statistical analysis research definition

What Is Statistical Analysis?

Statistical analysis helps you pull meaningful insights from data. The process involves working with data and deducing numbers to tell quantitative stories.

Statistical analysis is a technique we use to find patterns in data and make inferences about those patterns to describe variability in the results of a data set or an experiment. 

In its simplest form, statistical analysis answers questions about:

• Quantification — how big/small/tall/wide is it?
• Variability — growth, increase, decline
• The confidence level of these variabilities

What Are the 2 Types of Statistical Analysis?

• Descriptive Statistics:  Descriptive statistical analysis describes the quality of the data by summarizing large data sets into single measures. 
• Inferential Statistics:  Inferential statistical analysis allows you to draw conclusions from your sample data set and make predictions about a population using statistical tests.

What’s the Purpose of Statistical Analysis?

Using statistical analysis, you can determine trends in the data by calculating your data set’s mean or median. You can also analyze the variation between different data points from the mean to get the standard deviation . Furthermore, to test the validity of your statistical analysis conclusions, you can use hypothesis testing techniques, like P-value, to determine the likelihood that the observed variability could have occurred by chance.

More From Abdishakur Hassan The 7 Best Thematic Map Types for Geospatial Data

Statistical Analysis Methods

There are two major types of statistical data analysis: descriptive and inferential. 

Descriptive Statistical Analysis

Descriptive statistical analysis describes the quality of the data by summarizing large data sets into single measures. 

Within the descriptive analysis branch, there are two main types: measures of central tendency (i.e. mean, median and mode) and measures of dispersion or variation (i.e. variance , standard deviation and range). 

For example, you can calculate the average exam results in a class using central tendency or, in particular, the mean. In that case, you’d sum all student results and divide by the number of tests. You can also calculate the data set’s spread by calculating the variance. To calculate the variance, subtract each exam result in the data set from the mean, square the answer, add everything together and divide by the number of tests.

Inferential Statistics

On the other hand, inferential statistical analysis allows you to draw conclusions from your sample data set and make predictions about a population using statistical tests. 

There are two main types of inferential statistical analysis: hypothesis testing and regression analysis. We use hypothesis testing to test and validate assumptions in order to draw conclusions about a population from the sample data. Popular tests include Z-test, F-Test, ANOVA test and confidence intervals . On the other hand, regression analysis primarily estimates the relationship between a dependent variable and one or more independent variables. There are numerous types of regression analysis but the most popular ones include linear and logistic regression .  

Statistical Analysis Steps  

In the era of big data and data science, there is a rising demand for a more problem-driven approach. As a result, we must approach statistical analysis holistically. We may divide the entire process into five different and significant stages by using the well-known PPDAC model of statistics: Problem, Plan, Data, Analysis and Conclusion.

In the first stage, you define the problem you want to tackle and explore questions about the problem. 

2. Plan

Next is the planning phase. You can check whether data is available or if you need to collect data for your problem. You also determine what to measure and how to measure it. 

The third stage involves data collection, understanding the data and checking its quality. 

4. Analysis

Statistical data analysis is the fourth stage. Here you process and explore the data with the help of tables, graphs and other data visualizations.  You also develop and scrutinize your hypothesis in this stage of analysis. 

5. Conclusion

The final step involves interpretations and conclusions from your analysis. It also covers generating new ideas for the next iteration. Thus, statistical analysis is not a one-time event but an iterative process.

Statistical Analysis Uses

Statistical analysis is useful for research and decision making because it allows us to understand the world around us and draw conclusions by testing our assumptions. Statistical analysis is important for various applications, including:

• Statistical quality control and analysis in product development 
• Clinical trials
• Customer satisfaction surveys and customer experience research 
• Marketing operations management
• Process improvement and optimization
• Training needs 

More on Statistical Analysis From Built In Experts Intro to Descriptive Statistics for Machine Learning

Benefits of Statistical Analysis

Here are some of the reasons why statistical analysis is widespread in many applications and why it’s necessary:

Understand Data

Statistical analysis gives you a better understanding of the data and what they mean. These types of analyses provide information that would otherwise be difficult to obtain by merely looking at the numbers without considering their relationship.

Find Causal Relationships

Statistical analysis can help you investigate causation or establish the precise meaning of an experiment, like when you’re looking for a relationship between two variables.

Make Data-Informed Decisions

Businesses are constantly looking to find ways to improve their services and products . Statistical analysis allows you to make data-informed decisions about your business or future actions by helping you identify trends in your data, whether positive or negative. 

Determine Probability

Statistical analysis is an approach to understanding how the probability of certain events affects the outcome of an experiment. It helps scientists and engineers decide how much confidence they can have in the results of their research, how to interpret their data and what questions they can feasibly answer.

You’ve Got Questions. Our Experts Have Answers. Confidence Intervals, Explained!

What Are the Risks of Statistical Analysis?

Statistical analysis can be valuable and effective, but it’s an imperfect approach. Even if the analyst or researcher performs a thorough statistical analysis, there may still be known or unknown problems that can affect the results. Therefore, statistical analysis is not a one-size-fits-all process. If you want to get good results, you need to know what you’re doing. It can take a lot of time to figure out which type of statistical analysis will work best for your situation .

Thus, you should remember that our conclusions drawn from statistical analysis don’t always guarantee correct results. This can be dangerous when making business decisions. In marketing , for example, we may come to the wrong conclusion about a product . Therefore, the conclusions we draw from statistical data analysis are often approximated; testing for all factors affecting an observation is impossible.

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Connect with the latest insights on analytics through related articles and research., more on statistical analysis.

• Three trends for successful citizen data scientists
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• Get certified! Statistics for machine learning

Statistics is so unique because it can go from health outcomes research to marketing analysis to the longevity of a light bulb. It’s a fun field because you really can do so many different things with it.

Besa Smith President and Senior Scientist Analydata

Statistical Computing

Traditional methods for statistical analysis – from sampling data to interpreting results – have been used by scientists for thousands of years. But today’s data volumes make statistics ever more valuable and powerful. Affordable storage, powerful computers and advanced algorithms have all led to an increased use of computational statistics.

Whether you are working with large data volumes or running multiple permutations of your calculations, statistical computing has become essential for today’s statistician. Popular statistical computing practices include:

• Statistical programming – From traditional analysis of variance and linear regression to exact methods and statistical visualization techniques, statistical programming is essential for making data-based decisions in every field.
• Econometrics – Modeling, forecasting and simulating business processes for improved strategic and tactical planning. This method applies statistics to economics to forecast future trends.
• Operations research – Identify the actions that will produce the best results – based on many possible options and outcomes. Scheduling, simulation, and related modeling processes are used to optimize business processes and management challenges.
• Matrix programming – Powerful computer techniques for implementing your own statistical methods and exploratory data analysis using row operation algorithms.
• Statistical quality improvement – A mathematical approach to reviewing the quality and safety characteristics for all aspects of production.

Careers in Statistical Analysis

With everyone from The New York Times to Google’s Chief Economist Hal Varien proclaiming statistics to be the latest hot career field, who are we to argue? But why is there so much talk about careers in statistical analysis and data science? It could be the shortage of trained analytical thinkers. Or it could be the demand for managing the latest big data strains. Or, maybe it’s the excitement of applying mathematical concepts to make a difference in the world.

If you talk to statisticians about what first interested them in statistical analysis, you’ll hear a lot of stories about collecting baseball cards as a child. Or applying statistics to win more games of Axis and Allies. It is often these early passions that lead statisticians into the field. As adults, those passions can carry over into the workforce as a love of analysis and reasoning, where their passions are applied to everything from the influence of friends on purchase decisions to the study of endangered species around the world.

Learn more about the field of statistics and real-world statisticians:

• Read the personal stories of data scientists solving real-world problems.
• Learn about statistics and advanced analytics on the SAS Data Science blog .

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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 ].

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

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

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:

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:

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:

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:

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

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 ].

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.

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

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

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:

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

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:

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:

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

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:

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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Introduction to Statistical Analysis: A Beginner’s Guide.

Statistical analysis is a crucial component of research work across various disciplines, helping researchers derive meaningful insights from data. Whether you’re conducting scientific studies, social research, or data-driven investigations, having a solid understanding of statistical analysis is essential. In this beginner’s guide, we will explore the fundamental concepts and techniques of statistical analysis specifically tailored for research work, providing you with a strong foundation to enhance the quality and credibility of your research findings.

1. Importance of Statistical Analysis in Research:

Research aims to uncover knowledge and make informed conclusions. Statistical analysis plays a pivotal role in achieving this by providing tools and methods to analyze and interpret data accurately. It helps researchers identify patterns, test hypotheses, draw inferences, and quantify the strength of relationships between variables. Understanding the significance of statistical analysis empowers researchers to make evidence-based decisions.

2. Data Collection and Organization:

Before diving into statistical analysis, researchers must collect and organize their data effectively. We will discuss the importance of proper sampling techniques, data quality assurance, and data preprocessing. Additionally, we will explore methods to handle missing data and outliers, ensuring that your dataset is reliable and suitable for analysis.

3. Exploratory Data Analysis (EDA):

Exploratory Data Analysis is a preliminary step that involves visually exploring and summarizing the main characteristics of the data. We will cover techniques such as data visualization, descriptive statistics, and data transformations to gain insights into the distribution, central tendencies, and variability of the variables in your dataset. EDA helps researchers understand the underlying structure of the data and identify potential relationships for further investigation.

4. Statistical Inference and Hypothesis Testing:

Statistical inference allows researchers to make generalizations about a population based on a sample. We will delve into hypothesis testing, covering concepts such as null and alternative hypotheses, p-values, and significance levels. By understanding these concepts, you will be able to test your research hypotheses and determine if the observed results are statistically significant.

5. Parametric and Non-parametric Tests:

Parametric and non-parametric tests are statistical techniques used to analyze data based on different assumptions about the underlying population distribution. We will explore commonly used parametric tests, such as t-tests and analysis of variance (ANOVA), as well as non-parametric tests like the Mann-Whitney U test and Kruskal-Wallis test. Understanding when to use each type of test is crucial for selecting the appropriate analysis method for your research questions.

6. Correlation and Regression Analysis:

Correlation and regression analysis allow researchers to explore relationships between variables and make predictions. We will cover Pearson correlation coefficients, multiple regression analysis, and logistic regression. These techniques enable researchers to quantify the strength and direction of associations and identify predictive factors in their research.

7. Sample Size Determination and Power Analysis:

Sample size determination is a critical aspect of research design, as it affects the validity and reliability of your findings. We will discuss methods for estimating sample size based on statistical power analysis, ensuring that your study has sufficient statistical power to detect meaningful effects. Understanding sample size determination is essential for planning robust research studies.

Conclusion:

Statistical analysis is an indispensable tool for conducting high-quality research. This beginner’s guide has provided an overview of key concepts and techniques specifically tailored for research work, enabling you to enhance the credibility and reliability of your findings. By understanding the importance of statistical analysis, collecting and organizing data effectively, performing exploratory data analysis, conducting hypothesis testing, utilizing parametric and non-parametric tests, and considering sample size determination, you will be well-equipped to carry out rigorous research and contribute valuable insights to your field. Remember, continuous learning, practice, and seeking guidance from statistical experts will further enhance your skills in statistical analysis for research.

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Statistical Analysis: Definition, Examples

Statistics Definitions > What is Statistical Analysis?

• Summarize the data. For example, make a pie chart .
• Find key measures of location . For example, the mean tells you what the average (or “middling”) number is in a set of data.
• Calculate measures of spread : these tell you if your data is tightly clustered or more spread out. The standard deviation is one of the more commonly used measures of spread; it tells you how spread out your data is about the mean.
• Make future predictions based on past behavior. This is especially useful in retail, manufacturing, banking, sports or for any organization where knowing future trends would be a benefit.
• Test an experiment’s hypothesis . Collecting data from an experiment only tells a story when you analyze the data. This part of statistical analysis is more formally called “ Hypothesis Testing ,” where the null hypothesis (the commonly accepted theory) is either proved or disproved.

Statistical Analysis and the Scientific Method

Statistical analysis is used extensively in science, from physics to the social sciences. As well as testing hypotheses, statistics can provide an approximation for an unknown that is difficult or impossible to measure. For example, the field of quantum field theory , while providing success in the theoretical side of things, has proved challenging for empirical experimentation and measurement. Some social science topics, like the study of consciousness or choice, are practically impossible to measure; statistical analysis can shed light on what would be the most likely or the least likely scenario.

When Statistics Lie

While statistics can sound like a solid base to draw conclusions and present “facts,” be wary of the pitfalls of statistical analysis. They include deliberate and accidental manipulation of results. However, sometimes statistics are just plain wrong. A famous example of “plain wrong” statistics is Simpson’s Paradox , which shows us that even the best statistics can be completely useless. In a classic case of Simpson’s, averages from University of Berkeley admissions (correctly) showed their average admission rate was higher for women than men, when in fact it was the other way around. For a more detailed explanation of that brain bender, see Simpson’s Paradox .

For some examples of deliberate (or plain dumb) manipulation of statistics, see:

• Misleading Graphs: Real Life Examples
• Misleading Statistics Examples in Advertising and in the News

Table of Contents

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, understanding statistical analysis: techniques and applications.

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. 

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• Table Of Contents

data analysis , the process of systematically collecting, cleaning, transforming, describing, modeling, and interpreting data , generally employing statistical techniques. Data analysis is an important part of both scientific research and business, where demand has grown in recent years for data-driven decision making . Data analysis techniques are used to gain useful insights from datasets, which can then be used to make operational decisions or guide future research . With the rise of “Big Data,” the storage of vast quantities of data in large databases and data warehouses, there is increasing need to apply data analysis techniques to generate insights about volumes of data too large to be manipulated by instruments of low information-processing capacity.

Datasets are collections of information. Generally, data and datasets are themselves collected to help answer questions, make decisions, or otherwise inform reasoning. The rise of information technology has led to the generation of vast amounts of data of many kinds, such as text, pictures, videos, personal information, account data, and metadata, the last of which provide information about other data. It is common for apps and websites to collect data about how their products are used or about the people using their platforms. Consequently, there is vastly more data being collected today than at any other time in human history. A single business may track billions of interactions with millions of consumers at hundreds of locations with thousands of employees and any number of products. Analyzing that volume of data is generally only possible using specialized computational and statistical techniques.

The desire for businesses to make the best use of their data has led to the development of the field of business intelligence , which covers a variety of tools and techniques that allow businesses to perform data analysis on the information they collect.

For data to be analyzed, it must first be collected and stored. Raw data must be processed into a format that can be used for analysis and be cleaned so that errors and inconsistencies are minimized. Data can be stored in many ways, but one of the most useful is in a database . A database is a collection of interrelated data organized so that certain records (collections of data related to a single entity) can be retrieved on the basis of various criteria . The most familiar kind of database is the relational database , which stores data in tables with rows that represent records (tuples) and columns that represent fields (attributes). A query is a command that retrieves a subset of the information in the database according to certain criteria. A query may retrieve only records that meet certain criteria, or it may join fields from records across multiple tables by use of a common field.

Frequently, data from many sources is collected into large archives of data called data warehouses. The process of moving data from its original sources (such as databases) to a centralized location (generally a data warehouse) is called ETL (which stands for extract , transform , and load ).

• The extraction step occurs when you identify and copy or export the desired data from its source, such as by running a database query to retrieve the desired records.
• The transformation step is the process of cleaning the data so that they fit the analytical need for the data and the schema of the data warehouse. This may involve changing formats for certain fields, removing duplicate records, or renaming fields, among other processes.
• Finally, the clean data are loaded into the data warehouse, where they may join vast amounts of historical data and data from other sources.

After data are effectively collected and cleaned, they can be analyzed with a variety of techniques. Analysis often begins with descriptive and exploratory data analysis. Descriptive data analysis uses statistics to organize and summarize data, making it easier to understand the broad qualities of the dataset. Exploratory data analysis looks for insights into the data that may arise from descriptions of distribution, central tendency, or variability for a single data field. Further relationships between data may become apparent by examining two fields together. Visualizations may be employed during analysis, such as histograms (graphs in which the length of a bar indicates a quantity) or stem-and-leaf plots (which divide data into buckets, or “stems,” with individual data points serving as “leaves” on the stem).

Data analysis frequently goes beyond descriptive analysis to predictive analysis, making predictions about the future using predictive modeling techniques. Predictive modeling uses machine learning , regression analysis methods (which mathematically calculate the relationship between an independent variable and a dependent variable), and classification techniques to identify trends and relationships among variables. Predictive analysis may involve data mining , which is the process of discovering interesting or useful patterns in large volumes of information. Data mining often involves cluster analysis , which tries to find natural groupings within data, and anomaly detection , which detects instances in data that are unusual and stand out from other patterns. It may also look for rules within datasets, strong relationships among variables in the data.

• > Statistics

7 Types of Statistical Analysis: Definition and Explanation

• Neelam Tyagi
• Jan 19, 2021
• Updated on: Jul 19, 2022

Statistics is the branch of science that renders various tools and analytical techniques in order to deal with the huge extent of data, in simple terms, it is the science of assembling, classifying, analyzing and interpreting & manifesting the numeric form of data for making inferences about the population, from the picked out sample data that can be used by business experts to solve their problems.

Therefore, in the efforts to organize data and anticipates future trends, depending upon the information, many organizations heavily rely on statistical analysis.

More precisely, statistical data analysis concerns data collection, interpretation and presentation. It can be approached while handling data to solve complex problems. More precisely, the statistical analysis delivers significance to insignificant/irrelevant data or numbers. 

The Key types of Statistical Analysis are

In particular, statistical analysis is the process of consolidating and analyzing distinct samples of data to divulge patterns or trends and anticipating future events/situations to make appropriate decisions. 

The statistical analysis has the following types that considerably depends upon data types. 

7 types of statistical analysis

1. Descriptive Statistical Analysis

Fundamentally, it deals with organizing and summarizing data using numbers and graphs. It makes easy the massive quantities of data for intelligible interpretation even without forming conclusions beyond the analysis or responding to any hypotheses. 

Instead of processing data in its raw form, descriptive statistical analysis enables us to represent and interpret data more efficiently through numerical calculation, graphs or tables.

From all necessary preparatory steps to concluding analysis and interpretation, descriptive statistical analysis involves various processes such as tabulation, a measure of central tendency (mean, median, mode), a measure of dispersion or variance (range, variation, standard deviation), skewness measurements and time-series analysis .

Formula chart, source

Under descriptive analysis, the data is summarized in tabular form and managed & presented in the forms of charts and graphs for summing up data, assuming it for the whole population. 

Also Read |  Descriptive statistics in R

Moreover, it helps in extracting distinct characteristics of data and in summarizing and explaining the essential features of data. What’s more, no insights are drawn regarding the groups which are not observed/sampled.

2. Inferential Statistical Analysis

The inferential statistical analysis basically is used when the inspection of each unit from the population is not achievable, hence, it extrapolates, the information obtained, to the complete population. 

In simple words, inferential statistical analysis lets us test a hypothesis depending on a sample data from which we can extract inferences by applying probabilities and make generalizations about the whole data, and also can make conclusions with respect to future outcomes beyond the data available.

By this way, it is highly preferable while drawing conclusions and making decisions about the whole population on the basis of sample data. As such, this method involves the sampling theory, various tests of significance, statistical control etc.

Descriptive vs Inferential Statistical Analysis 

3. Predictive Analysis

Predictive analysis is implemented to make a prediction of future events, or what is likely to take place next, based on current and past facts and figures. 

In simple terms, predictive analytics uses statistical techniques and machine learning algorithms to describe the possibility of future outcomes, behaviour, and trends depending on recent and previous data. Widely used techniques under predictive analysis include data mining, data modelling, artificial intelligence, machine learning and etc. to make imperative predictions. 

In the current business system, this analysis is approached by marketing companies, insurance organizations, online service providers, data-driven marketing , and financial corporations, however, any business can take advantage of it by planning for an unpredictable future, such as to gain the competitive advantage and narrow down the risk connected with an unpredictable future event.

The predictive analysis converges on forecasting upcoming events using data and ascertaining the likelihood of several trends in data behaviour. Therefore, businesses use this approach to get the answer “what might happen?” where the basis of making predictions is a probability measure.

4. Prescriptive Analysis

The prescriptive analysis examines the data In order to find out what should be done, it is widely used in business analysis for identifying the best possible action for a situation. 

While other statistical analysis might be deployed for driving exclusions, it provides the actual answer. Basically, it focuses on discovering the optimal suggestion for a process of decision making.

Several techniques, implemented under prescriptive analysis are simulation, graph analysis, algorithms, complex event processing, machine learning, recommendation engine , business rules, etc.

However, it is nearly related to descriptive and predictive analysis, where descriptive analysis explains data in terms of what has happened, predictive analysis anticipates what could happen, and here prescriptive analysis deals in providing appropriate suggestions among the available preferences.

5. Exploratory Data Analysis (EDA)

Exploratory data analysis , or EDA as it is known, is a counterpart of inferential statistics, and greatly implemented by data experts. It is generally the first step of the data analysis process that is conducted prior to any other statistical analysis techniques.

EDA is not deployed alone for predicting or generalizing, it renders a preview of data and assists in getting some key insights into it. 

This method fully focuses on analyzing patterns in the data to recognize potential relationships. EDA can be approached for discovering unknown associations within data, inspecting missing data from collected data and obtaining maximum insights, examining assumptions and hypotheses. 

6. Causal Analysis

In general, causal analysis assists in understanding and determining the reasons behind “why” things occur, or why things are as such, as they appear. 

For example, in the present business environment, many ideas, or businesses are there that get failed due to some events’ happening, in that condition, the causal analysis identifies the root cause of failures, or simply the basic reason why something could happen. 

In the IT industry , this is used to check the quality assurance of particular software, like why that software failed, if there was a bug, a data breach, etc, and prevents companies from major setbacks.  

We can consider the causal analysis when;

• Identifying significant problem-areas inside the data,
• Examining and identifying the root causes of the problem, or failure, 
• Understanding what will be happening to a provided variable if one another variable changes.

7. Mechanistic Analysis

Among the above statistical analysis, mechanistic is the least common type, however, it is worthy in the process of big data analytics and biological science. It is deployed to understand and explain how things happen rather than how specific things will take place ulteriorly.

It uses the clear concept of understanding individual changes in variables that cause changes in other variables correspondingly while excluding external influences and considering the assumption that the entire system gets influenced via its own internal elements’ interaction.

The fundamental objectives of mechanistic analysis involve;

• Understanding the definite changes in that could make changes in other variables
• A clear explanation of the happening of a past event in the context of data, especially when the particular subject/concern deals with specific activities.

For example, in biological science, when studying and inspecting how various parts of the virus are affected by making changes in medicine. 

Besides the above statistical analysis types, it is worth discussing here that these statistical treatments, or statistical data analysis techniques , profoundly rely on the way, the data is being used. While counting on the function and requirement of a particular study, data and statistical analysis can be employed for many purposes, for example, medical scientists can use a variety of statistical analysis for testing the drug effectiveness, or potency.

What’s more, plenty of available data can inform numerous things, data professionalists want to explore, therefore statistical analysis is able to get some informative outcomes and make some inferences. Also, in some cases, statistical analysis can be approached to accumulate information regarding the preference of people and their habits. 

For example, user data, at sites like Facebook and Instagram , can be used by analysts for understanding user perception, like what uses are doing and what motivates them. This information can benefit commercial ads where a particular group of users are targeted to sell them things. It is also helpful for the application developers to understand users’ response and habits and make changes in products accordingly.

A deeper understanding of data can widen the numerous opportunities for a business, with the implementation of business analytics , an organization can achieve while scrutinizing data, for driving, for example, predictions, insights, or conclusions from data and this is what statistical analysis can do, for example;

• Compiling and manifesting data in the form of graphs or charts to show key findings,
• Exploring significant elements/ measurements from data, like mean, variance, skewness, etc,
• Testing a hypothesis from multiple experiments,
• Anticipating coming foresight on the basis of past data behaviour, and many more.

And hence, a business can take advantages of statistical analysis in various ways, for example, to determine the down performance of sales, to uncover trends from customer data, conducting financial audits, etc.  

Also Read |  Bayesian Statistics

We have seen, preferably, two main types “ descriptive and inferential statistical analysis ” is to choose while applying statistical analysis to a business problem, however, other types of statistical analysis will become a priority while addressing some other business needs, organizations are looking for, depending on the complete intent or queries.

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An Overview of Statistical Data Analysis

• August 2019

• University of Porto

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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.

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.

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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Understanding statistical analysis: A beginner’s guide to data interpretation

Statistical analysis is a crucial part of research in many fields. It is used to analyze data and draw conclusions about the population being studied. However, statistical analysis can be complex and intimidating for beginners. In this article, we will provide a beginner’s guide to statistical analysis and data interpretation, with the aim of helping researchers understand the basics of statistical methods and their application in research.

What is Statistical Analysis?

Statistical analysis is a collection of methods used to analyze data. These methods are used to summarize data, make predictions, and draw conclusions about the population being studied. Statistical analysis is used in a variety of fields, including medicine, social sciences, economics, and more.

Statistical analysis can be broadly divided into two categories: descriptive statistics and inferential statistics. Descriptive statistics are used to summarize data, while inferential statistics are used to draw conclusions about the population based on a sample of data.

Descriptive Statistics

Descriptive statistics are used to summarize data. This includes measures such as the mean, median, mode, and standard deviation. These measures provide information about the central tendency and variability of the data. For example, the mean provides information about the average value of the data, while the standard deviation provides information about the variability of the data.

Inferential Statistics

Inferential statistics are used to draw conclusions about the population based on a sample of data. This involves making inferences about the population based on the sample data. For example, a researcher might use inferential statistics to test whether there is a significant difference between two groups in a study.

Statistical Analysis Techniques

There are many different statistical analysis techniques that can be used in research. Some of the most common techniques include:

Correlation Analysis: This involves analyzing the relationship between two or more variables.

Regression Analysis: This involves analyzing the relationship between a dependent variable and one or more independent variables.

T-Tests: This is a statistical test used to compare the means of two groups.

Analysis of Variance (ANOVA): This is a statistical test used to compare the means of three or more groups.

Chi-Square Test: This is a statistical test used to determine whether there is a significant association between two categorical variables.

Data Interpretation

Once data has been analyzed, it must be interpreted. This involves making sense of the data and drawing conclusions based on the results of the analysis. Data interpretation is a crucial part of statistical analysis, as it is used to draw conclusions and make recommendations based on the data.

When interpreting data, it is important to consider the context in which the data was collected. This includes factors such as the sample size, the sampling method, and the population being studied. It is also important to consider the limitations of the data and the statistical methods used.

Best Practices for Statistical Analysis

To ensure that statistical analysis is conducted correctly and effectively, there are several best practices that should be followed. These include:

Clearly define the research question : This is the foundation of the study and will guide the analysis.

Choose appropriate statistical methods: Different statistical methods are appropriate for different types of data and research questions.

Use reliable and valid data: The data used for analysis should be reliable and valid. This means that it should accurately represent the population being studied and be collected using appropriate methods.

Ensure that the data is representative: The sample used for analysis should be representative of the population being studied. This helps to ensure that the results of the analysis are applicable to the population.

Follow ethical guidelines : Researchers should follow ethical guidelines when conducting research. This includes obtaining informed consent from participants, protecting their privacy, and ensuring that the study does not cause harm.

Statistical analysis and data interpretation are essential tools for any researcher. Whether you are conducting research in the social sciences, natural sciences, or humanities, understanding statistical methods and interpreting data correctly is crucial to drawing accurate conclusions and making informed decisions. By following the best practices for statistical analysis and data interpretation outlined in this article, you can ensure that your research is based on sound statistical principles and is therefore more credible and reliable. Remember to start with a clear research question, use appropriate statistical methods, and always interpret your data in context. With these guidelines in mind, you can confidently approach statistical analysis and data interpretation and make meaningful contributions to your field of study.

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What is ANOVA Test? Definition, Types, Examples

Appinio Research · 13.08.2024 · 31min read

Have you ever wondered how researchers determine if different groups in a study have significantly different outcomes? Analysis of Variance, commonly known as ANOVA, is a statistical method that helps answer this crucial question. Whether comparing the effectiveness of various treatments, understanding the impact of different teaching methods, or evaluating marketing strategies, ANOVA is a powerful tool that allows us to compare the means of three or more groups to see if at least one is significantly different. This guide will walk you through the fundamental concepts, mathematical foundations, types, and practical applications of ANOVA, ensuring you understand how to design experiments, check assumptions, perform analyses, and interpret results effectively. By the end, you'll see why ANOVA is a cornerstone of statistical analysis across numerous fields, providing a robust framework for making informed decisions based on data.

What is ANOVA?

ANOVA, or Analysis of Variance, is a statistical method used to compare the means of three or more groups to see if at least one of them is significantly different from the others. This technique helps to determine whether observed differences in sample means are due to actual differences in population means or merely the result of random variation.

Purpose of ANOVA

• Comparing Multiple Groups : ANOVA allows you to simultaneously compare the means of three or more independent groups. This is more efficient and informative than conducting multiple t-tests, which increases the risk of Type I error (false positives).
• Identifying Significant Differences : By testing for differences in group means, ANOVA helps to determine whether any of the groups are significantly different from each other. This is essential in experiments where treatments or interventions are compared.
• Partitioning Variance : ANOVA partitions the total variance observed in the data into variance between groups and variance within groups. This helps in understanding the sources of variability in the data.
• Evaluating Interactions : Two-way ANOVA can assess interactions between factors. This means you can see if the effect of one factor depends on the level of another factor.
• Guiding Further Analysis : When ANOVA shows significant differences, it often leads to further analysis, such as post-hoc tests, to identify which specific groups differ from each other.

Importance of ANOVA in Statistical Analysis

ANOVA is a cornerstone of statistical analysis in many fields, including psychology, medicine, agriculture, marketing, and education. Its importance lies in its versatility and robustness in comparing multiple groups and understanding complex data structures.

• Enhanced Accuracy : ANOVA controls for the Type I error rate better than multiple t-tests, providing more reliable results when comparing multiple groups.
• Comprehensive Analysis : It offers a systematic approach to understanding the variability in data by decomposing the total variance into meaningful components.
• Flexibility : ANOVA can handle different experimental designs, including one-way, two-way, and multivariate designs, making it adaptable to various research questions and data structures.
• Insight into Interactions : By assessing interactions between factors, ANOVA provides deeper insights into how different variables jointly affect the outcome.
• Foundation for Advanced Methods : ANOVA forms the basis for more complex statistical methods like MANOVA (Multivariate ANOVA), ANCOVA (Analysis of Covariance), and repeated measures ANOVA, which are essential for analyzing more complex data sets.
• Widespread Application : Its principles are widely applied across diverse disciplines, making it a fundamental tool for researchers and analysts aiming to draw meaningful conclusions from their data.

ANOVA Mathematical Foundations

ANOVA is grounded in several key mathematical concepts. A solid grasp of these foundations will deepen your understanding and enhance your ability to apply ANOVA effectively.

Understanding Variance and its Components

Variance measures how much the data points in a set differ from the mean of the set. It's crucial for ANOVA because the technique relies on partitioning this variance to understand differences between groups.

• Total Variance : The overall variability in the data.
• Between-Group Variance : The variability due to differences between the group means.
• Within-Group Variance : The variability within each group.

To illustrate, imagine you have test scores from three different classes. Total variance includes all score variations, but between-group variance focuses on differences between the classes' average scores, and within-group variance looks at the score spread within each class.

Formulae and Calculations

The core of ANOVA lies in calculating the F-ratio, which compares the variance between groups to the variance within groups.

ANOVA Calculation

• Calculate Group Means : Compute the mean for each group.
• Overall Mean : Calculate the mean of all data points combined. X_total = (ΣX_i) / n
• Sum of Squares Between (SSB) : This measures the variation due to the interaction between the groups. SSB = Σ (n_j * (X_j - X_total)^2) Where n_j is the number of observations in group j, X_j is the mean of group j, and k is the number of groups.
• Sum of Squares Within (SSW) : This measures the variation within each group. SSW = Σ Σ (X_ij - X_j)^2 Where X_ij is the observation i in group j.
• Degrees of Freedom : Calculate the degrees of freedom for between groups (dfB) and within groups (dfW). dfB = k - 1 dfW = N - k Where N is the total number of observations.
• Mean Squares : Compute the mean squares for between groups (MSB) and within groups (MSW). MSB = SSB / dfB MSW = SSW / dfW
• F-Ratio : Finally, calculate the F-ratio. F = MSB / MSW

The F-ratio tells you if the between-group variance is significantly greater than the within-group variance, indicating significant differences among group means.

The F-Distribution

The F-distribution is essential for determining the statistical significance of your ANOVA results. It's a probability distribution that arises frequently when dealing with variances.

• Shape : The F-distribution is right-skewed and varies based on the degrees of freedom for the numerator (between groups) and the denominator (within groups).
• Critical Values : These are determined by the F-distribution and your chosen significance level (usually 0.05). If your calculated F-ratio exceeds the critical value, you reject the null hypothesis, concluding that significant differences exist among the group means.

To use the F-distribution, you typically refer to F-tables or use statistical software, which will provide the p-value associated with your F-ratio. This p-value helps in deciding whether to accept or reject the null hypothesis.

Understanding these mathematical foundations equips you to use ANOVA effectively, ensuring accurate and meaningful statistical analysis.

Types of ANOVA

ANOVA comes in various forms, each suited for different experimental designs and research questions. Understanding these types will help you choose the proper method for your analysis.

One-Way ANOVA

One-Way ANOVA is the simplest form of ANOVA, used when comparing the means of three or more independent groups based on one factor. It's advantageous when assessing whether there are any statistically significant differences between the means of independent (unrelated) groups.

One-Way ANOVA is used when there is a single independent variable with multiple levels and one dependent variable. For example, you might want to compare the test scores of students taught using three different teaching methods.

Example Scenarios

Suppose you are investigating the effect of different fertilizers on plant growth. You have three types of fertilizers (A, B, and C), and you measure the growth of plants using each type.

One-Way ANOVA Calculation

• Calculate Group Means : Compute the mean for each fertilizer type.
• Sum of Squares Between (SSB) : This measures the variation due to the interaction between the groups. SSB = Σ (n_j * (X_j - X_total)^2)
• Sum of Squares Within (SSW) : This measures the variation within each group. SSW = Σ Σ (X_ij - X_j)^2
• F-Ratio : Calculate the F-ratio to determine if the variance between group means is significantly larger than the variance within the groups. F = MSB / MSW

Two-Way ANOVA

Two-Way ANOVA extends the one-way ANOVA by incorporating two independent variables. This method allows you to examine the interaction between these variables and their individual effects on the dependent variable.

Two-Way ANOVA is used when you have two independent variables. For example, you might want to examine the effects of different diets and exercise regimes on weight loss.

Interaction Effects

Interaction effects occur when the impact of one independent variable on the dependent variable depends on the level of the other independent variable. Understanding these interactions can provide deeper insights into the data.

Two-Way ANOVA Calculation

• Calculate Group Means : Compute the mean for each combination of levels of the two factors.
• Sum of Squares : Compute the sum of squares for each main effect and the interaction effect. SS_A = Σ n_ij * (X_A - X_total)^2 SS_B = Σ n_ij * (X_B - X_total)^2 SS_AB = Σ n_ij * (X_AB - X_total)^2
• Mean Squares : Compute the mean squares for each source of variation. MS_A = SS_A / dfA MS_B = SS_B / dfB MS_AB = SS_AB / dfAB
• F-Ratios : Calculate the F-ratios for each main effect and the interaction effect. F_A = MS_A / MSW F_B = MS_B / MSW F_AB = MS_AB / MSW
• Interpretation : Determine if the F-ratios are significant to understand the main and interaction effects.

MANOVA (Multivariate ANOVA)

MANOVA extends ANOVA by analyzing multiple dependent variables simultaneously. This method is useful when you need to understand the effect of independent variables on several outcomes.

MANOVA is used when you have more than one dependent variable. For example, you might want to study the impact of a training program on both employee performance and job satisfaction.

While ANOVA examines one dependent variable at a time, MANOVA assesses multiple dependent variables, accounting for their correlations and providing a more comprehensive analysis.

MANOVA Calculation

• Calculate Mean Vectors : Compute the mean vector for each group. X_mean_vector = (X1_mean, X2_mean, ..., Xk_mean)
• Covariance Matrices : Calculate the within-group and between-group covariance matrices. W = Σ (X_i - X_mean) * (X_i - X_mean)^T B = Σ n_j * (X_j - X_total)^2
• Multivariate Test Statistics : To evaluate the multivariate significance, use test statistics like Wilks' Lambda, Pillai's Trace, or Hotelling's Trace. Wilks' Lambda = |W| / |W + B|
• Significance Testing : Compare the test statistics to critical values from the multivariate F-distribution to determine significance.

Understanding these types of ANOVA and their applications will help you design better experiments and analyze data more effectively, providing deeper insights and more accurate conclusions.

ANOVA Assumptions and Preconditions

To ensure the validity of your ANOVA results, it's essential to understand and meet certain assumptions. These assumptions underpin the accuracy and reliability of the analysis.

Normality refers to the assumption that the data within each group follows a normal distribution. This assumption is crucial because ANOVA relies on the mean and variance of the data, and normality ensures that these statistics are reliable.

When the data are normally distributed, the statistical tests used in ANOVA are more accurate. This assumption is particularly important for smaller sample sizes, where deviations from normality can significantly impact the results.

Testing for Normality

Several methods can help you assess normality:

• Q-Q Plots : These plots compare your data's quantiles against a theoretical normal distribution. If the data points fall approximately along a straight line, the data are likely normal.
• Shapiro-Wilk Test : This statistical test checks for normality. A non-significant result (p > 0.05) suggests that the data do not significantly deviate from normality.
• Kolmogorov-Smirnov Test : Another test for normality, comparing the sample distribution with a normal distribution.

If your data deviates from normality, consider transforming the data (e.g., log transformation) or using non-parametric alternatives like the Kruskal-Wallis test.

Homogeneity of Variances

Homogeneity of variances, or homoscedasticity, means that the variances within each group are approximately equal. This assumption ensures that the comparison of means across groups is fair and accurate.

When variances are equal, the pooled estimate of the variance used in ANOVA calculations is accurate. Unequal variances can lead to biased results and incorrect conclusions.

Testing for Homogeneity

Several tests can check for homogeneity of variances:

• Levene's Test : This test assesses whether variances are equal across groups. A non-significant result (p > 0.05) indicates equal variances.
• Bartlett's Test : Another test for equal variances, more sensitive to departures from normality.
• Hartley's F-max Test : This test compares the largest and smallest variances among groups.

If variances are unequal, consider using a different version of ANOVA, such as Welch's ANOVA, which is more robust to heteroscedasticity.

Independence of Observations

Independence means that the observations within each group are not related to each other. This assumption ensures that the variance within groups reflects true individual differences rather than patterns or correlations.

Ensuring Independence

Independence is typically ensured through the study design:

• Randomization : Randomly assign subjects to groups to prevent bias.
• Blinding : Use single or double-blind designs to reduce biases that could correlate observations.
• Proper Sampling Techniques : Ensure that samples are drawn independently and represent the population accurately.

Violations of independence can severely affect ANOVA results, making them unreliable. If observations are not independent, consider using techniques like mixed-effects models that account for the lack of independence.

Checking Assumptions with Tests and Diagnostics

Before conducting ANOVA, verifying all assumptions are met is crucial. Here are some practical steps and tools:

Visual Diagnostics

• Box Plots : These plots can help you visualize the spread and identify potential outliers.
• Histograms : Assess the distribution of data within each group.
• Scatter Plots : Check for patterns that might suggest violations of independence.

Statistical Tests

• Shapiro-Wilk Test : Assess normality.
• Levene's Test : Check homogeneity of variances.
• Durbin-Watson Statistic : Evaluate independence in time-series data.

Software Tools

Statistical software packages (like SPSS, R, and Python libraries) offer built-in functions to perform these tests and generate diagnostic plots. For example:

• R : Functions like shapiro.test(), leveneTest(), and durbinWatsonTest() are available in various packages.
• SPSS : Offers point-and-click options to perform these tests.
• Python : Libraries such as scipy.stats and statsmodels provide functions for these diagnostics.

Ensuring that these assumptions are met is critical for the validity of ANOVA results. By rigorously checking assumptions, you can trust that your analysis is both accurate and reliable.

How to Conduct an ANOVA Test?

Conducting ANOVA involves several well-defined steps, from designing your experiment to interpreting the results. Here's a detailed guide to help you navigate this process effectively.

1. Designing an Experiment

A well-designed experiment is the foundation of a successful ANOVA. Start by clearly defining your research question and identifying your independent and dependent variables. Determine the number of levels for your independent variable(s) and ensure you have a sufficient sample size to detect meaningful differences.

Randomization is crucial to eliminate bias and ensure that the groups are comparable. Consider assigning subjects to different treatment groups using random assignment. If possible, incorporate blinding methods to reduce any potential influence of expectations on the outcomes.

2. Data Collection and Preparation

Accurate data collection is vital. Ensure that your measurement tools are reliable and valid. Collect data systematically and consistently across all groups. Be diligent in recording your data to prevent any errors.

Once data collection is complete, prepare your data for analysis. This includes checking for missing values, outliers, and errors. Clean your data to ensure it is ready for ANOVA. Coding your variables appropriately is also essential; for instance, assigning numerical values to categorical variables can streamline the analysis process.

3. Performing ANOVA with Statistical Software

Several statistical software packages, including SPSS, R, and Python, can perform ANOVA. Here's a brief overview of how to conduct ANOVA using these tools:

• Enter Data : Input your data into the SPSS data editor.
• Define Variables : Go to the Variable View and define your independent and dependent variables.
• Run ANOVA : Navigate to Analyze > Compare Means > One-Way ANOVA for a one-way ANOVA, or Analyze > General Linear Model > Univariate for more complex designs.
• Set Options : Choose the appropriate options for post-hoc tests and effect size if needed.
• Interpret Output : Review the ANOVA table for the F-value and p-value to determine significance.
• Load Data : Import your dataset using functions like read.csv() or read.table().
• Fit ANOVA Model : Use the aov() function for a one-way ANOVA or anova() for more complex designs. Example: model <- aov(dependent_variable ~ independent_variable, data = dataset).
• Summary : Generate a summary of the model using summary(model).
• Post-Hoc Tests : Conduct post-hoc tests using functions like TukeyHSD(model).
• Load Libraries : Import necessary libraries like pandas, scipy.stats, and statsmodels.
• Load Data : Use pandas to read your data, e.g., data = pd.read_csv('yourfile.csv').
• Fit ANOVA Model : Use stats.f_oneway() for a one-way ANOVA or ols() from statsmodels for more complex designs.
• Interpret Results : Examine the output for F-values and p-values.

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4. Interpreting Results

After performing ANOVA, interpreting the results correctly is crucial:

• F-Statistic : This value indicates the ratio of between-group variance to within-group variance. A higher F-value suggests a more significant difference between groups.
• P-Value : This value helps determine the statistical significance. A p-value less than 0.05 typically indicates significant differences between groups.
• Post-Hoc Tests : If your ANOVA results are significant, post-hoc tests (like Tukey's HSD) can identify which specific groups differ from each other.
• Effect Size : Consider calculating the effect size to understand the magnitude of the differences, not just their significance. Standard measures include eta-squared and Cohen's d.

5. Reporting Results

Presenting your ANOVA results clearly and accurately is essential. Include the following in your report:

• Descriptive Statistics : Mean and standard deviation for each group.
• ANOVA Table : F-value, degrees of freedom, and p-value.
• Post-Hoc Test Results : Detailed results of any post-hoc analyses.
• Interpretation : A clear interpretation of the findings, including any practical or theoretical implications.

By following these steps, you can confidently conduct ANOVA, ensuring that your results are accurate and meaningful. Whether you're using SPSS, R, or Python, the principles remain the same: a robust design, meticulous data preparation, and thorough analysis and interpretation.

ANOVA vs. Other Statistical Tests

Choosing a suitable statistical test is crucial for accurate analysis and meaningful results. ANOVA is a powerful tool, but understanding how it compares to other statistical tests will help you make the best choice for your data.

ANOVA vs. T-Test

The t-test is another widely used statistical test, primarily for comparing the means of two groups. Here's how ANOVA and t-test differ and when to use each:

Number of Groups

• T-Test : Ideal for comparing the means of two groups. There are two types of t-tests: independent (for two separate groups) and paired (for two related groups).
• ANOVA : Designed to compare the means of three or more groups. It can handle more complex experimental designs with multiple groups and factors.

Consider a study comparing the effects of two diets on weight loss. A t-test is suitable here since there are only two groups. However, if you introduce a third diet, ANOVA becomes the appropriate choice.

Assumptions

Both tests share similar assumptions, including normality, homogeneity of variances, and independence. If these assumptions are violated, consider using non-parametric alternatives like the Mann-Whitney U test for the t-test and the Kruskal-Wallis test for ANOVA.

ANOVA vs. Regression Analysis

Regression analysis explores the relationship between dependent and independent variables. It's versatile and can handle various types of data and relationships. Here's a comparison:

• ANOVA : Primarily focuses on comparing means across different groups and determining if those means are significantly different.
• Regression : Examines the relationship between dependent and independent variables, predicting the dependent variable based on one or more predictors.
• ANOVA : Easier to use for simple comparisons and experimental designs with categorical independent variables.
• Regression : More flexible and can include both categorical and continuous variables, interaction effects, and polynomial terms to model complex relationships.

Suppose you're studying the impact of education level and work experience on salary. Regression analysis allows you to include both factors and their interaction, providing a detailed model of how they influence salary.

ANOVA vs. Chi-Square Test

The chi-square test is used for categorical data to assess the association between variables. Here's how it compares to ANOVA:

• ANOVA : Used for continuous data where you're interested in comparing group means.
• Chi-Square Test : Suitable for categorical data, where you're examining the relationship or independence between categorical variables.

If you want to compare the average scores of students in different schools, ANOVA is appropriate. However, if you're interested in whether the distribution of students' preferred study methods (e.g., online, in-person, hybrid) differs by school, the chi-square test is the right choice.

• ANOVA : Assumes normality, homogeneity of variances, and independence.
• Chi-Square Test : Assumes a sufficiently large sample size and that the data are categorical.

ANOVA vs. MANOVA

MANOVA (Multivariate ANOVA) is an extension of ANOVA that handles multiple dependent variables. Here's the distinction:

Number of Dependent Variables

• ANOVA : Used when there is one dependent variable.
• MANOVA : Suitable for analyzing multiple dependent variables simultaneously, considering the correlation between them.

If you're evaluating the effect of a training program on employee performance, ANOVA is suitable for a single performance metric. However, if you want to assess performance, job satisfaction, and retention simultaneously, MANOVA provides a more comprehensive analysis.

Practical Considerations

When deciding between these tests, consider the following:

• Research Question : Clearly define what you're trying to discover. Are you comparing means, exploring relationships, or examining associations?
• Data Type : Ensure your data matches the requirements of the test (continuous vs. categorical).
• Assumptions : Check if your data meet the assumptions of the test. If not, look for robust or non-parametric alternatives.
• Complexity : Choose a test that matches your statistical knowledge and the complexity of your data.

Understanding the differences between ANOVA and other statistical tests allows you to choose the most appropriate method for your analysis. This ensures accurate, reliable, and meaningful results, ultimately leading to better-informed decisions and insights.

Conclusion for ANOVA

ANOVA is an essential tool in the statistician's toolkit, providing a robust method for comparing multiple groups and understanding the variability within data. By partitioning variance into meaningful components, ANOVA helps us determine whether observed differences in group means are statistically significant or merely the result of random chance. This guide has explored the foundational concepts, mathematical underpinnings, various types of ANOVA, and the importance of meeting assumptions for accurate results. Whether you're using one-way, two-way, or multivariate ANOVA, the principles remain the same: a rigorous approach to analyzing data and drawing reliable conclusions. Understanding ANOVA's application in real-world scenarios, from clinical trials to market research, underscores its versatility and importance. Mastering ANOVA allows you to design better experiments, make more informed decisions, and contribute valuable insights to your field. This guide aims to demystify ANOVA and equip you with the knowledge and tools needed to apply this technique confidently. As you continue to work with data, remember that ANOVA is not just a statistical test but a gateway to deeper insights and more effective strategies based on empirical evidence.

How to Do an ANOVA Test in Minutes?

Appinio revolutionizes the way businesses conduct ANOVA by offering a real-time market research platform that makes gathering consumer insights quick, intuitive, and exciting. With Appinio, companies can effortlessly perform ANOVA to compare multiple groups and derive actionable insights without the hassle of lengthy, complicated, expensive research processes.

The platform's user-friendly interface and rapid data collection capabilities ensure that anyone, regardless of their research background, can conduct sophisticated statistical analyses like ANOVA and make data-driven decisions in minutes.

• Rapid Insights : From formulating questions to obtaining comprehensive insights, Appinio delivers results in under 23 minutes for up to 1,000 respondents, ensuring you have the data you need swiftly to make timely business decisions.
• Intuitive Platform : Designed for ease of use, Appinio's platform allows anyone to conduct thorough market research and ANOVA without needing advanced research expertise or a PhD. The intuitive interface guides users through the process seamlessly.
• Global Reach and Precision : With the ability to define target groups from over 1,200 characteristics and survey respondents in more than 90 countries, Appinio ensures your research is both precise and globally representative, providing the breadth and depth of insights required to inform strategic decisions.

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• Knowledge Base
• Choosing the Right Statistical Test | Types & Examples

Choosing the Right Statistical Test | Types & Examples

Published on January 28, 2020 by Rebecca Bevans . Revised on June 22, 2023.

Statistical tests are used in hypothesis testing . They can be used to:

• determine whether a predictor variable has a statistically significant relationship with an outcome variable.
• estimate the difference between two or more groups.

Statistical tests assume a null hypothesis of no relationship or no difference between groups. Then they determine whether the observed data fall outside of the range of values predicted by the null hypothesis.

If you already know what types of variables you’re dealing with, you can use the flowchart to choose the right statistical test for your data.

Statistical tests flowchart

Table of contents

What does a statistical test do, when to perform a statistical test, choosing a parametric test: regression, comparison, or correlation, choosing a nonparametric test, flowchart: choosing a statistical test, other interesting articles, frequently asked questions about statistical tests.

Statistical tests work by calculating a test statistic – a number that describes how much the relationship between variables in your test differs from the null hypothesis of no relationship.

It then calculates a p value (probability value). The p -value estimates how likely it is that you would see the difference described by the test statistic if the null hypothesis of no relationship were true.

If the value of the test statistic is more extreme than the statistic calculated from the null hypothesis, then you can infer a statistically significant relationship between the predictor and outcome variables.

If the value of the test statistic is less extreme than the one calculated from the null hypothesis, then you can infer no statistically significant relationship between the predictor and outcome variables.

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You can perform statistical tests on data that have been collected in a statistically valid manner – either through an experiment , or through observations made using probability sampling methods .

For a statistical test to be valid , your sample size needs to be large enough to approximate the true distribution of the population being studied.

To determine which statistical test to use, you need to know:

• whether your data meets certain assumptions.
• the types of variables that you’re dealing with.

Statistical assumptions

Statistical tests make some common assumptions about the data they are testing:

• Independence of observations (a.k.a. no autocorrelation): The observations/variables you include in your test are not related (for example, multiple measurements of a single test subject are not independent, while measurements of multiple different test subjects are independent).
• Homogeneity of variance : the variance within each group being compared is similar among all groups. If one group has much more variation than others, it will limit the test’s effectiveness.
• Normality of data : the data follows a normal distribution (a.k.a. a bell curve). This assumption applies only to quantitative data .

If your data do not meet the assumptions of normality or homogeneity of variance, you may be able to perform a nonparametric statistical test , which allows you to make comparisons without any assumptions about the data distribution.

If your data do not meet the assumption of independence of observations, you may be able to use a test that accounts for structure in your data (repeated-measures tests or tests that include blocking variables).

Types of variables

The types of variables you have usually determine what type of statistical test you can use.

Quantitative variables represent amounts of things (e.g. the number of trees in a forest). Types of quantitative variables include:

• Continuous (aka ratio variables): represent measures and can usually be divided into units smaller than one (e.g. 0.75 grams).
• Discrete (aka integer variables): represent counts and usually can’t be divided into units smaller than one (e.g. 1 tree).

Categorical variables represent groupings of things (e.g. the different tree species in a forest). Types of categorical variables include:

• Ordinal : represent data with an order (e.g. rankings).
• Nominal : represent group names (e.g. brands or species names).
• Binary : represent data with a yes/no or 1/0 outcome (e.g. win or lose).

Choose the test that fits the types of predictor and outcome variables you have collected (if you are doing an experiment , these are the independent and dependent variables ). Consult the tables below to see which test best matches your variables.

Parametric tests usually have stricter requirements than nonparametric tests, and are able to make stronger inferences from the data. They can only be conducted with data that adheres to the common assumptions of statistical tests.

The most common types of parametric test include regression tests, comparison tests, and correlation tests.

Regression tests

Regression tests look for cause-and-effect relationships . They can be used to estimate the effect of one or more continuous variables on another variable.

Comparison tests

Comparison tests look for differences among group means . They can be used to test the effect of a categorical variable on the mean value of some other characteristic.

T-tests are used when comparing the means of precisely two groups (e.g., the average heights of men and women). ANOVA and MANOVA tests are used when comparing the means of more than two groups (e.g., the average heights of children, teenagers, and adults).

Correlation tests

Correlation tests check whether variables are related without hypothesizing a cause-and-effect relationship.

These can be used to test whether two variables you want to use in (for example) a multiple regression test are autocorrelated.

Non-parametric tests don’t make as many assumptions about the data, and are useful when one or more of the common statistical assumptions are violated. However, the inferences they make aren’t as strong as with parametric tests.

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This flowchart helps you choose among parametric tests. For nonparametric alternatives, check the table above.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient
• Null hypothesis

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Statistical tests commonly assume that:

• the data are normally distributed
• the groups that are being compared have similar variance
• the data are independent

If your data does not meet these assumptions you might still be able to use a nonparametric statistical test , which have fewer requirements but also make weaker inferences.

A test statistic is a number calculated by a  statistical test . It describes how far your observed data is from the  null hypothesis  of no relationship between  variables or no difference among sample groups.

The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis . Different test statistics are used in different statistical tests.

Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.

Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .

When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.

Quantitative variables are any variables where the data represent amounts (e.g. height, weight, or age).

Categorical variables are any variables where the data represent groups. This includes rankings (e.g. finishing places in a race), classifications (e.g. brands of cereal), and binary outcomes (e.g. coin flips).

You need to know what type of variables you are working with to choose the right statistical test for your data and interpret your results .

Discrete and continuous variables are two types of quantitative variables :

• Discrete variables represent counts (e.g. the number of objects in a collection).
• Continuous variables represent measurable amounts (e.g. water volume or weight).

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