Chi Square Test: Introduction, How to calculate, When to use

In statistics, the chi-square test is used to analyse data from observations of a normally distributed collection of variables. Typically, this involves contrasting two sets of numerical information. Karl Pearson first proposed this method of analysing and distributing categorical data, naming it Pearson’s chi-square test.

The chi-square test developed by Pearson is used in a contingency table to evaluate whether there is a significant statistical difference between the predicted and actual frequencies in one or more of the categories of the chi-square table.

Statistically, statisticians use the chi-square test to determine how well a model fits the data. Chi-square statistics need a random, mutually exclusive, raw, independent variable data sample of sufficient size.

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Chi-square test basic terminologies

The standard formula for calculating a chi-square test is the summation of square mistakes or false positives divided by the sample variance. There are a few terms that are implemented when using the Chi-square test. These terms have been defined below:


The p-value is the likelihood of achieving a chi-square that is equal to or greater than that in the present experiment, and the data still supports the hypothesis. This probability is expressed as a percentage. It refers to the likelihood that anticipated variations are caused by nothing more than random occurrences.
If the p-value is less than equal to 0.05, then the hypothesis taken into consideration is accepted. If the value is more than 0.05, then the hypothesis is rejected.

Degree of Freedom 

An estimation problem has a certain degree of freedom equal to the number of independent variables. Although there are no hard limits on the values of these variables, they do impose limits on other variables if we want our data set to be consistent with the estimated parameters.

One definition of “degree of freedom” is the greatest number of values in the data set that are logically independent of one another and hence subject to change. Deducting one from the total number of observations in a data set yields the degree of freedom.

One prominent context in which the concept of degree of freedom is addressed is in the context of statistical hypothesis tests like the chi-square.

Understanding the significance of a chi-square statistic and the robustness of the null hypothesis relies heavily on accurately calculating the degree of freedom.


The variance of a random number sample is a measure of its dispersion around its mean. It is calculated by squaring the value of the standard deviation.

Properties to perform the Chi-square Test

The Chi-square test has the following properties:

  • Mean distribution equals the number of degrees of freedom.
  • The variance should be equal to twice the degree of freedom.
  • As the degree of freedom grows, the chi-square distribution curve begins to resemble the normal distribution curve, i.e. a bell curve.

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How to perform the Chi-square Test?

The Chi-square for distribution is calculated using the formula below:

2= [(Observed value – Expected Value)2/ Expected Value]

Steps to follow to calculate the Chi-square statistic

  1. Calculate the observed and the expected value.
  2. Subtract each of the expected values from the observed value in the distribution table.
  3. Square the value for each observation you get in Step 2.
  4. Divide each of these square values by its corresponding expected values.
  5. Adding up all the values that we get in Step 4 gives a value that defines the chi-square statistic.
  6. Calculate the degree of freedom to check for the aforementioned property satisfaction of chi-square tests.

Types of Chi-Square Test

Goodness of Fit

If you want to see how well a sample of the population represents the whole, you may apply the Chi-square goodness-of-fit test. The sample population and the projected sample population are compared using this technique.

Test for Independence

This Chi-square test for independence one population to determine whether there is a correlation among two categorical variables. The independent test differs from the goodness-of-fit test as it does not compare a single observed parameter to a theoretical population. Instead, the test for independence compares two values within a sample set to each other.

Test for Homogeneity

As with the independence test, the test for homogeneity follows the same format and procedure. The critical distinction between the two is that the test for homogeneity examines if a variable has the same distribution across many populations. In contrast, the test for independence examines the presence of a link between two categorical variables within a similar population. 

When should you use a Chi-square test?

The Chi-Square Test determines whether actual values are consistent with theoretical probabilities. Chi-Square is the most reliable test to use when the data being analyzed comes from a random sample and the variable in issue is categorical.

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Where is the Chi-square test used?

Let us take the example of a marketing company.
A marketing company is looking at the correlation between consumer geography and brand choices. Consequently, chi-square plays a significant role, and the value of the statistic will inform how the corporation can adapt its marketing approach across geographies in order to maximise revenues.
When analysing data, the Chi-square test comes in handy for checking the consistency or independence of categorical variables, as well as the goodness-of-fit model under consideration.

Similarly, the chi-square statistic may find use in the medical profession. The chi-square test is suitable for determining the efficacy of a medicine in comparison to a control group.

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In this article, you learned about Chi-square statistics and how to calculate its values. Since Chi-square works with categorical variables, it is often employed by academics investigating survey response data. This form of study is common in many fields, including sociology, psychology, economics, political science, and marketing.

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How is the p-value related to the Chi-square test?

The p-value is the region under the chi-square density curve that is to the right of the test statistic's value. Whether or not the chi-square test statistic is sufficiently big to reject the null hypothesis is the last step in the chi-square test of significance. The p-value is used for this purpose.

Are there any limitations or drawbacks to using the Chi-square test?

All individuals being studied must be unique; else, the results would be meaningless. A chi-square test should not be used if a given respondent may be classified into two distinct groups. Yet another restriction of chi-square is that it can only be used for frequency data. Additionally, the sum of all predicted persons across all classes should be larger than 5.

What are the strengths of the Chi-square test?

One of its primary strengths is that chi-square can be calculated quickly and easily. Nominal data may also be utilised using this method. It may also be used to compare more than two groups of categorical variables for statistical significance.

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