A chi-square test, also written as a χ2 test, is a statistical hypothesis test valid for performing when the chi-square test statistics are distributed under the null hypothesis, specifically the chi-square test of Pearson and its variants. In this article, we will learn about what is chi-square test, uses of chi-square test, application of chi-square test, chi-square test definition, when to use chi-square test, limitations of chi-square test, and chi-square test formula.

In this article let us look at:

  1. What is the Chi-square Test?
  2. Uses of the Chi-square Test
  3. Assumptions of the Chi-square Test
  4. Advantages and Limitations of the Chi-square Test
  5. Chi-square Test in R
  6. Types of the Chi-square Test

1. What is the Chi-square Test?

Chi-square test definition: A chi-square (χ2) statistic is a test that tests the contrast of a model with real data observed. Data used to measure a chi-square test statistic must be random, raw, mutually exclusive, derived from independent variables, and taken from a sufficiently large sample. The outcomes of flipping a fair coin, for instance, follow these conditions.

In hypothesis testing, the chi-square test is also used. Given the size of the sample and the number of variables in the relationship, the chi-square statistics compare the size of any differences between the predicted results and the actual results. For these tests, degrees of freedom are used to determine if a certain null hypothesis can be discounted, depending on the total number of variables and samples within the experiment. As for other data, the more specific the findings are the larger the sample size.

The Chi-Square test is a statistical method that researchers use to analyze the variations in the same population between categorical variables.

chi-square test example: assume that a research group is interested in whether or not the level of education and marital status are connected to all individuals in the U.S. The researchers were first able to manually observe the frequency distribution of marital status and education categories within their sample after gathering a simple random sample of 500 U.S. people and conducting a survey to this sample. The researchers could then conduct a Chi-Square test for these observed frequencies to verify or provide additional background.

2. Uses of the Chi-square Test

Here are some of the uses of the chi-square test in different fields and works:

  • When they find themselves in one of the following conditions, market analysts use the Chi-Square test:
  • They need to estimate how exactly a distribution observed corresponds to a predicted distribution. This is referred to as a measure for ‘goodness-of-fit.’
  • They need to estimate whether there are two independent random variables.
  • When analyzing the cross-tabulations of survey response results, the Chi-Square test is most helpful. Since cross-tabulations show the frequency and percentage of responses by different segments or categories of respondents to questions (gender, occupation, level of education, etc the Chi-Square test tells researchers whether or not there is a statistically significant difference in how a given question was answered by the different segments or categories.

3. Assumptions of the Chi-square Test

The assumptions of the chi-square test are:

  • The variables’ levels (or categories) are mutually exclusive. That is, a specific topic falls into one level of each of the variables, and only one level.
  • Data may be contributed by each subject to one and only one cell in the χ2. If for instance, the same subjects are tested over time in such a way that at Time 1, Time 2, Time 3, etc the comparisons are of the same subjects, then χ2 can not be used.
  • Study groups must be autonomous. This implies that if the two groups are connected, a separate test must be used. For example, if the researcher’s data consists of paired samples, such as in studies in which a parent is paired with his or her infant, a different test must be used.
  • There are 2 variables, and both, typically at the nominal level, are calculated as groups. The data, however, could be ordinary data. It is also possible to use interval or ratio data that has collapsed into ordinal categories. While Chi-square does not have the rule to restrict the number of cells (by restricting the number of categories for each variable), it can be difficult for a very large number of cells (over 20) to satisfy the below assumption #6 and to interpret the meaning of the results.
  • In at least 80% of cells, the expected cell value should be 5 or more and no cell should have an expected cell value of less than one (3). This assumption is more likely to be fulfilled if at least the number of cells multiplied by 5 equals the sample size. This statement effectively defines the number of cases (sample size) required for χ2 to be used for any number of cells in χ2.

4. Advantages and Limitations of the Chi-square Test

Advantages of the Chi-square test include its robustness in terms of data distribution, its ease of calculation, the extensive knowledge that can be obtained from the test, its use in studies for which parametric assumptions cannot be met, and its versatility in managing data from two or more group studies. Limitations of the chi-square test include the sample size criteria, the complexity of analysis when the independent or dependent variables contain large numbers of categories (20 or more and the propensity of Cramer’s V to generate relatively low correlation measurements, except for highly significant results.

5. Chi-square Test in R

A statistical approach used to assess whether two categorical variables have a meaningful association between them is the Chi-Square test in R. The two variables from the same population are chosen. Also, these considerations are then graded as Male/Female, Red/Green, Yes/No, etc.

For instance:

With observations on the cake buying pattern of individuals, we can create a dataset. And, try to compare a person’s gender with the cake flavor they want. However, if a connection is found, by knowing the number of people visiting concerning gender, we can prepare for a suitable stock of flavors.

Syntax of a test of chi-square:


6. Types of the Chi-square Test

Two types of chi-square tests exist. For various purposes, they both use chi-square statistics and distribution:

  • Chi-square goodness of fit test decides whether a population fits sample data. See Goodness of Fit Test for more information concerning this kind.
  • In a contingency table, a chi-square test for independence compares two variables to see if they are related. It checks to see if distributions of categorical variables vary from each other in a more general sense.
    • A very small statistic of the chi-square test indicates that your observed data matches extremely well with your expected knowledge. In other terms, a partnership exists.
    • A very broad statistic of the chi-square test suggests that knowledge does not suit very well. There isn’t in other words, a friendship.


One way to illustrate a relationship between two categorical variables is a chi-square statistic. There are two kinds of variables in statistics: numerical (countable) variables and (categorical) non-numerical variables. A chi-squared statistic is a single number that tells you how much variation there is between the counts you have observed and the counts you would predict if the population had no relationship at all.

The chi-square statistic has a few variants. Which one you use depends on how the knowledge is gathered and which theory is evaluated. All the variants, however, use the same principle, which is that you equate the estimated values with the values that you currently obtain.

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