institute of Medical Sciences, Bellary, Karnataka, India

Statistical methods involved in carrying out a study include planning, designing, collecting data, analysing, drawing meaningful interpretation and reporting of the research findings. The statistical analysis gives meaning to the meaningless numbers, thereby breathing life into a lifeless data. The results and inferences are precise only if proper statistical tests are used. This article will try to acquaint the reader with the basic research tools that are utilised while conducting various studies. The article covers a brief outline of the variables, an understanding of quantitative and qualitative variables and the measures of central tendency. An idea of the sample size estimation, power analysis and the statistical errors is given. Finally, there is a summary of parametric and non-parametric tests used for data analysis.

INTRODUCTION

Statistics is a branch of science that deals with the collection, organisation, analysis of data and drawing of inferences from the samples to the whole population.[ 1 ] This requires a proper design of the study, an appropriate selection of the study sample and choice of a suitable statistical test. An adequate knowledge of statistics is necessary for proper designing of an epidemiological study or a clinical trial. Improper statistical methods may result in erroneous conclusions which may lead to unethical practice.[ 2 ]

Variable is a characteristic that varies from one individual member of population to another individual.[ 3 ] Variables such as height and weight are measured by some type of scale, convey quantitative information and are called as quantitative variables. Sex and eye colour give qualitative information and are called as qualitative variables[ 3 ] [ Figure 1 ].

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Classification of variables

Quantitative variables

Quantitative or numerical data are subdivided into discrete and continuous measurements. Discrete numerical data are recorded as a whole number such as 0, 1, 2, 3,… (integer), whereas continuous data can assume any value. Observations that can be counted constitute the discrete data and observations that can be measured constitute the continuous data. Examples of discrete data are number of episodes of respiratory arrests or the number of re-intubations in an intensive care unit. Similarly, examples of continuous data are the serial serum glucose levels, partial pressure of oxygen in arterial blood and the oesophageal temperature.

A hierarchical scale of increasing precision can be used for observing and recording the data which is based on categorical, ordinal, interval and ratio scales [ Figure 1 ].

Categorical or nominal variables are unordered. The data are merely classified into categories and cannot be arranged in any particular order. If only two categories exist (as in gender male and female), it is called as a dichotomous (or binary) data. The various causes of re-intubation in an intensive care unit due to upper airway obstruction, impaired clearance of secretions, hypoxemia, hypercapnia, pulmonary oedema and neurological impairment are examples of categorical variables.

Ordinal variables have a clear ordering between the variables. However, the ordered data may not have equal intervals. Examples are the American Society of Anesthesiologists status or Richmond agitation-sedation scale.

Interval variables are similar to an ordinal variable, except that the intervals between the values of the interval variable are equally spaced. A good example of an interval scale is the Fahrenheit degree scale used to measure temperature. With the Fahrenheit scale, the difference between 70° and 75° is equal to the difference between 80° and 85°: The units of measurement are equal throughout the full range of the scale.

Ratio scales are similar to interval scales, in that equal differences between scale values have equal quantitative meaning. However, ratio scales also have a true zero point, which gives them an additional property. For example, the system of centimetres is an example of a ratio scale. There is a true zero point and the value of 0 cm means a complete absence of length. The thyromental distance of 6 cm in an adult may be twice that of a child in whom it may be 3 cm.

STATISTICS: DESCRIPTIVE AND INFERENTIAL STATISTICS

Descriptive statistics[ 4 ] try to describe the relationship between variables in a sample or population. Descriptive statistics provide a summary of data in the form of mean, median and mode. Inferential statistics[ 4 ] use a random sample of data taken from a population to describe and make inferences about the whole population. It is valuable when it is not possible to examine each member of an entire population. The examples if descriptive and inferential statistics are illustrated in Table 1 .

Example of descriptive and inferential statistics

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Descriptive statistics

The extent to which the observations cluster around a central location is described by the central tendency and the spread towards the extremes is described by the degree of dispersion.

Measures of central tendency

The measures of central tendency are mean, median and mode.[ 6 ] Mean (or the arithmetic average) is the sum of all the scores divided by the number of scores. Mean may be influenced profoundly by the extreme variables. For example, the average stay of organophosphorus poisoning patients in ICU may be influenced by a single patient who stays in ICU for around 5 months because of septicaemia. The extreme values are called outliers. The formula for the mean is

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where x = each observation and n = number of observations. Median[ 6 ] is defined as the middle of a distribution in a ranked data (with half of the variables in the sample above and half below the median value) while mode is the most frequently occurring variable in a distribution. Range defines the spread, or variability, of a sample.[ 7 ] It is described by the minimum and maximum values of the variables. If we rank the data and after ranking, group the observations into percentiles, we can get better information of the pattern of spread of the variables. In percentiles, we rank the observations into 100 equal parts. We can then describe 25%, 50%, 75% or any other percentile amount. The median is the 50 th percentile. The interquartile range will be the observations in the middle 50% of the observations about the median (25 th -75 th percentile). Variance[ 7 ] is a measure of how spread out is the distribution. It gives an indication of how close an individual observation clusters about the mean value. The variance of a population is defined by the following formula:

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where σ 2 is the population variance, X is the population mean, X i is the i th element from the population and N is the number of elements in the population. The variance of a sample is defined by slightly different formula:

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where s 2 is the sample variance, x is the sample mean, x i is the i th element from the sample and n is the number of elements in the sample. The formula for the variance of a population has the value ‘ n ’ as the denominator. The expression ‘ n −1’ is known as the degrees of freedom and is one less than the number of parameters. Each observation is free to vary, except the last one which must be a defined value. The variance is measured in squared units. To make the interpretation of the data simple and to retain the basic unit of observation, the square root of variance is used. The square root of the variance is the standard deviation (SD).[ 8 ] The SD of a population is defined by the following formula:

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where σ is the population SD, X is the population mean, X i is the i th element from the population and N is the number of elements in the population. The SD of a sample is defined by slightly different formula:

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where s is the sample SD, x is the sample mean, x i is the i th element from the sample and n is the number of elements in the sample. An example for calculation of variation and SD is illustrated in Table 2 .

Example of mean, variance, standard deviation

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Normal distribution or Gaussian distribution

Most of the biological variables usually cluster around a central value, with symmetrical positive and negative deviations about this point.[ 1 ] The standard normal distribution curve is a symmetrical bell-shaped. In a normal distribution curve, about 68% of the scores are within 1 SD of the mean. Around 95% of the scores are within 2 SDs of the mean and 99% within 3 SDs of the mean [ Figure 2 ].

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Normal distribution curve

Skewed distribution

It is a distribution with an asymmetry of the variables about its mean. In a negatively skewed distribution [ Figure 3 ], the mass of the distribution is concentrated on the right of Figure 1 . In a positively skewed distribution [ Figure 3 ], the mass of the distribution is concentrated on the left of the figure leading to a longer right tail.

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Curves showing negatively skewed and positively skewed distribution

Inferential statistics

In inferential statistics, data are analysed from a sample to make inferences in the larger collection of the population. The purpose is to answer or test the hypotheses. A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. Hypothesis tests are thus procedures for making rational decisions about the reality of observed effects.

Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty).

In inferential statistics, the term ‘null hypothesis’ ( H 0 ‘ H-naught ,’ ‘ H-null ’) denotes that there is no relationship (difference) between the population variables in question.[ 9 ]

Alternative hypothesis ( H 1 and H a ) denotes that a statement between the variables is expected to be true.[ 9 ]

The P value (or the calculated probability) is the probability of the event occurring by chance if the null hypothesis is true. The P value is a numerical between 0 and 1 and is interpreted by researchers in deciding whether to reject or retain the null hypothesis [ Table 3 ].

P values with interpretation

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If P value is less than the arbitrarily chosen value (known as α or the significance level), the null hypothesis (H0) is rejected [ Table 4 ]. However, if null hypotheses (H0) is incorrectly rejected, this is known as a Type I error.[ 11 ] Further details regarding alpha error, beta error and sample size calculation and factors influencing them are dealt with in another section of this issue by Das S et al .[ 12 ]

Illustration for null hypothesis

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PARAMETRIC AND NON-PARAMETRIC TESTS

Numerical data (quantitative variables) that are normally distributed are analysed with parametric tests.[ 13 ]

Two most basic prerequisites for parametric statistical analysis are:

  • The assumption of normality which specifies that the means of the sample group are normally distributed
  • The assumption of equal variance which specifies that the variances of the samples and of their corresponding population are equal.

However, if the distribution of the sample is skewed towards one side or the distribution is unknown due to the small sample size, non-parametric[ 14 ] statistical techniques are used. Non-parametric tests are used to analyse ordinal and categorical data.

Parametric tests

The parametric tests assume that the data are on a quantitative (numerical) scale, with a normal distribution of the underlying population. The samples have the same variance (homogeneity of variances). The samples are randomly drawn from the population, and the observations within a group are independent of each other. The commonly used parametric tests are the Student's t -test, analysis of variance (ANOVA) and repeated measures ANOVA.

Student's t -test

Student's t -test is used to test the null hypothesis that there is no difference between the means of the two groups. It is used in three circumstances:

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where X = sample mean, u = population mean and SE = standard error of mean

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where X 1 − X 2 is the difference between the means of the two groups and SE denotes the standard error of the difference.

  • To test if the population means estimated by two dependent samples differ significantly (the paired t -test). A usual setting for paired t -test is when measurements are made on the same subjects before and after a treatment.

The formula for paired t -test is:

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where d is the mean difference and SE denotes the standard error of this difference.

The group variances can be compared using the F -test. The F -test is the ratio of variances (var l/var 2). If F differs significantly from 1.0, then it is concluded that the group variances differ significantly.

Analysis of variance

The Student's t -test cannot be used for comparison of three or more groups. The purpose of ANOVA is to test if there is any significant difference between the means of two or more groups.

In ANOVA, we study two variances – (a) between-group variability and (b) within-group variability. The within-group variability (error variance) is the variation that cannot be accounted for in the study design. It is based on random differences present in our samples.

However, the between-group (or effect variance) is the result of our treatment. These two estimates of variances are compared using the F-test.

A simplified formula for the F statistic is:

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where MS b is the mean squares between the groups and MS w is the mean squares within groups.

Repeated measures analysis of variance

As with ANOVA, repeated measures ANOVA analyses the equality of means of three or more groups. However, a repeated measure ANOVA is used when all variables of a sample are measured under different conditions or at different points in time.

As the variables are measured from a sample at different points of time, the measurement of the dependent variable is repeated. Using a standard ANOVA in this case is not appropriate because it fails to model the correlation between the repeated measures: The data violate the ANOVA assumption of independence. Hence, in the measurement of repeated dependent variables, repeated measures ANOVA should be used.

Non-parametric tests

When the assumptions of normality are not met, and the sample means are not normally, distributed parametric tests can lead to erroneous results. Non-parametric tests (distribution-free test) are used in such situation as they do not require the normality assumption.[ 15 ] Non-parametric tests may fail to detect a significant difference when compared with a parametric test. That is, they usually have less power.

As is done for the parametric tests, the test statistic is compared with known values for the sampling distribution of that statistic and the null hypothesis is accepted or rejected. The types of non-parametric analysis techniques and the corresponding parametric analysis techniques are delineated in Table 5 .

Analogue of parametric and non-parametric tests

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Median test for one sample: The sign test and Wilcoxon's signed rank test

The sign test and Wilcoxon's signed rank test are used for median tests of one sample. These tests examine whether one instance of sample data is greater or smaller than the median reference value.

This test examines the hypothesis about the median θ0 of a population. It tests the null hypothesis H0 = θ0. When the observed value (Xi) is greater than the reference value (θ0), it is marked as+. If the observed value is smaller than the reference value, it is marked as − sign. If the observed value is equal to the reference value (θ0), it is eliminated from the sample.

If the null hypothesis is true, there will be an equal number of + signs and − signs.

The sign test ignores the actual values of the data and only uses + or − signs. Therefore, it is useful when it is difficult to measure the values.

Wilcoxon's signed rank test

There is a major limitation of sign test as we lose the quantitative information of the given data and merely use the + or – signs. Wilcoxon's signed rank test not only examines the observed values in comparison with θ0 but also takes into consideration the relative sizes, adding more statistical power to the test. As in the sign test, if there is an observed value that is equal to the reference value θ0, this observed value is eliminated from the sample.

Wilcoxon's rank sum test ranks all data points in order, calculates the rank sum of each sample and compares the difference in the rank sums.

Mann-Whitney test

It is used to test the null hypothesis that two samples have the same median or, alternatively, whether observations in one sample tend to be larger than observations in the other.

Mann–Whitney test compares all data (xi) belonging to the X group and all data (yi) belonging to the Y group and calculates the probability of xi being greater than yi: P (xi > yi). The null hypothesis states that P (xi > yi) = P (xi < yi) =1/2 while the alternative hypothesis states that P (xi > yi) ≠1/2.

Kolmogorov-Smirnov test

The two-sample Kolmogorov-Smirnov (KS) test was designed as a generic method to test whether two random samples are drawn from the same distribution. The null hypothesis of the KS test is that both distributions are identical. The statistic of the KS test is a distance between the two empirical distributions, computed as the maximum absolute difference between their cumulative curves.

Kruskal-Wallis test

The Kruskal–Wallis test is a non-parametric test to analyse the variance.[ 14 ] It analyses if there is any difference in the median values of three or more independent samples. The data values are ranked in an increasing order, and the rank sums calculated followed by calculation of the test statistic.

Jonckheere test

In contrast to Kruskal–Wallis test, in Jonckheere test, there is an a priori ordering that gives it a more statistical power than the Kruskal–Wallis test.[ 14 ]

Friedman test

The Friedman test is a non-parametric test for testing the difference between several related samples. The Friedman test is an alternative for repeated measures ANOVAs which is used when the same parameter has been measured under different conditions on the same subjects.[ 13 ]

Tests to analyse the categorical data

Chi-square test, Fischer's exact test and McNemar's test are used to analyse the categorical or nominal variables. The Chi-square test compares the frequencies and tests whether the observed data differ significantly from that of the expected data if there were no differences between groups (i.e., the null hypothesis). It is calculated by the sum of the squared difference between observed ( O ) and the expected ( E ) data (or the deviation, d ) divided by the expected data by the following formula:

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A Yates correction factor is used when the sample size is small. Fischer's exact test is used to determine if there are non-random associations between two categorical variables. It does not assume random sampling, and instead of referring a calculated statistic to a sampling distribution, it calculates an exact probability. McNemar's test is used for paired nominal data. It is applied to 2 × 2 table with paired-dependent samples. It is used to determine whether the row and column frequencies are equal (that is, whether there is ‘marginal homogeneity’). The null hypothesis is that the paired proportions are equal. The Mantel-Haenszel Chi-square test is a multivariate test as it analyses multiple grouping variables. It stratifies according to the nominated confounding variables and identifies any that affects the primary outcome variable. If the outcome variable is dichotomous, then logistic regression is used.

SOFTWARES AVAILABLE FOR STATISTICS, SAMPLE SIZE CALCULATION AND POWER ANALYSIS

Numerous statistical software systems are available currently. The commonly used software systems are Statistical Package for the Social Sciences (SPSS – manufactured by IBM corporation), Statistical Analysis System ((SAS – developed by SAS Institute North Carolina, United States of America), R (designed by Ross Ihaka and Robert Gentleman from R core team), Minitab (developed by Minitab Inc), Stata (developed by StataCorp) and the MS Excel (developed by Microsoft).

There are a number of web resources which are related to statistical power analyses. A few are:

  • StatPages.net – provides links to a number of online power calculators
  • G-Power – provides a downloadable power analysis program that runs under DOS
  • Power analysis for ANOVA designs an interactive site that calculates power or sample size needed to attain a given power for one effect in a factorial ANOVA design
  • SPSS makes a program called SamplePower. It gives an output of a complete report on the computer screen which can be cut and paste into another document.

It is important that a researcher knows the concepts of the basic statistical methods used for conduct of a research study. This will help to conduct an appropriately well-designed study leading to valid and reliable results. Inappropriate use of statistical techniques may lead to faulty conclusions, inducing errors and undermining the significance of the article. Bad statistics may lead to bad research, and bad research may lead to unethical practice. Hence, an adequate knowledge of statistics and the appropriate use of statistical tests are important. An appropriate knowledge about the basic statistical methods will go a long way in improving the research designs and producing quality medical research which can be utilised for formulating the evidence-based guidelines.

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  • How to Write a Results Section | Tips & Examples

How to Write a Results Section | Tips & Examples

Published on August 30, 2022 by Tegan George . Revised on July 18, 2023.

A results section is where you report the main findings of the data collection and analysis you conducted for your thesis or dissertation . You should report all relevant results concisely and objectively, in a logical order. Don’t include subjective interpretations of why you found these results or what they mean—any evaluation should be saved for the discussion section .

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Table of contents

How to write a results section, reporting quantitative research results, reporting qualitative research results, results vs. discussion vs. conclusion, checklist: research results, other interesting articles, frequently asked questions about results sections.

When conducting research, it’s important to report the results of your study prior to discussing your interpretations of it. This gives your reader a clear idea of exactly what you found and keeps the data itself separate from your subjective analysis.

Here are a few best practices:

  • Your results should always be written in the past tense.
  • While the length of this section depends on how much data you collected and analyzed, it should be written as concisely as possible.
  • Only include results that are directly relevant to answering your research questions . Avoid speculative or interpretative words like “appears” or “implies.”
  • If you have other results you’d like to include, consider adding them to an appendix or footnotes.
  • Always start out with your broadest results first, and then flow into your more granular (but still relevant) ones. Think of it like a shoe store: first discuss the shoes as a whole, then the sneakers, boots, sandals, etc.

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If you conducted quantitative research , you’ll likely be working with the results of some sort of statistical analysis .

Your results section should report the results of any statistical tests you used to compare groups or assess relationships between variables . It should also state whether or not each hypothesis was supported.

The most logical way to structure quantitative results is to frame them around your research questions or hypotheses. For each question or hypothesis, share:

  • A reminder of the type of analysis you used (e.g., a two-sample t test or simple linear regression ). A more detailed description of your analysis should go in your methodology section.
  • A concise summary of each relevant result, both positive and negative. This can include any relevant descriptive statistics (e.g., means and standard deviations ) as well as inferential statistics (e.g., t scores, degrees of freedom , and p values ). Remember, these numbers are often placed in parentheses.
  • A brief statement of how each result relates to the question, or whether the hypothesis was supported. You can briefly mention any results that didn’t fit with your expectations and assumptions, but save any speculation on their meaning or consequences for your discussion  and conclusion.

A note on tables and figures

In quantitative research, it’s often helpful to include visual elements such as graphs, charts, and tables , but only if they are directly relevant to your results. Give these elements clear, descriptive titles and labels so that your reader can easily understand what is being shown. If you want to include any other visual elements that are more tangential in nature, consider adding a figure and table list .

As a rule of thumb:

  • Tables are used to communicate exact values, giving a concise overview of various results
  • Graphs and charts are used to visualize trends and relationships, giving an at-a-glance illustration of key findings

Don’t forget to also mention any tables and figures you used within the text of your results section. Summarize or elaborate on specific aspects you think your reader should know about rather than merely restating the same numbers already shown.

A two-sample t test was used to test the hypothesis that higher social distance from environmental problems would reduce the intent to donate to environmental organizations, with donation intention (recorded as a score from 1 to 10) as the outcome variable and social distance (categorized as either a low or high level of social distance) as the predictor variable.Social distance was found to be positively correlated with donation intention, t (98) = 12.19, p < .001, with the donation intention of the high social distance group 0.28 points higher, on average, than the low social distance group (see figure 1). This contradicts the initial hypothesis that social distance would decrease donation intention, and in fact suggests a small effect in the opposite direction.

Example of using figures in the results section

Figure 1: Intention to donate to environmental organizations based on social distance from impact of environmental damage.

In qualitative research , your results might not all be directly related to specific hypotheses. In this case, you can structure your results section around key themes or topics that emerged from your analysis of the data.

For each theme, start with general observations about what the data showed. You can mention:

  • Recurring points of agreement or disagreement
  • Patterns and trends
  • Particularly significant snippets from individual responses

Next, clarify and support these points with direct quotations. Be sure to report any relevant demographic information about participants. Further information (such as full transcripts , if appropriate) can be included in an appendix .

When asked about video games as a form of art, the respondents tended to believe that video games themselves are not an art form, but agreed that creativity is involved in their production. The criteria used to identify artistic video games included design, story, music, and creative teams.One respondent (male, 24) noted a difference in creativity between popular video game genres:

“I think that in role-playing games, there’s more attention to character design, to world design, because the whole story is important and more attention is paid to certain game elements […] so that perhaps you do need bigger teams of creative experts than in an average shooter or something.”

Responses suggest that video game consumers consider some types of games to have more artistic potential than others.

Your results section should objectively report your findings, presenting only brief observations in relation to each question, hypothesis, or theme.

It should not  speculate about the meaning of the results or attempt to answer your main research question . Detailed interpretation of your results is more suitable for your discussion section , while synthesis of your results into an overall answer to your main research question is best left for your conclusion .

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I have completed my data collection and analyzed the results.

I have included all results that are relevant to my research questions.

I have concisely and objectively reported each result, including relevant descriptive statistics and inferential statistics .

I have stated whether each hypothesis was supported or refuted.

I have used tables and figures to illustrate my results where appropriate.

All tables and figures are correctly labelled and referred to in the text.

There is no subjective interpretation or speculation on the meaning of the results.

You've finished writing up your results! Use the other checklists to further improve your thesis.

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The results chapter of a thesis or dissertation presents your research results concisely and objectively.

In quantitative research , for each question or hypothesis , state:

  • The type of analysis used
  • Relevant results in the form of descriptive and inferential statistics
  • Whether or not the alternative hypothesis was supported

In qualitative research , for each question or theme, describe:

  • Recurring patterns
  • Significant or representative individual responses
  • Relevant quotations from the data

Don’t interpret or speculate in the results chapter.

Results are usually written in the past tense , because they are describing the outcome of completed actions.

The results chapter or section simply and objectively reports what you found, without speculating on why you found these results. The discussion interprets the meaning of the results, puts them in context, and explains why they matter.

In qualitative research , results and discussion are sometimes combined. But in quantitative research , it’s considered important to separate the objective results from your interpretation of them.

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Statistical Analysis: Types, Examples and Process

Home Blog Data Science Statistical Analysis: Types, Examples and Process

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Data Science is an interdisciplinary field of study that doesn’t require one to work in a certain domain to thrive. Professionals from any domain could solve business problems in the industry by leveraging the available data.  

To solve these problems, often, a certain set of tools and techniques are implied, which helps to extract meaningful information from the data. Such a process of uncovering trends and patterns in the data by usage of various tools and functionalities could be referred to as Statistical Analysis. The  Data Scientist  online course  provides a detailed understanding of various statistical analysis methods.      

What is Statistical Analysis in Data Science?

Statistics is a science concerned with collection, analysis, interpretation, and presentation of data. In Statistics, we generally want to study a population. You may consider a population as a collection of things, persons, or objects under experiment or study. It is usually not possible to gain access to all of the information from the entire population due to logistical reasons. So, when we want to study a population, we generally select a sample.  

In sampling, we select a portion (or subset) of the larger population and then study the portion (or the sample) to learn about the population. Data is the result of sampling from a population. 

Types of Statistical Analysis

In the modern world, data comes in the form of 3V’s – Volume, Velocity, and Variety. The advancement of technology has resulted in businesses generating tremendous volumes of data across various sources at a very rapid pace. Companies like Google and Meta have their data servers to store dynamic data.  

To extract rich information from all these diverse high-volume datasets, there are certain statistical analysis types that are in place. The following is a list of seven such statistical analysis techniques:

Statistical Analysis Technique

Source: Researchgate.com

1. Descriptive Statistical Analysis

As mentioned earlier, every company stores large chunks of historical data, which carries a rich set of information. Descriptive Analysis is a way to analyze historical data through a series of basic statistical analysis. It gives a more holistic view of how the business has operated by identifying the weaknesses and strengths of its operation.  

Some of common examples could be monthly sales growth, yearly price changes, and so on. Any business could get its answer of “what has happened?” through descriptive analysis. 

2. Inferential Statistical Analysis

In the real world, often, the sheer volume of data makes it challenging for an analyst to draw conclusions about the entire population. Instead, you fetch a sample from the population and try to validate some of the basic assumptions from the data. This process of generating inferences about the population from the fetched sample by leveraging some of the statistical analysis tools is referred to as Inferential statistics. 

To generate inferences about the population, several statistical testing methods such as mean, variance, etc., could be performed. Additionally, you would need to use some sampling techniques to fetch relevant samples from the population. Hypothesis testing and Regression Analysis could be termed as two main types of inferential statistics.  

3. Associational Statistical Analysis

Causality is an important field in Data Science and Statistics. The urge to confidently say the ‘Why’ behind any drawn inference drives an organization and brings business value. Such an analysis that helps in identifying the relationship between multiple variables could be referred to as associational statistical analysis. 

Since this type of analysis is a bit advanced, it requires the latest cutting-edge statistical analysis software. Techniques such as regression analysis, co-relation, etc., are widely used for associational statistics. 

Other Types of Statistical Analysis  

1. predictive analysis.

Most businesses nowadays are looking to set up a system that helps them reduce the uncertainty in an event to a large extent. For e.g., many retail stores would like to forecast the demand of their products to plan for labor and inventory accordingly. To build such systems, you need to understand the relation in the data and predict the unseen event. This entire process is known as predictive analysis.  

To perform predictive analysis, knowledge of Machine Learning is required which is capable to capture the relation in huge volumes of data and generate predictions. 

2. Prescriptive Analysis

Often a business wants to understand what it needs to do to achieve a certain event. This type of analysis is referred to as prescriptive analysis. 

The decisions made in the prescriptive analysis are based on facts instead of instinct. Graph analysis, simulation, etc., could be used to perform prescriptive analysis. 

3. Exploratory Data Analysis

When you work in a Data Science project, one of the key steps you need to perform before going to predictive modeling is exploratory data analysis.  

It gives a deeper understanding of the historical data. You could perform several analysis such as checking missing values, duplicates, univariate, bi-variate, multi-variate relations, and so on. 

4. Causal Analysis

Many organizations want to know the reason behind the model predictions. E.g.- A bank would want to know why a loan has been defaulted, or a HR would like a reason behind employees leaving. All these reasoning could be determined using causal analysis. 

A lot of research is going on around model interpretability and causal analysis. Causal analysis is one such statistical research techniques that could fetch rich dividends for any company if applied correctly.  You can go for  KnowledgeHut Data Scientist online course  to further enhance your learning and knowledge on Data Science.    

Statistical Analysis Process [Step-by-Step]

While building a statistical pipeline, it is important to follow a few steps to ensure the analysis is conducted smoothly and relevant important steps are captured: 

Step 1: Write your Hypothesis and Plan Research Design

Defining the hypothesis is the first step to understand what kind of validation is needed and the relevant data to be captured. It could be classified into Null and Alternate hypotheses. The null hypothesis is referred to the condition where any observed event is happening by chance, and it is not statistically driven. On the other hand, the alternate hypothesis tends to contradict the null hypothesis stating that the observed event is statistically driven.  

Additionally, a research design could be classified into an experimental design which identifies a causal relationship, a correlation design which captures the bi-variate relation, and a descriptive design which identifies the statistical properties in historical data. 

Step 2: Collect Data from a Sample

Once the hypothesis is drawn, relevant data could be captured from various sources. Sometimes, fetching the entire corpus of data could be challenging.  

Hence, samples are drawn from the population by means of various sampling techniques such as random sampling, stratified sampling, periodic sampling, and so on. As a rule of thumb, a sample size of 30 per sub-groups is recommended.  

Step 3: Summarize Data with Descriptive Statistics

The first step to understanding the information carried by the raw data is to perform a descriptive analysis on it. These would tell us how the historical data is behaving.  

As part of descriptive statistics, you can find the distribution of numeric variables and frequency plots for categorical data and calculate statistical measures like mean, median, mode, standard deviation , various percentiles, and so on.  

As part of descriptive statistics, you can find the distribution of numeric variables and frequency plots for categorical data and calculate statistical measures like mean, median, mode, standard deviation, various percentiles, and so on.  

Step 4: Test the Hypothesis or Make Estimates with Inferential Statistics

You can estimate parameters as well as perform hypothesis testing on the data. It could be a point estimate or a range of estimates. A point estimate gives the exact value of a certain parameter, whereas in a range of estimate you will get a range within which a certain parameter is expected to lie. 

For hypothesis testing, you can calculate the p-value which would tell whether the observed event is statistically significant given the null hypothesis is true. Beyond that, there are other comparison tests such as z and t tests which will help to identify if two samples belong to the same population. 

Step 5: Interpret your Results

Calculating the p-value and identifying whether the event is statistically significant is an important part of result interpretation. 

You can also calculate Type I or Type II errors while rejecting or accepting a null hypothesis.

Top Five Considerations for Statistical Data Analysis

Data can be messy. Even a small blunder may cost you a fortune. Therefore, special care when working with statistical data is of utmost importance. Here are a few key takeaways you must consider minimizing errors and improve accuracy.  

  • Define the purpose and determine the location where the publication will take place.   
  • Understand the assets to undertake the investigation.  
  • Understand the individual capability of appropriately managing and understanding the analysis.   
  • Determine whether there is a need to repeat the process.   
  • Know the expectation of the individuals evaluating reviewing, committee, and supervision. 

Statistical Analysis Methods

To carry out statistical analysis, there are certain methods which give more robust information about the data.

statistical analysis paper example

It is nothing but average of a numeric variable. A mean value is often used to impute missing data or get a rough estimate about the magnitude of the numeric variable. However, it is affected by outliers in the data. E.g. – You have few points 3, 4, 6, 2, 9, 6, 5, 8, 1, then the average would sum of all these points divided by 9.  

mean = (3+4+6+2+9+6+5+8+1) / 9 = 4.88 

2. Standard Deviation

It shows how the data is varying around the mean. The spread of data around the mean is captured by standard deviation.  

Taking the previous example, the std would be sqrt((3-4.88)^2 +….+ (1-4.88)^2)/9) which is 2.514. 

3. Regression

Regression analysis in statistics could be defined as a line drawn to determine the relation between independent variables with a dependent variable. E.g., in case of simple linear regression, the equation would be y = mx + c, where m is slope and c is intercepted, and ‘x’ is an independent variable here whereas ‘y’ is dependent on ‘x’. 

4. Hypothesis Testing

In hypothesis testing, you calculate the p-value and set the significance level based on the use case. E.g., in simple linear regression, you can use p values to determine if an independent variable is statistically significant.  

The null hypothesis is that the independent variable doesn’t capture any variance in the dependent variable which would be rejected if the p-value is less than the significance level which is generally kept at 0.05 by the rule of thumb. 

5. Sample Size Determination

In situations like twitter sentiment analysis, where the dataset is so enormous, getting a sample is recommended. Hence, getting the right sample size is important which could be based on several sampling techniques or determined by the business objectives. 

Terms Used to Describe Data

When working with data, you will need to search, inspect, and characterize them. To understand the data in a tech-savvy and straightforward way, we use a few statistical terms to denote them individually or in groups.   

The most frequently used terms used to describe data include data point, quantitative variables, indicator, statistic, time-series data, variable, data aggregation, time series, dataset, and database. Let us define each one of them in brief:  

  • Data points: These are the numerical files formed and organized for interpretations.  
  • Quantitative variables: These variables present the information in digit form.   
  • Indicator: An indicator explains the action of a community's social-economic surroundings.   
  • Time-series data: The time-series defines the sequential data.   
  • Data aggregation: A group of data points and data set.  
  • Database: A group of arranged information for examination and recovery.   
  • Time-series: A set of measures of a variable documented over a specified time.

Benefits of Statistical Analysis

  • Statistical Analysis helps a business to uncover some of the hidden patterns in the data. It helps to validate certain hypothesis and make informed decisions. On top of that, executing any Data Science workflow requires a thorough exploratory data analysis wherein you perform several descriptive as well model agnostic causal analysis.  
  • Even the application of statistics in business is so heavy that many organizations completely rely on simple statistical analysis to boost their workplace performance and drive business growth.  

Statistical Analysis Software/Tools

You can use languages like Python and R to execute various statistical techniques. Additionally, you can also perform statistical analysis in Excel. However, there are a few software available in the market which readily allow you to implement the statistical analysis. 

1. SPSS Statistics

Helps in analyzing large datasets for quick insights and decision-making. SPSS Statistics is developed by IBM. 

A cloud-based platform that helps in analysis and visualization of data. The SAS statistical analysis system helps in predictive modelling as well. 

Used by Data Scientists for manipulation and exploration of data. It is available in four different versions depending on the data size. 

Statistical Analysis Examples

Given a dataset with both numeric and categorical features, some of the examples of statistical analysis that could be performed are-

  • Calculation of Central Tendencies– Helps to reduce the data to one representative value which helps to understand a large population.  
  • Outliers in Numeric Data– Outliers often impact model performance. Hence, identifying and dealing with them would be helpful before executing an ML model.  
  • Frequency Distribution of Ordinal Variable– To get a better understanding of an ordinal variable, and for better feature engineering, a frequency distribution plot is useful.  
  • Inter-quartile Ranges– It is often used to identify outliers in a numeric variable.  
  • Correlation- Relation between two variables could be understood by looking at their co-relation coefficient values. E.g.- In linear regression, we check how two variables are correlated during feature selection.

Chi-square  T est

Chi-square test records the contrast of a model to actual experimental data. Data is unsystematic, underdone, equally limited, obtained from independent variables, and a sufficient sample.    

It relates the size of any inconsistencies among the expected outcomes and the actual outcomes, provided with the sample size and the number of variables in the connection.    

Types of Variables and Frequencies

Types of variables.

A variable is any digit, amount, or feature that is countable or measurable. Simply put, it is a variable characteristic that varies. The six types of variables include the following:   

1. Dependent variable 

A dependent variable has values that vary according to the value of another variable known as the independent variable.    

2. Independent variable 

An independent variable on the other side is controllable by experts. Its reports are recorded and equated.    

3. Intervening variable

An intervening variable explicates fundamental relations between variables.   

4. Moderator variable

A moderator variable upsets the power of the connection between dependent and independent variables.    

5. Control variable 

A control variable is anything restricted to a research study. The values are constant throughout the experiment.   

6. Extraneous variable 

Extraneous variable refers to the entire variables that are dependent but can upset experimental outcomes.  

Types of Frequencies

Frequency refers to the number of repetitions of reading in an experiment in a given time. Three types of frequency distribution include the following:    

  • Grouped, ungrouped.    
  • Cumulative, relative    
  • Relative cumulative frequency distribution.    

Features of Frequencies

  • The calculation of central tendency and position (median, mean, and mode).    
  • The measure of dispersion (range, variance, and standard deviation).    
  • Degree of symmetry (skewness).    
  • Peakedness (kurtosis).    

Correlation Matrix

The correlation matrix is a table that shows the correlation coefficients of unique variables. It is a powerful tool that summarizes datasets points and picture sequences in the provided data. A correlation matrix includes rows and columns that display variables. Additionally, the correlation matrix exploits in aggregation with other varieties of statistical analysis.    

Inferential Statistics

Inferential statistics use random data samples for demonstration and to create inferences. They are  measured  when analysis of each individual of a whole group is not likely to happen.    

Applications of Inferential Statistics

Inferential statistics in educational research is not likely to sample the entire population that has summaries. For instance, the aim of an investigation study may be to obtain whether a new method of learning mathematics develops mathematical accomplishment for all students in a class.   

  • Marketing organizations: Marketing organizations use inferential statistics to dispute a survey and request inquiries. It is because carrying out surveys for all the individuals about merchandise is not likely.    
  • Finance departments: Financial departments apply inferential statistics for expected financial plan and resources expenses, especially when there are several indefinite aspects. However, economists cannot estimate all that use possibility.    
  • Economic planning: In economic planning, there are potent methods like index figures, time series investigation, and estimation. Inferential statistics measures national income and its components. It gathers info about revenue, investment, saving, and spending to establish links among them.  

Statistical Analysis is one of the crucial aspects of any business who are looking to leverage the full potential of data for maximum value. The latest tools and software allow organizations to perform several analysis and generate real-time insights. These insights are then used by stakeholders for better decision-making. KnowledgeHut’s Data Science Bootcamp covers statistical analysis in depth. 

Frequently Asked Questions (FAQs)

1. what are the five basic methods of statistical analysis.

The 5 basic methods of statistical analysis are: Mean, Standard deviation, Regression, Hypothesis testing and Sample size determination. 

2. What is the difference between data analysis and statistical analysis?

Data Analysis helps in inspecting and reporting data to non-technical people. Statistical Analysis gives more in-depth representation of the large population of data. 

3. Why do we need to do statistical analysis?

Statistical Analysis gives a robust understanding of the data. It helps in generating insights which brings business value. 

4. Is statistical analysis quantitative or qualitative?

Statistical analysis is quantitative in nature as it is applied on numeric data. It generates rich information by performing various descriptive and predictive analysis. 

5. Which tool can be used for statistical analysis?

There are several tools or software available for statistical analysis. Some of them are SPSS statistics, SAS, Stata. 

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Suman is a Data Scientist working for a Fortune Top 5 company. His expertise lies in the field of Machine Learning, Time Series & NLP. He has built scalable solutions for retail & manufacturing organisations.

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The Beginner's Guide to Statistical Analysis | 5 Steps & Examples

Statistical analysis means investigating trends, patterns, and relationships using quantitative data . It is an important research tool used by scientists, governments, businesses, and other organisations.

To draw valid conclusions, statistical analysis requires careful planning from the very start of the research process . You need to specify your hypotheses and make decisions about your research design, sample size, and sampling procedure.

After collecting data from your sample, you can organise and summarise the data using descriptive statistics . Then, you can use inferential statistics to formally test hypotheses and make estimates about the population. Finally, you can interpret and generalise your findings.

This article is a practical introduction to statistical analysis for students and researchers. We’ll walk you through the steps using two research examples. The first investigates a potential cause-and-effect relationship, while the second investigates a potential correlation between variables.

Table of contents

Step 1: write your hypotheses and plan your research design, step 2: collect data from a sample, step 3: summarise your data with descriptive statistics, step 4: test hypotheses or make estimates with inferential statistics, step 5: interpret your results, frequently asked questions about statistics.

To collect valid data for statistical analysis, you first need to specify your hypotheses and plan out your research design.

Writing statistical hypotheses

The goal of research is often to investigate a relationship between variables within a population . You start with a prediction, and use statistical analysis to test that prediction.

A statistical hypothesis is a formal way of writing a prediction about a population. Every research prediction is rephrased into null and alternative hypotheses that can be tested using sample data.

While the null hypothesis always predicts no effect or no relationship between variables, the alternative hypothesis states your research prediction of an effect or relationship.

  • Null hypothesis: A 5-minute meditation exercise will have no effect on math test scores in teenagers.
  • Alternative hypothesis: A 5-minute meditation exercise will improve math test scores in teenagers.
  • Null hypothesis: Parental income and GPA have no relationship with each other in college students.
  • Alternative hypothesis: Parental income and GPA are positively correlated in college students.

Planning your research design

A research design is your overall strategy for data collection and analysis. It determines the statistical tests you can use to test your hypothesis later on.

First, decide whether your research will use a descriptive, correlational, or experimental design. Experiments directly influence variables, whereas descriptive and correlational studies only measure variables.

  • In an experimental design , you can assess a cause-and-effect relationship (e.g., the effect of meditation on test scores) using statistical tests of comparison or regression.
  • In a correlational design , you can explore relationships between variables (e.g., parental income and GPA) without any assumption of causality using correlation coefficients and significance tests.
  • In a descriptive design , you can study the characteristics of a population or phenomenon (e.g., the prevalence of anxiety in U.S. college students) using statistical tests to draw inferences from sample data.

Your research design also concerns whether you’ll compare participants at the group level or individual level, or both.

  • In a between-subjects design , you compare the group-level outcomes of participants who have been exposed to different treatments (e.g., those who performed a meditation exercise vs those who didn’t).
  • In a within-subjects design , you compare repeated measures from participants who have participated in all treatments of a study (e.g., scores from before and after performing a meditation exercise).
  • In a mixed (factorial) design , one variable is altered between subjects and another is altered within subjects (e.g., pretest and posttest scores from participants who either did or didn’t do a meditation exercise).
  • Experimental
  • Correlational

First, you’ll take baseline test scores from participants. Then, your participants will undergo a 5-minute meditation exercise. Finally, you’ll record participants’ scores from a second math test.

In this experiment, the independent variable is the 5-minute meditation exercise, and the dependent variable is the math test score from before and after the intervention. Example: Correlational research design In a correlational study, you test whether there is a relationship between parental income and GPA in graduating college students. To collect your data, you will ask participants to fill in a survey and self-report their parents’ incomes and their own GPA.

Measuring variables

When planning a research design, you should operationalise your variables and decide exactly how you will measure them.

For statistical analysis, it’s important to consider the level of measurement of your variables, which tells you what kind of data they contain:

  • Categorical data represents groupings. These may be nominal (e.g., gender) or ordinal (e.g. level of language ability).
  • Quantitative data represents amounts. These may be on an interval scale (e.g. test score) or a ratio scale (e.g. age).

Many variables can be measured at different levels of precision. For example, age data can be quantitative (8 years old) or categorical (young). If a variable is coded numerically (e.g., level of agreement from 1–5), it doesn’t automatically mean that it’s quantitative instead of categorical.

Identifying the measurement level is important for choosing appropriate statistics and hypothesis tests. For example, you can calculate a mean score with quantitative data, but not with categorical data.

In a research study, along with measures of your variables of interest, you’ll often collect data on relevant participant characteristics.

Population vs sample

In most cases, it’s too difficult or expensive to collect data from every member of the population you’re interested in studying. Instead, you’ll collect data from a sample.

Statistical analysis allows you to apply your findings beyond your own sample as long as you use appropriate sampling procedures . You should aim for a sample that is representative of the population.

Sampling for statistical analysis

There are two main approaches to selecting a sample.

  • Probability sampling: every member of the population has a chance of being selected for the study through random selection.
  • Non-probability sampling: some members of the population are more likely than others to be selected for the study because of criteria such as convenience or voluntary self-selection.

In theory, for highly generalisable findings, you should use a probability sampling method. Random selection reduces sampling bias and ensures that data from your sample is actually typical of the population. Parametric tests can be used to make strong statistical inferences when data are collected using probability sampling.

But in practice, it’s rarely possible to gather the ideal sample. While non-probability samples are more likely to be biased, they are much easier to recruit and collect data from. Non-parametric tests are more appropriate for non-probability samples, but they result in weaker inferences about the population.

If you want to use parametric tests for non-probability samples, you have to make the case that:

  • your sample is representative of the population you’re generalising your findings to.
  • your sample lacks systematic bias.

Keep in mind that external validity means that you can only generalise your conclusions to others who share the characteristics of your sample. For instance, results from Western, Educated, Industrialised, Rich and Democratic samples (e.g., college students in the US) aren’t automatically applicable to all non-WEIRD populations.

If you apply parametric tests to data from non-probability samples, be sure to elaborate on the limitations of how far your results can be generalised in your discussion section .

Create an appropriate sampling procedure

Based on the resources available for your research, decide on how you’ll recruit participants.

  • Will you have resources to advertise your study widely, including outside of your university setting?
  • Will you have the means to recruit a diverse sample that represents a broad population?
  • Do you have time to contact and follow up with members of hard-to-reach groups?

Your participants are self-selected by their schools. Although you’re using a non-probability sample, you aim for a diverse and representative sample. Example: Sampling (correlational study) Your main population of interest is male college students in the US. Using social media advertising, you recruit senior-year male college students from a smaller subpopulation: seven universities in the Boston area.

Calculate sufficient sample size

Before recruiting participants, decide on your sample size either by looking at other studies in your field or using statistics. A sample that’s too small may be unrepresentative of the sample, while a sample that’s too large will be more costly than necessary.

There are many sample size calculators online. Different formulas are used depending on whether you have subgroups or how rigorous your study should be (e.g., in clinical research). As a rule of thumb, a minimum of 30 units or more per subgroup is necessary.

To use these calculators, you have to understand and input these key components:

  • Significance level (alpha): the risk of rejecting a true null hypothesis that you are willing to take, usually set at 5%.
  • Statistical power : the probability of your study detecting an effect of a certain size if there is one, usually 80% or higher.
  • Expected effect size : a standardised indication of how large the expected result of your study will be, usually based on other similar studies.
  • Population standard deviation: an estimate of the population parameter based on a previous study or a pilot study of your own.

Once you’ve collected all of your data, you can inspect them and calculate descriptive statistics that summarise them.

Inspect your data

There are various ways to inspect your data, including the following:

  • Organising data from each variable in frequency distribution tables .
  • Displaying data from a key variable in a bar chart to view the distribution of responses.
  • Visualising the relationship between two variables using a scatter plot .

By visualising your data in tables and graphs, you can assess whether your data follow a skewed or normal distribution and whether there are any outliers or missing data.

A normal distribution means that your data are symmetrically distributed around a center where most values lie, with the values tapering off at the tail ends.

Mean, median, mode, and standard deviation in a normal distribution

In contrast, a skewed distribution is asymmetric and has more values on one end than the other. The shape of the distribution is important to keep in mind because only some descriptive statistics should be used with skewed distributions.

Extreme outliers can also produce misleading statistics, so you may need a systematic approach to dealing with these values.

Calculate measures of central tendency

Measures of central tendency describe where most of the values in a data set lie. Three main measures of central tendency are often reported:

  • Mode : the most popular response or value in the data set.
  • Median : the value in the exact middle of the data set when ordered from low to high.
  • Mean : the sum of all values divided by the number of values.

However, depending on the shape of the distribution and level of measurement, only one or two of these measures may be appropriate. For example, many demographic characteristics can only be described using the mode or proportions, while a variable like reaction time may not have a mode at all.

Calculate measures of variability

Measures of variability tell you how spread out the values in a data set are. Four main measures of variability are often reported:

  • Range : the highest value minus the lowest value of the data set.
  • Interquartile range : the range of the middle half of the data set.
  • Standard deviation : the average distance between each value in your data set and the mean.
  • Variance : the square of the standard deviation.

Once again, the shape of the distribution and level of measurement should guide your choice of variability statistics. The interquartile range is the best measure for skewed distributions, while standard deviation and variance provide the best information for normal distributions.

Using your table, you should check whether the units of the descriptive statistics are comparable for pretest and posttest scores. For example, are the variance levels similar across the groups? Are there any extreme values? If there are, you may need to identify and remove extreme outliers in your data set or transform your data before performing a statistical test.

From this table, we can see that the mean score increased after the meditation exercise, and the variances of the two scores are comparable. Next, we can perform a statistical test to find out if this improvement in test scores is statistically significant in the population. Example: Descriptive statistics (correlational study) After collecting data from 653 students, you tabulate descriptive statistics for annual parental income and GPA.

It’s important to check whether you have a broad range of data points. If you don’t, your data may be skewed towards some groups more than others (e.g., high academic achievers), and only limited inferences can be made about a relationship.

A number that describes a sample is called a statistic , while a number describing a population is called a parameter . Using inferential statistics , you can make conclusions about population parameters based on sample statistics.

Researchers often use two main methods (simultaneously) to make inferences in statistics.

  • Estimation: calculating population parameters based on sample statistics.
  • Hypothesis testing: a formal process for testing research predictions about the population using samples.

You can make two types of estimates of population parameters from sample statistics:

  • A point estimate : a value that represents your best guess of the exact parameter.
  • An interval estimate : a range of values that represent your best guess of where the parameter lies.

If your aim is to infer and report population characteristics from sample data, it’s best to use both point and interval estimates in your paper.

You can consider a sample statistic a point estimate for the population parameter when you have a representative sample (e.g., in a wide public opinion poll, the proportion of a sample that supports the current government is taken as the population proportion of government supporters).

There’s always error involved in estimation, so you should also provide a confidence interval as an interval estimate to show the variability around a point estimate.

A confidence interval uses the standard error and the z score from the standard normal distribution to convey where you’d generally expect to find the population parameter most of the time.

Hypothesis testing

Using data from a sample, you can test hypotheses about relationships between variables in the population. Hypothesis testing starts with the assumption that the null hypothesis is true in the population, and you use statistical tests to assess whether the null hypothesis can be rejected or not.

Statistical tests determine where your sample data would lie on an expected distribution of sample data if the null hypothesis were true. These tests give two main outputs:

  • A test statistic tells you how much your data differs from the null hypothesis of the test.
  • A p value tells you the likelihood of obtaining your results if the null hypothesis is actually true in the population.

Statistical tests come in three main varieties:

  • Comparison tests assess group differences in outcomes.
  • Regression tests assess cause-and-effect relationships between variables.
  • Correlation tests assess relationships between variables without assuming causation.

Your choice of statistical test depends on your research questions, research design, sampling method, and data characteristics.

Parametric tests

Parametric tests make powerful inferences about the population based on sample data. But to use them, some assumptions must be met, and only some types of variables can be used. If your data violate these assumptions, you can perform appropriate data transformations or use alternative non-parametric tests instead.

A regression models the extent to which changes in a predictor variable results in changes in outcome variable(s).

  • A simple linear regression includes one predictor variable and one outcome variable.
  • A multiple linear regression includes two or more predictor variables and one outcome variable.

Comparison tests usually compare the means of groups. These may be the means of different groups within a sample (e.g., a treatment and control group), the means of one sample group taken at different times (e.g., pretest and posttest scores), or a sample mean and a population mean.

  • A t test is for exactly 1 or 2 groups when the sample is small (30 or less).
  • A z test is for exactly 1 or 2 groups when the sample is large.
  • An ANOVA is for 3 or more groups.

The z and t tests have subtypes based on the number and types of samples and the hypotheses:

  • If you have only one sample that you want to compare to a population mean, use a one-sample test .
  • If you have paired measurements (within-subjects design), use a dependent (paired) samples test .
  • If you have completely separate measurements from two unmatched groups (between-subjects design), use an independent (unpaired) samples test .
  • If you expect a difference between groups in a specific direction, use a one-tailed test .
  • If you don’t have any expectations for the direction of a difference between groups, use a two-tailed test .

The only parametric correlation test is Pearson’s r . The correlation coefficient ( r ) tells you the strength of a linear relationship between two quantitative variables.

However, to test whether the correlation in the sample is strong enough to be important in the population, you also need to perform a significance test of the correlation coefficient, usually a t test, to obtain a p value. This test uses your sample size to calculate how much the correlation coefficient differs from zero in the population.

You use a dependent-samples, one-tailed t test to assess whether the meditation exercise significantly improved math test scores. The test gives you:

  • a t value (test statistic) of 3.00
  • a p value of 0.0028

Although Pearson’s r is a test statistic, it doesn’t tell you anything about how significant the correlation is in the population. You also need to test whether this sample correlation coefficient is large enough to demonstrate a correlation in the population.

A t test can also determine how significantly a correlation coefficient differs from zero based on sample size. Since you expect a positive correlation between parental income and GPA, you use a one-sample, one-tailed t test. The t test gives you:

  • a t value of 3.08
  • a p value of 0.001

The final step of statistical analysis is interpreting your results.

Statistical significance

In hypothesis testing, statistical significance is the main criterion for forming conclusions. You compare your p value to a set significance level (usually 0.05) to decide whether your results are statistically significant or non-significant.

Statistically significant results are considered unlikely to have arisen solely due to chance. There is only a very low chance of such a result occurring if the null hypothesis is true in the population.

This means that you believe the meditation intervention, rather than random factors, directly caused the increase in test scores. Example: Interpret your results (correlational study) You compare your p value of 0.001 to your significance threshold of 0.05. With a p value under this threshold, you can reject the null hypothesis. This indicates a statistically significant correlation between parental income and GPA in male college students.

Note that correlation doesn’t always mean causation, because there are often many underlying factors contributing to a complex variable like GPA. Even if one variable is related to another, this may be because of a third variable influencing both of them, or indirect links between the two variables.

Effect size

A statistically significant result doesn’t necessarily mean that there are important real life applications or clinical outcomes for a finding.

In contrast, the effect size indicates the practical significance of your results. It’s important to report effect sizes along with your inferential statistics for a complete picture of your results. You should also report interval estimates of effect sizes if you’re writing an APA style paper .

With a Cohen’s d of 0.72, there’s medium to high practical significance to your finding that the meditation exercise improved test scores. Example: Effect size (correlational study) To determine the effect size of the correlation coefficient, you compare your Pearson’s r value to Cohen’s effect size criteria.

Decision errors

Type I and Type II errors are mistakes made in research conclusions. A Type I error means rejecting the null hypothesis when it’s actually true, while a Type II error means failing to reject the null hypothesis when it’s false.

You can aim to minimise the risk of these errors by selecting an optimal significance level and ensuring high power . However, there’s a trade-off between the two errors, so a fine balance is necessary.

Frequentist versus Bayesian statistics

Traditionally, frequentist statistics emphasises null hypothesis significance testing and always starts with the assumption of a true null hypothesis.

However, Bayesian statistics has grown in popularity as an alternative approach in the last few decades. In this approach, you use previous research to continually update your hypotheses based on your expectations and observations.

Bayes factor compares the relative strength of evidence for the null versus the alternative hypothesis rather than making a conclusion about rejecting the null hypothesis or not.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

The research methods you use depend on the type of data you need to answer your research question .

  • If you want to measure something or test a hypothesis , use quantitative methods . If you want to explore ideas, thoughts, and meanings, use qualitative methods .
  • If you want to analyse a large amount of readily available data, use secondary data. If you want data specific to your purposes with control over how they are generated, collect primary data.
  • If you want to establish cause-and-effect relationships between variables , use experimental methods. If you want to understand the characteristics of a research subject, use descriptive methods.

Statistical analysis is the main method for analyzing quantitative research data . It uses probabilities and models to test predictions about a population from sample data.

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1. Introduction

Statistics is a body of quantitative methods associated with empirical observation. A primary goal of these methods is coping with uncertainty. Most formal statistical methods rely on probability theory to express this uncertainty and to provide a formal mathematical basis for data description and for analysis. The notion of variability associated with data, expressed through probability, plays a fundamental role in this theory. As a consequence, much statistical effort is focused on how to control and measure variability and/or how to assign it to its sources.

Almost all characterizations of statistics as a field include the following elements:

(a) Designing experiments, surveys, and other systematic forms of empirical study.

(b) Summarizing and extracting information from data.

(c) Drawing formal inferences from empirical data through the use of probability.

(d) Communicating the results of statistical investigations to others, including scientists, policy makers, and the public.

This research paper describes a number of these elements, and the historical context out of which they grew. It provides a broad overview of the field, that can serve as a starting point to many of the other statistical entries in this encyclopedia.

2. The Origins Of The Field of Statistics

The word ‘statistics’ is related to the word ‘state’ and the original activity that was labeled as statistics was social in nature and related to elements of society through the organization of economic, demographic, and political facts. Paralleling this work to some extent was the development of the probability calculus and the theory of errors, typically associated with the physical sciences. These traditions came together in the nineteenth century and led to the notion of statistics as a collection of methods for the analysis of scientific data and the drawing of inferences therefrom.

As Hacking (1990) has noted: ‘By the end of the century chance had attained the respectability of a Victorian valet, ready to be the logical servant of the natural, biological and social sciences’ ( p. 2). At the beginning of the twentieth century, we see the emergence of statistics as a field under the leadership of Karl Pearson, George Udny Yule, Francis Y. Edgeworth, and others of the ‘English’ statistical school. As Stigler (1986) suggests:

Before 1900 we see many scientists of different fields developing and using techniques we now recognize as belonging to modern statistics. After 1900 we begin to see identifiable statisticians developing such techniques into a unified logic of empirical science that goes far beyond its component parts. There was no sharp moment of birth; but with Pearson and Yule and the growing number of students in Pearson’s laboratory, the infant discipline may be said to have arrived. (p. 361)

Pearson’s laboratory at University College, London quickly became the first statistics department in the world and it was to influence subsequent developments in a profound fashion for the next three decades. Pearson and his colleagues founded the first methodologically-oriented statistics journal, Biometrika, and they stimulated the development of new approaches to statistical methods. What remained before statistics could legitimately take on the mantle of a field of inquiry, separate from mathematics or the use of statistical approaches in other fields, was the development of the formal foundations of theories of inference from observations, rooted in an axiomatic theory of probability.

Beginning at least with the Rev. Thomas Bayes and Pierre Simon Laplace in the eighteenth century, most early efforts at statistical inference used what was known as the method of inverse probability to update a prior probability using the observed data in what we now refer to as Bayes’ Theorem. (For a discussion of who really invented Bayes’ Theorem, see Stigler 1999, Chap. 15). Inverse probability came under challenge in the nineteenth century, but viable alternative approaches gained little currency. It was only with the work of R. A. Fisher on statistical models, estimation, and significance tests, and Jerzy Neyman and Egon Pearson, in the 1920s and 1930s, on tests of hypotheses, that alternative approaches were fully articulated and given a formal foundation. Neyman’s advocacy of the role of probability in the structuring of a frequency-based approach to sample surveys in 1934 and his development of confidence intervals further consolidated this effort at the development of a foundation for inference (cf. Statistical Methods, History of: Post- 1900 and the discussion of ‘The inference experts’ in Gigerenzer et al. 1989).

At about the same time Kolmogorov presented his famous axiomatic treatment of probability, and thus by the end of the 1930s, all of the requisite elements were finally in place for the identification of statistics as a field. Not coincidentally, the first statistical society devoted to the mathematical underpinnings of the field, The Institute of Mathematical Statistics, was created in the United States in the mid-1930s. It was during this same period that departments of statistics and statistical laboratories and groups were first formed in universities in the United States.

3. Emergence Of Statistics As A Field

3.1 the role of world war ii.

Perhaps the greatest catalysts to the emergence of statistics as a field were two major social events: the Great Depression of the 1930s and World War II. In the United States, one of the responses to the depression was the development of large-scale probability-based surveys to measure employment and unemployment. This was followed by the institutionalization of sampling as part of the 1940 US decennial census. But with World War II raging in Europe and in Asia, mathematicians and statisticians were drawn into the war effort, and as a consequence they turned their attention to a broad array of new problems. In particular, multiple statistical groups were established in both England and the US specifically to develop new methods and to provide consulting. (See Wallis 1980, on statistical groups in the US; Barnard and Plackett 1985, for related efforts in the United Kingdom; and Fienberg 1985). These groups not only created imaginative new techniques such as sequential analysis and statistical decision theory, but they also developed a shared research agenda. That agenda led to a blossoming of statistics after the war, and in the 1950s and 1960s to the creation of departments of statistics at universities—from coast to coast in the US, and to a lesser extent in England and elsewhere.

3.2 The Neo-Bayesian Revival

Although inverse probability came under challenge in the 1920s and 1930s, it was not totally abandoned. John Maynard Keynes (1921) wrote A Treatise on Probability that was rooted in this tradition, and Frank Ramsey (1926) provided an early effort at justifying the subjective nature of prior distributions and suggested the importance of utility functions as an adjunct to statistical inference. Bruno de Finetti provided further development of these ideas in the 1930s, while Harold Jeffreys (1938) created a separate ‘objective’ development of these and other statistical ideas on inverse probability.

Yet as statistics flourished in the post-World War II era, it was largely based on the developments of Fisher, Neyman and Pearson, as well as the decision theory methods of Abraham Wald (1950). L. J. Savage revived interest in the inverse probability approach with The Foundations of Statistics (1954) in which he attempted to provide the axiomatic foundation from the subjective perspective. In an essentially independent effort, Raiffa and Schlaifer (1961) attempted to provide inverse probability counterparts to many of the then existing frequentist tools, referring to these alternatives as ‘Bayesian.’ By 1960, the term ‘Bayesian inference’ had become standard usage in the statistical literature, the theoretical interest in the development of Bayesian approaches began to take hold, and the neo-Bayesian revival was underway. But the movement from Bayesian theory to statistical practice was slow, in large part because the computations associated with posterior distributions were an overwhelming stumbling block for those who were interested in the methods. Only in the 1980s and 1990s did new computational approaches revolutionize both Bayesian methods, and the interest in them, in a broad array of areas of application.

3.3 The Role Of Computation In Statistics

From the days of Pearson and Fisher, computation played a crucial role in the development and application of statistics. Pearson’s laboratory employed dozens of women who used mechanical devices to carry out the careful and painstaking calculations required to tabulate values from various probability distributions. This effort ultimately led to the creation of the Biometrika Tables for Statisticians that were so widely used by others applying tools such as chisquare tests and the like. Similarly, Fisher also developed his own set of statistical tables with Frank Yates when he worked at Rothamsted Experiment Station in the 1920s and 1930s. One of the most famous pictures of Fisher shows him seated at Whittingehame Lodge, working at his desk calculator (see Box 1978).

The development of the modern computer revolutionized statistical calculation and practice, beginning with the creation of the first statistical packages in the 1960s—such as the BMDP package for biological and medical applications, and Datatext for statistical work in the social sciences. Other packages soon followed—such as SAS and SPSS for both data management and production-like statistical analyses, and MINITAB for the teaching of statistics. In 2001, in the era of the desktop personal computer, almost everyone has easy access to interactive statistical programs that can implement complex statistical procedures and produce publication-quality graphics. And there is a new generation of statistical tools that rely upon statistical simulation such as the bootstrap and Markov Chain Monte Carlo methods. Complementing the traditional production-like packages for statistical analysis are more methodologically oriented languages such as S and S-PLUS, and symbolic and algebraic calculation packages. Statistical journals and those in various fields of application devote considerable space to descriptions of such tools.

4. Statistics At The End Of The Twentieth Century

It is widely recognized that any statistical analysis can only be as good as the underlying data. Consequently, statisticians take great care in the the design of methods for data collection and in their actual implementation. Some of the most important modes of statistical data collection include censuses, experiments, observational studies, and sample Surveys, all of which are discussed elsewhere in this encyclopedia. Statistical experiments gain their strength and validity both through the random assignment of treatments to units and through the control of nontreatment variables. Similarly sample surveys gain their validity for generalization through the careful design of survey questionnaires and probability methods used for the selection of the sample units. Approaches to cope with the failure to fully implement randomization in experiments or random selection in sample surveys are discussed in Experimental Design: Compliance and Nonsampling Errors.

Data in some statistical studies are collected essentially at a single point in time (cross-sectional studies), while in others they are collected repeatedly at several time points or even continuously, while in yet others observations are collected sequentially, until sufficient information is available for inferential purposes. Different entries discuss these options and their strengths and weaknesses.

After a century of formal development, statistics as a field has developed a number of different approaches that rely on probability theory as a mathematical basis for description, analysis, and statistical inference. We provide an overview of some of these in the remainder of this section and provide some links to other entries in this encyclopedia.

4.1 Data Analysis

The least formal approach to inference is often the first employed. Its name stems from a famous article by John Tukey (1962), but it is rooted in the more traditional forms of descriptive statistical methods used for centuries.

Today, data analysis relies heavily on graphical methods and there are different traditions, such as those associated with

(a) The ‘exploratory data analysis’ methods suggested by Tukey and others.

(b) The more stylized correspondence analysis techniques of Benzecri and the French school.

(c) The alphabet soup of computer-based multivariate methods that have emerged over the past decade such as ACE, MARS, CART, etc.

No matter which ‘school’ of data analysis someone adheres to, the spirit of the methods is typically to encourage the data to ‘speak for themselves.’ While no theory of data analysis has emerged, and perhaps none is to be expected, the flexibility of thought and method embodied in the data analytic ideas have influenced all of the other approaches.

4.2 Frequentism

The name of this group of methods refers to a hypothetical infinite sequence of data sets generated as was the data set in question. Inferences are to be made with respect to this hypothetical infinite sequence. (For details, see Frequentist Inference).

One of the leading frequentist methods is significance testing, formalized initially by R. A. Fisher (1925) and subsequently elaborated upon and extended by Neyman and Pearson and others (see below). Here a null hypothesis is chosen, for example, that the mean, µ, of a normally distributed set of observations is 0. Fisher suggested the choice of a test statistic, e.g., based on the sample mean, x, and the calculation of the likelihood of observing an outcome as or more extreme as x is from µ 0, a quantity usually labeled as the p-value. When p is small (e.g., less than 5 percent), either a rare event has occurred or the null hypothesis is false. Within this theory, no probability can be given for which of these two conclusions is the case.

A related set of methods is testing hypotheses, as proposed by Neyman and Pearson (1928, 1932). In this approach, procedures are sought having the property that, for an infinite sequence of such sets, in only (say) 5 percent for would the null hypothesis be rejected if the null hypothesis were true. Often the infinite sequence is restricted to sets having the same sample size, but this is unnecessary. Here, in addition to the null hypothesis, an alternative hypothesis is specified. This permits the definition of a power curve, reflecting the frequency of rejecting the null hypothesis when the specified alternative is the case. But, as with the Fisherian approach, no probability can be given to either the null or the alternative hypotheses.

The construction of confidence intervals, following the proposal of Neyman (1934), is intimately related to testing hypotheses; indeed a 95 percent confidence interval may be regarded as the set of null hypotheses which, had they been tested at the 5 percent level of significance, would not have been rejected. A confidence interval is a random interval, having the property that the specified proportion (say 95 percent) of the infinite sequence, of random intervals would have covered the true value. For example, an interval that 95 percent of the time (by auxiliary randomization) is the whole real line, and 5 percent of the time is the empty set, is a valid 95 percent confidence interval.

Estimation of parameters—i.e., choosing a single value of the parameters that is in some sense best—is also an important frequentist method. Many methods have been proposed, both for particular models and as general approaches regardless of model, and their frequentist properties explored. These methods usually extended to intervals of values through inversion of test statistics or via other related devices. The resulting confidence intervals share many of the frequentist theoretical properties of the corresponding test procedures.

Frequentist statisticians have explored a number of general properties thought to be desirable in a procedure, such as invariance, unbiasedness, sufficiency, conditioning on ancillary statistics, etc. While each of these properties has examples in which it appears to produce satisfactory recommendations, there are others in which it does not. Additionally, these properties can conflict with each other. No general frequentist theory has emerged that proposes a hierarchy of desirable properties, leaving a frequentist without guidance in facing a new problem.

4.3 Likelihood Methods

The likelihood function (first studied systematically by R. A. Fisher) is the probability density of the data, viewed as a function of the parameters. It occupies an interesting middle ground in the philosophical debate, as it is used both by frequentists (as in maximum likelihood estimation) and by Bayesians in the transition from prior distributions to posterior distributions. A small group of scholars (among them G. A. Barnard, A. W. F. Edwards, R. Royall, D. Sprott) have proposed the likelihood function as an independent basis for inference. The issue of nuisance parameters has perplexed this group, since maximization, as would be consistent with maximum likelihood estimation, leads to different results in general than does integration, which would be consistent with Bayesian ideas.

4.4 Bayesian Methods

Both frequentists and Bayesians accept Bayes’ Theorem as correct, but Bayesians use it far more heavily. Bayesian analysis proceeds from the idea that probability is personal or subjective, reflecting the views of a particular person at a particular point in time. These views are summarized in the prior distribution over the parameter space. Together the prior distribution and the likelihood function define the joint distribution of the parameters and the data. This joint distribution can alternatively be factored as the product of the posterior distribution of the parameter given the data times the predictive distribution of the data.

In the past, Bayesian methods were deemed to be controversial because of the avowedly subjective nature of the prior distribution. But the controversy surrounding their use has lessened as recognition of the subjective nature of the likelihood has spread. Unlike frequentist methods, Bayesian methods are, in principle, free of the paradoxes and counterexamples that make classical statistics so perplexing. The development of hierarchical modeling and Markov Chain Monte Carlo (MCMC) methods have further added to the current popularity of the Bayesian approach, as they allow analyses of models that would otherwise be intractable.

Bayesian decision theory, which interacts closely with Bayesian statistical methods, is a useful way of modeling and addressing decision problems of experimental designs and data analysis and inference. It introduces the notion of utilities and the optimum decision combines probabilities of events with utilities by the calculation of expected utility and maximizing the latter (e.g., see the discussion in Lindley 2000).

Current research is attempting to use the Bayesian approach to hypothesis testing to provide tests and pvalues with good frequentist properties (see Bayarri and Berger 2000).

4.5 Broad Models: Nonparametrics And Semiparametrics

These models include parameter spaces of infinite dimensions, whether addressed in a frequentist or Bayesian manner. In a sense, these models put more inferential weight on the assumption of conditional independence than does an ordinary parametric model.

4.6 Some Cross-Cutting Themes

Often different fields of application of statistics need to address similar issues. For example, dimensionality of the parameter space is often a problem. As more parameters are added, the model will in general fit better (at least no worse). Is the apparent gain in accuracy worth the reduction in parsimony? There are many different ways to address this question in the various applied areas of statistics.

Another common theme, in some sense the obverse of the previous one, is the question of model selection and goodness of fit. In what sense can one say that a set of observations is well-approximated by a particular distribution? (cf. Goodness of Fit: Overview). All statistical theory relies at some level on the use of formal models, and the appropriateness of those models and their detailed specification are of concern to users of statistical methods, no matter which school of statistical inference they choose to work within.

5. Statistics In The Twenty-first Century

5.1 adapting and generalizing methodology.

Statistics as a field provides scientists with the basis for dealing with uncertainty, and, among other things, for generalizing from a sample to a population. There is a parallel sense in which statistics provides a basis for generalization: when similar tools are developed within specific substantive fields, such as experimental design methodology in agriculture and medicine, and sample surveys in economics and sociology. Statisticians have long recognized the common elements of such methodologies and have sought to develop generalized tools and theories to deal with these separate approaches (see e.g., Fienberg and Tanur 1989).

One hallmark of modern statistical science is the development of general frameworks that unify methodology. Thus the tools of Generalized Linear Models draw together methods for linear regression and analysis of various models with normal errors and those log-linear and logistic models for categorical data, in a broader and richer framework. Similarly, graphical models developed in the 1970s and 1980s use concepts of independence to integrate work in covariance section, decomposable log-linear models, and Markov random field models, and produce new methodology as a consequence. And the latent variable approaches from psychometrics and sociology have been tied with simultaneous equation and measurement error models from econometrics into a broader theory of covariance analysis and structural equations models.

Another hallmark of modern statistical science is the borrowing of methods in one field for application in another. One example is provided by Markov Chain Monte Carlo methods, now used widely in Bayesian statistics, which were first used in physics. Survival analysis, used in biostatistics to model the disease-free time or time-to-mortality of medical patients, and analyzed as reliability in quality control studies, are now used in econometrics to measure the time until an unemployed person gets a job. We anticipate that this trend of methodological borrowing will continue across fields of application.

5.2 Where Will New Statistical Developments Be Focused ?

In the issues of its year 2000 volume, the Journal of the American Statistical Association explored both the state of the art of statistics in diverse areas of application, and that of theory and methods, through a series of vignettes or short articles. These essays provide an excellent supplement to the entries of this encyclopedia on a wide range of topics, not only presenting a snapshot of the current state of play in selected areas of the field but also affecting some speculation on the next generation of developments. In an afterword to the last set of these vignettes, Casella (2000) summarizes five overarching themes that he observed in reading through the entire collection:

(a) Large datasets.

(b) High-dimensional/nonparametric models.

(c) Accessible computing.

(d) Bayes/frequentist/who cares?

(e) Theory/applied/why differentiate?

Not surprisingly, these themes fit well those that one can read into the statistical entries in this encyclopedia. The coming together of Bayesian and frequentist methods, for example, is illustrated by the movement of frequentists towards the use of hierarchical models and the regular consideration of frequentist properties of Bayesian procedures (e.g., Bayarri and Berger 2000). Similarly, MCMC methods are being widely used in non-Bayesian settings and, because they focus on long-run sequences of dependent draws from multivariate probability distributions, there are frequentist elements that are brought to bear in the study of the convergence of MCMC procedures. Thus the oft-made distinction between the different schools of statistical inference (suggested in the preceding section) is not always clear in the context of real applications.

5.3 The Growing Importance Of Statistics Across The Social And Behavioral Sciences

Statistics touches on an increasing number of fields of application, in the social sciences as in other areas of scholarship. Historically, the closest links have been with economics; together these fields share parentage of econometrics. There are now vigorous interactions with political science, law, sociology, psychology, anthropology, archeology, history, and many others.

In some fields, the development of statistical methods has not been universally welcomed. Using these methods well and knowledgeably requires an understanding both of the substantive field and of statistical methods. Sometimes this combination of skills has been difficult to develop.

Statistical methods are having increasing success in addressing questions throughout the social and behavioral sciences. Data are being collected and analyzed on an increasing variety of subjects, and the analyses are becoming increasingly sharply focused on the issues of interest.

We do not anticipate, nor would we find desirable, a future in which only statistical evidence was accepted in the social and behavioral sciences. There is room for, and need for, many different approaches. Nonetheless, we expect the excellent progress made in statistical methods in the social and behavioral sciences in recent decades to continue and intensify.

Bibliography:

  • Barnard G A, Plackett R L 1985 Statistics in the United Kingdom, 1939–1945. In: Atkinson A C, Fienberg S E (eds.) A Celebration of Statistics: The ISI Centennial Volume. Springer-Verlag, New York, pp. 31–55
  • Bayarri M J, Berger J O 2000 P values for composite null models (with discussion). Journal of the American Statistical Association 95: 1127–72
  • Box J 1978 R. A. Fisher, The Life of a Scientist. Wiley, New York
  • Casella G 2000 Afterword. Journal of the American Statistical Association 95: 1388
  • Fienberg S E 1985 Statistical developments in World War II: An international perspective. In: Anthony C, Atkinson A C, Fienberg S E (eds.) A Celebration of Statistics: The ISI Centennial Volume. Springer-Verlag, New York, pp. 25–30
  • Fienberg S E, Tanur J M 1989 Combining cognitive and statistical approaches to survey design. Science 243: 1017–22
  • Fisher R A 1925 Statistical Methods for Research Workers. Oliver and Boyd, London
  • Gigerenzer G, Swijtink Z, Porter T, Daston L, Beatty J, Kruger L 1989 The Empire of Chance. Cambridge University Press, Cambridge, UK
  • Hacking I 1990 The Taming of Chance. Cambridge University Press, Cambridge, UK
  • Jeffreys H 1938 Theory of Probability, 2nd edn. Clarendon Press, Oxford, UK
  • Keynes J 1921 A Treatise on Probability. Macmillan, London
  • Lindley D V 2000/1932 The philosophy of statistics (with discussion). The Statistician 49: 293–337
  • Neyman J 1934 On the two different aspects of the representative method: the method of stratified sampling and the method of purposive selection (with discussion). Journal of the Royal Statistical Society 97: 558–625
  • Neyman J, Pearson E S 1928 On the use and interpretation of certain test criteria for purposes of statistical inference. Part I. Biometrika 20A: 175–240
  • Neyman J, Pearson E S 1932 On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society, Series. A 231: 289–337
  • Raiffa H, Schlaifer R 1961 Applied Statistical Decision Theory. Harvard Business School, Boston
  • Ramsey F P 1926 Truth and probability. In: The Foundations of Mathematics and Other Logical Essays. Kegan Paul, London, pp.
  • Savage L J 1954 The Foundations of Statistics. Wiley, New York
  • Stigler S M 1986 The History of Statistics: The Measurement of Uncertainty Before 1900. Harvard University Press, Cambridge, MA
  • Stigler S M 1999 Statistics on the Table: The History of Statistical Concepts and Methods. Harvard University Press, Cambridge, MA
  • Tukey John W 1962 The future of data analysis. Annals of Mathematical Statistics 33: 1–67
  • Wald A 1950 Statistical Decision Functions. Wiley, New York
  • Wallis W 1980 The Statistical Research Group, 1942–1945 (with discussion). Journal of the American Statistical Association 75: 320–35

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How To Write A Statistics Research Paper?

Haiden Malecot

Table of Contents

Statistics Research Paper

Naturally, all-encompassing information about the slightest details of the statistical paper writing cannot be stuffed into one guideline. Still, we will provide a glimpse of the basics of the stats research paper.

What is a stats research paper?

One of the main problems of stats academic research papers is that not all students understand what it is. Put it bluntly, it is an essay that provides an analysis of the gathered statistical data to induce the key points of a specified research issue. Thus, the author of the paper creates a construct of the topic by explaining the statistical data.

Writing a statistics research paper is quite challenging because the sources of data for statistical analysis are quite numerous. These are data mining, biostatistics, quality control, surveys, statistical modelling, etc.

Collecting data for the college research paper analysis is another headache. Research papers of this type call for the data taken from the most reliable and relevant sources because no indeterminate information is inadmissible here.

How to create the perfect statistics research paper example?

If you want to create the paper that can serve as a research paper writing example of well-written statistics research paper example, then here is a guideline that will help you to master this task.

Select the topic

Obviously, work can’t be written without a topic. Therefore, it is essential to come up with the theme that promises interesting statistics, and a possibility to gather enough data for the research. Access to the reliable sources of the research data is also a must.

If you are not confident about the availability of several sources concerning the chosen topic, you’d better choose something else.

Remember to jot down all the needed information for the proper referencing when you use a resource

Data collection

The duration of this stage depends on the number of data sources and the chosen methodology of the data collection. Mind that once you have chosen the method, you should stick to it. Naturally, it is essential to explain your choice of the methodology in your statistics research paper.

Outlining the paper

Creating a rough draft of the paper is your chance to save some time and nerves. Once you’ve done it, you get a clear picture of what to write about and what points should be worked through.

The intro section

This is, perhaps, the most important part of the paper. As this is the most scientific paper from all the papers you will have to write in your studies, it calls for the most logical and clear approach. Thus, your intro should consist of:

  • Opening remarks about the field of the research.
  • Credits to other researchers who worked on this theme.
  • The scientific motivation for the new research .
  • An explanation of why existing researches are not sufficient.
  • The thesis statement , aka the core idea of the text.

The body of the text (research report, as they say in statistics)

Believe it or not, but many professional writers start such papers from the body. Here you have to place the Methodology Section where you establish the methods of data collection and the results of it. Usually, all main graphs or charts are placed here as a way to convey the results. All additional materials are gathered in the appendices.

The next paragraph of the paper will be the Evaluation of the gathered data . And that’s where the knowledge on how to read statistics in a research paper can come in handy. If you have no clue how to do it, you’re in trouble, to be honest. At least, you should know three concepts: odds ratios, confidence intervals, and p values. You can start searching for them on the web or in B.S.Everitt’s Dictionary of Statistics.

And the last section of the body is Discussion . Here, as the name suggests, you have to discuss the analysis and the results of the research.

The conclusion

This section requires only several sentences where you summarise the findings and highlight the importance of the research. You may also include a suggestion on how to continue or deepen the research of the issue.

Tips on how to write a statistics paper example

Here are some life hacks and shortcuts that you may use to boost your paper:

  • Many sources where you take the statistical data , do offer it with the interpretation. Do not waste time on calculations and take the interpretation from there.
  • Visuals are the must: always include a graph, chart, or a table to visualize your words.
  • If you do not know the statistical procedure and how to interpret the results , never use it in the paper.
  • Always put the statistics at the end of the sentence.
  • If your paper requires the presentation of your calculations and you are not confident with it, ask a pro to help you.
  • Different types of statistical data require proper formatting. Cite statistics properly according to the chosen format.

…Final thoughts

We hope that our guideline on how to write a statistics paper example unveiled the mystery of writing such papers.

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Americans’ social media use, youtube and facebook are by far the most used online platforms among u.s. adults; tiktok’s user base has grown since 2021.

To better understand Americans’ social media use, Pew Research Center surveyed 5,733 U.S. adults from May 19 to Sept. 5, 2023. Ipsos conducted this National Public Opinion Reference Survey (NPORS) for the Center using address-based sampling and a multimode protocol that included both web and mail. This way nearly all U.S. adults have a chance of selection. The survey is weighted to be representative of the U.S. adult population by gender, race and ethnicity, education and other categories.

Polls from 2000 to 2021 were conducted via phone. For more on this mode shift, read our Q&A .

Here are the questions used for this analysis , along with responses, and  its methodology ­­­.

A note on terminology: Our May-September 2023 survey was already in the field when Twitter changed its name to “X.” The terms  Twitter  and  X  are both used in this report to refer to the same platform.

Social media platforms faced a range of controversies in recent years, including concerns over misinformation and data privacy . Even so, U.S. adults use a wide range of sites and apps, especially YouTube and Facebook. And TikTok – which some Congress members previously called to ban – saw growth in its user base.

These findings come from a Pew Research Center survey of 5,733 U.S. adults conducted May 19-Sept. 5, 2023.

Which social media sites do Americans use most?

A horizontal bar chart showing that most U.S. adults use YouTube and Facebook; about half use Instagram.

YouTube by and large is the most widely used online platform measured in our survey. Roughly eight-in-ten U.S. adults (83%) report ever using the video-based platform.

While a somewhat lower share reports using it, Facebook is also a dominant player in the online landscape. Most Americans (68%) report using the social media platform.

Additionally, roughly half of U.S. adults (47%) say they use Instagram .

The other sites and apps asked about are not as widely used , but a fair portion of Americans still use them:

  • 27% to 35% of U.S. adults use Pinterest, TikTok, LinkedIn, WhatsApp and Snapchat.
  • About one-in-five say they use Twitter (recently renamed “X”) and Reddit.  

This year is the first time we asked about BeReal, a photo-based platform launched in 2020. Just 3% of U.S. adults report using it.

Recent Center findings show that YouTube also dominates the social media landscape among U.S. teens .

TikTok sees growth since 2021

One platform – TikTok – stands out for growth of its user base. A third of U.S. adults (33%) say they use the video-based platform, up 12 percentage points from 2021 (21%).

A line chart showing that a third of U.S. adults say they use TikTok, up from 21% in 2021.

The other sites asked about had more modest or no growth over the past couple of years. For instance, while YouTube and Facebook dominate the social media landscape, the shares of adults who use these platforms has remained stable since 2021.

The Center has been tracking use of online platforms for many years. Recently, we shifted from gathering responses via telephone to the web and mail. Mode changes can affect study results in a number of ways, therefore we have to take a cautious approach when examining how things have – or have not – changed since our last study on these topics in 2021. For more details on this shift, please read our Q&A .

Stark age differences in who uses each app or site

Adults under 30 are far more likely than their older counterparts to use many of the online platforms. These findings are consistent with previous Center data .

A dot plot showing that the youngest U.S. adults are far more likely to use Instagram, Snapchat and TikTok; age differences are less pronounced for Facebook.

Age gaps are especially large for Instagram, Snapchat and TikTok – platforms that are used by majorities of adults under 30. For example:

  • 78% of 18- to 29-year-olds say they use Instagram, far higher than the share among those 65 and older (15%).
  • 65% of U.S. adults under 30 report using Snapchat, compared with just 4% of the oldest age cohort.
  • 62% of 18- to 29-year-olds say they use TikTok, much higher than the share among adults ages 65 years and older (10%).
  • Americans ages 30 to 49 and 50 to 64 fall somewhere in between for all three platforms.

YouTube and Facebook are the only two platforms that majorities of all age groups use. That said, there is still a large age gap between the youngest and oldest adults when it comes to use of YouTube. The age gap for Facebook, though, is much smaller.

Americans ages 30 to 49 stand out for using three of the platforms – LinkedIn, WhatsApp and Facebook – at higher rates. For instance, 40% of this age group uses LinkedIn, higher than the roughly three-in-ten among those ages 18 to 29 and 50 to 64. And just 12% of those 65 and older say the same. 

Overall, a large majority of the youngest adults use multiple sites and apps. About three-quarters of adults under 30 (74%) use at least five of the platforms asked about. This is far higher than the shares of those ages 30 to 49 (53%), 50 to 64 (30%), and ages 65 and older (8%) who say the same.  

Refer to our social media fact sheet for more detailed data by age for each site and app.

Other demographic differences in use of online platforms

A number of demographic differences emerge in who uses each platform. Some of these include the following:

  • Race and ethnicity: Roughly six-in-ten Hispanic (58%) and Asian (57%) adults report using Instagram, somewhat higher than the shares among Black (46%) and White (43%) adults. 1
  • Gender: Women are more likely than their male counterparts to say they use the platform.
  • Education: Those with some college education and those with a college degree report using it at somewhat higher rates than those who have a high school degree or less education.
  • Race and ethnicity: Hispanic adults are particularly likely to use TikTok, with 49% saying they use it, higher than Black adults (39%). Even smaller shares of Asian (29%) and White (28%) adults say the same.
  • Gender: Women use the platform at higher rates than men (40% vs. 25%).
  • Education: Americans with higher levels of formal education are especially likely to use LinkedIn. For instance, 53% of Americans with at least a bachelor’s degree report using the platform, far higher than among those who have some college education (28%) and those who have a high school degree or less education (10%). This is the largest educational difference measured across any of the platforms asked about.

Twitter (renamed “X”)

  • Household income: Adults with higher household incomes use Twitter at somewhat higher rates. For instance, 29% of U.S. adults who have an annual household income of at least $100,000 say they use the platform. This compares with one-in-five among those with annual household incomes of $70,000 to $99,999, and around one-in-five among those with annual incomes of less than $30,000 and those between $30,000 and $69,999.
  • Gender: Women are far more likely to use Pinterest than men (50% vs. 19%).
  • Race and ethnicity: 54% of Hispanic adults and 51% of Asian adults report using WhatsApp. This compares with 31% of Black adults and even smaller shares of those who are White (20%).

A heat map showing how use of online platforms – such as Facebook, Instagram or TikTok – differs among some U.S. demographic groups.

  • Estimates for Asian adults are representative of English speakers only. ↩

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Report Materials

Table of contents, q&a: how – and why – we’re changing the way we study tech adoption, americans’ use of mobile technology and home broadband, social media fact sheet, internet/broadband fact sheet, mobile fact sheet, most popular.

About Pew Research Center Pew Research Center is a nonpartisan fact tank that informs the public about the issues, attitudes and trends shaping the world. It conducts public opinion polling, demographic research, media content analysis and other empirical social science research. Pew Research Center does not take policy positions. It is a subsidiary of The Pew Charitable Trusts .

Journals Logo

1. Introduction

2. materials and methods, 4. conclusions, 5. related literature, supporting information.

statistical analysis paper example

research papers \(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

Open Access

Structural analysis of nanocrystals by pair distribution function combining electron diffraction with crystal tilting

a School of Physical Science and Technology, and Shanghai Key Laboratory of High-resolution Electron Microscopy, ShanghaiTech University, Shanghai 201210, People's Republic of China, and b State Key Laboratory of High Performance Ceramics and Superfine Microstructure, Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai 200050, People's Republic of China * Correspondence e-mail: [email protected] , [email protected] , [email protected]

As an important characterization method, pair distribution function (PDF) has been extensively used in structural analysis of nanomaterials, providing key insights into the degree of crystallinity, atomic structure, local disorder etc . The collection of scattering signals with good statistics is necessary for a reliable structural analysis. However, current conventional electron diffraction experiments using PDF (ePDF) are limited in their ability to acquire continuous diffraction rings for large nanoparticles. Herein, a new method – tilt-ePDF – is proposed to improve the data quality and compatibility of ePDF by a combination of electron diffraction and specimen tilting. In the present work, a tilt-series of electron diffraction patterns was collected from gold nanoparticles with three different sizes and a standard sample polycrystalline aluminium film for ePDF analysis. The results show that tilt-ePDF can not only enhance the continuity of diffraction rings, but can also improve the signal-to-noise ratio in the high scattering angle range. As a result, compared with conventional ePDF data, tilt-ePDF data provide structure parameters with a better accuracy and lower residual factors in the refinement against the crystal structure. This method provides a new way of utilizing ePDF to obtain accurate local structure information from nanoparticles.

Keywords: electron pair distribution functions ; local structures ; crystal tilting ; nanoparticles ; PDF refinement .

Herein, to increase the continuity of diffraction rings of nanoparticles with large particle sizes and obtain high-quality ePDF data, we propose a new approach that involves the collection of electron diffraction rings via crystal tilting, termed tilt-ePDF. A tilt-series of ED patterns was collected and overlapped to obtain diffraction data for gold nanoparticles (AuNPs) of different sizes and polycrystalline aluminium film. Compared with conventional ED and precession ED (PED), the diffraction data obtained through tilting showed a significant improvement in the signal-to-noise ratio and enhanced continuity of diffraction rings. The refinement of these samples was then conducted against standard structural models, which showed that the tilt-ePDF data can decrease the residual factors of refinement, suggesting the promising application of tilt-ePDF.

2.1. AUNP and aluminium film preparation

2.1.1. synthesis of aunps of different sizes.

The second-generation gold nanoparticles (second AuNPs) were synthesized based on the first AuNPs. The remaining solution in the three-neck flask was cooled to 363 K before adding 1 ml of HAuCl 4 ·3H 2 O (25 m M ). After 30 min, 2 ml of HAuCl 4 ·3H 2 O (25 m M ) was added to the solution. When the reaction in the solution had proceeded for 30 min at 363 K, 3 ml of the solution was removed for cooling, and the second AuNPs were obtained. To synthesize third-generation gold nanoparticles (third AuNPs), 55 ml was removed from the second solution from the flask and 53 ml ultrapure water and 2 ml of Na 3 Cit (2.2 m M ) was added. 1 ml of HAuCl 4 ·3H 2 O (25 m M ) was added when the solution was stabilized at 363 K. The subsequent operation was the same as the synthesis of the second AuNPs: the solution reaction took 30 min and then 2 ml HAuCl 4 ·3H 2 O (25 m M ) was added. After waiting for 30 min, 3 ml of the solution was removed to cool down in air, and the third AuNPs were obtained.

2.1.2. Evaporated aluminium film

Polycrystalline aluminium (Al) film was purchased from Ted Pella Inc (Table S1 of the supporting information ) and distributed on a 3 mm TEM grid. The specimen was used without further purification. In addition, three samples of Al films with different thicknesses were prepared. TEM grids coated with ultra-thin carbon films were placed into a magnetron sputtering instrument and the sputter times were set to 121, 363 and 909 s to produce Al films on the TEM grids; as a result, three Al film samples with thicknesses of ∼20, 60 and 150 nm, respectively, were obtained.

2.2. Sample preparation

The three AuNPs samples (first, second and third AuNPs) of different sizes were all colloidal suspensions. However, because of their high concentrations, the particles distributed in the solution agglomerated easily. Therefore, the suspensions were pretreated in an ultrasonic cleaner for 30 min to preserve the suspension and avoid agglomeration of nanoparticles on the carbon films. Then, 2 µl of the suspension was dropped onto a copper grid loaded with ultra-thin carbon film (200 mesh), and the copper grids were used in TEM characterization after waiting 24 h for them to dry in air.

2.3. Data collection of single-ePDF, PED-ePDF and tilt-ePDF

The copper grids were loaded on a high-tilt specimen holder, which enables a large tilting angle range (±70°). Since the gold and aluminium specimens are rather stable under electron beams, a high electron dose (Table S2) was applied to collect diffraction patterns to ensure a high resolution and good signal-to-noise ratio. A beam stopper was used to block the central spot to avoid damage to the camera, and the exposure time was set to 500 ms. The diffraction patterns were recorded using 32-bit or 16-bit images to reach a high dynamic range (Table S3). All original data, including TEM images and diffraction patterns, were obtained using a Rio-16 detector equipped on a Jeol JEM-F200 TEM and a TVIPS (XF416) camera on a Jeol JEM-2100Plus at room temperature. The accelerating voltage was 200 kV ( λ = 0.0251 Å).

3.1. TEM images and electron diffraction

3.2. the error range of the epdf results.

Before processing diffraction data of three AuNPs samples and polycrystalline Al film, it is necessary to perform a stability test on the ePDF data in order to reduce the instability factor and specify the error of the ePDF results. For stability testing, sputtered AuNPs were selected and prepared using an Ion Sputter SBC-12. The deposition process was completed once the entire carbon film was fully covered by the AuNPs, with a deposition time of approximately 20 s.

The deposited AuNPs on carbon film could be seen in Fig. S7, and most nanoparticles in the sample were uniformly distributed. Eight different areas on the carbon film were selected to collect the corresponding polycrystalline diffraction rings. The exposure time of each SAED frame is 10 s. The electron dose rate is kept at 1.900 e Å −2  s − 1 . The SAED patterns collected from eight areas show high consistency (Fig. S8) and little difference could be identified among these data. These diffraction data were further processed and refined against the standard structure model of Au (ICSD no. 44362).

From Table S4 and Fig. S9, the ePDF stability test results show that the absolute deviation Δ a (%) is less than ±0.35% and the error range of Δ R w (%) is less than ±1%. It can be seen that the ePDF data obtained by collecting the diffraction rings of nanoparticles from different regions have a considerably high consistency and low deviation, and the results are reproducible.

3.3. ePDF refinement analysis

After Fourier transform, ePDF curves corresponding to different samples were obtained (for details, see the supporting information ). The two major factors that determine the accuracy of G ( r ) are the diffraction intensity and the Q range. The larger the range of Q values recorded, the more accurate the diffraction intensity is, resulting in a more accurate result. The Q max of our electron diffraction data is about 16.50 Å −1 and the optimal low scattering angle Q min is chosen to reduce noise input from the central spot.

The refined parameters include particle size, cell parameters, atomic isotropic parameters (ADPs) U and decay factor Q damp . Since the particles size is already known, the diameter of the Au samples could be fixed at 12 nm (first AuNPs), 22 nm (second AuNPs) and 31 nm (third AuNPs) during the refinement. Meanwhile, to reduce the affecting influences, all datasets were processed using a similar fitting Q range with approximately 2.3–16.0 Å −1 .

Given that the ePDF refinement results for the three different sizes of nanoparticles lead to nearly a 3% w reduction in R w , it had proven that the method of tilt-ePDF has the capability to improve data quality and produce more accurate structure refinement results. In addition, we also calculate the refined PDF data (Tables S11–S13) by merging different numbers of diffraction patterns from a tilt-series. Apparently, compared with a single ED pattern, the merging of multiple diffraction patterns reduced the residual factors. However, there is no universal rule for how many patterns should be used for merging, which might vary depending on the samples.

To overcome the size limitation for nanoparticles in the implementation of ePDF, a new method, tilt-ePDF, was proposed by combining ED with specimen tilting. A tilt-series of ED patterns was collected from multiple nanocrystals with continuous tilting of the specimen. As a result, diffraction rings became more consecutive compared with those in single ED and PED patterns, and the signal-to-noise ratio was also improved, especially in the high-scattering-angle range. A better fitting with the structural model was obtained during the following refinement. These results confirm that the tilt-ePDF method facilitates the application of the ePDF method to large-sized nanoparticles, thus broadening the scope of this technique. With the rapid development of ePDF, the new method proposed here might provide an additional way to obtain quantitative structural information from nanoparticles.

Supporting information file. DOI: https://doi.org/10.1107/S2052252524001064/of5003sup1.pdf

Funding information

The following funding is acknowledged: National Natural Science Foundation of China (grant Nos. 22222108; 12027804 awarded to YM); Shanghai Science and Technology Plan (grant No. 21DZ2260400 awarded to YM); Shanghai Government (grant No. 23JC1404000); National Key Research and Development Program of China (grant No. 2022YFA1506000 awarded to YM). The authors appreciate CℏEM, School of Physical Sciences and Technology, ShanghaiTech University (grant No. EM02161943) for supporting the EM facilities, and the support from the Analytical Instrumentation Center (grant No. SPSTAIC10112914), School of Physical Sciences and Technology, ShanghaiTech University.

This is an open-access article distributed under the terms of the Creative Commons Attribution (CC-BY) Licence , which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.

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