Chi-Square Statistic

What is the Chi-Square Statistic?

Quick Links

  1. What is the Chi-square statistic
  2. Types of Chi-Square Tests
  3. Steps in the Chi-Square test for independence
  4. Assumptions in the Chi-Square Statistic
  5. How to perform a Chi-square test
  6. Running the Chi-square test with SPSS
  7. Interpretation of the Output
  8. Limitations of the Chi-square Statistic
  9. Statistics in APA
  10. Professional Services for your Statistical Queries
  11. Chi-Square test Questions and Answers

The chi-square (χ2) statistic is commonly applied when testing the relationship between categorized variables. It is also called Pearson’s Chi-square test. It was first investigated in 1900 by Karl Pearson.

The purpose of a chi test is to measure and evaluate the association between categorized variables.

Definition of Chi-square statistic

Definition of Chi-square statistic

Chi-Square Test

Chi-Square Test









To calculate a chi-square statistic:

Raw and mutually exclusive data is randomly drawn from a large sample of independent variables.

For example, tossing a coin more than one hundred times represents a chi-square (χ2) statistic because the null hypothesis of the chi-square test is that the coin has equal chances of landing on the tail or head every time it is tossed. Therefore, a person will get 50 tails and 50 heads.

Hence a chi-square (χ2) statistic allows the researcher to input random variables which in turn yield both categorized and numerical data.

A chi-square (X2) statistic is a procedure for investigating whether there is an association between two different categorized variables.

Categorical variables show data in the groupings while numerical variables present data in numbers. Chi-square (χ2) statistic calculation helps to answer questions such as what is your gender and do you have a bank account. These questions can be analyzed and results presented in either discrete or continuous model.

As discussed above, the Chi-Square statistic test is carried out if two categorized variables in a given population are related in any way.  It includes the null hypothesis which mostly indicates that there is no relationship between the categorized variables in the population and each variable is independent of the other.  A chi-square statistic model can answer the following question: Is there a significant association between political transparency and economic development in your country?

the image shows the logic in Chi-square statistic

the logic in Chi-square statistic

The discussion above highlights that a chi-square (χ2) statistic can be used to assess whether there is a significant association between the expected and observed data.

How to perform a chi-square test

The main application of the chi-square statistic is in the Test of Independence analysis. In this test, the cross-tabulation (contingency table) is used to interpret data. The cross-tabulation is a table that simultaneously presents the distributions of two grouped variables. One category appears in the rows while the other category is presented in the columns. Additionally, each cell in the table shows the connection present in the rows and columns.

To determine whether there is a significant relationship between the two variables, an individual compares the expected and observed results provided the one variable had no effect on the other variable.  The Chi-Square statistic is then calculated using the formula below:

Formula for finding the Chi-Square Value

The formula for finding the Chi-Square Value


The calculated value is then compared with the critical value in the Chi-Square distribution tables thus allowing the investigator to conclude on the relationship that exists between the variables; expected and observed data. So the Chi-Square statistic represents the variance between the actual observed data and the expected data based on the assumption that there is no existent relationship between the two variables.

Types of chi-square tests are:

  1. The test of independence for data

The test of independence for data evaluates the relationship between categorical variables by comparing the observed results to the expected data using a contingency table. In a nutshell, the test of independence assesses whether unpaired observations on two variables are independent of each other.

Steps in the Chi-Square test for independence:

Draw the Contingency table

Draw the Contingency table










calculate totals

calculate totals











Find the Expected Frequencies

Find the Expected Frequencies











Formula for finding the Chi-Square Value

Formula for finding the Chi-Square Value











the Chi-Square table

the Chi-Square table

2. The test of Homogeneity

The test of Homogeneity compares how counts for two or more groups are distributed  using a similar categorical variable

     3. Tests of goodness of fit for a model.

The tests for goodness of fit, on the other hand, assesses whether the observed results differ significantly from the expected results. Both tests can be used to support or reject a given null hypothesis.

Therefore, the test tells us whether a categorical variable is able to fit the theoretical distribution or not. We are able to evaluate if the frequencies can distribute as per the random variation or obtain more meaningful results.

Other Chi-Square tests include:

  • Test for Trend
  • Cochran-Mantel-Haenszel Test

Assumptions in the Chi-square goodness of fit:

  1. The variable must either be ordinal or nominal. The data should also be represented as counts or frequencies of cases.
  2. The sample is simple and random.
  3. The categories of variables used are mutually exclusive.
  4. Every count is independent. One observation or respondent should not make more than one contribution to the table.
  5. The total score count should not be greater than the sample size. i.e., one person gets one count.
  6. Your sample size should not be less than 20. The more the sample size, the better.
  7. All your expected frequencies in the cells should be more than five.

Running the Chi-Square Statistic in SPSS 

SPSS provides an option for the Chi-Square statistic within the cross-tabulation feature. The output for the chi-square test results is called Chi-Square Tests. Moreover, the fundamental result in the Chi-Square statistic table is the Pearson Chi-Square. This statistic is assessed by linking it to the corresponding p-value SPSS provides.

The following steps should guide you on how to run a Chi-Square test on SPPS:

  • Open the Crosstabs dialog by clicking Analyze > Descriptive Statistics > Crosstabs
  • Selecting the row and column variables.
  • Click on Statistics, then check Chi-square, and click Continue.
  • Check the box for Display clustered bar charts.
  • Click OK.

Interpretation of the output

When making a conclusion on the hypothesis at 95% confidence level, compare the p-value of the Chi-Square statistic with the chosen significance level. In that case, the value should be less than .05 (i.e. the alpha level attached to the 95% confidence level). If the p-value is greater than the alpha level value, it implies that there is a statistical association between the categorized variables.

Take a look at the example below which aims to test the association between fundamentals and opinions on learners being taught sex education in public schools using a Chi-Square Test of Independence. (a = 0.05).

A good example that indicates the application of Chi-Square Test is where one uses statistics to show the relationship between political stability and economic development around the world. The opinions of leaders and experts can be considered in this regard to help determine if political stability is promoting economic stability around the world. Chi-Square Test can then be used to show the level of relationship between political stability and the level of economic development. If the results indicate that more than 15% of the target population supports political stability as part of programs that promote economic stability around the world, then policymakers can adopt political stability in promoting social and economic development around the world. However, before making interpretations and analysis in the Chi-Square Test, it is important that one considers the p-value and indicate the significant association between the categorical variables under the study. In this case, the main variables under study include political stability and the economic development of countries around the world. These two variables must be run in the Chi-Square Test to provide the correct values.

Limitations of Chi-Square Statistics

Chi-Square statistic as a model of research has its limitations. One of the limitations is that Chi-Square statistic is extremely complex when it comes to sampling size which means that you need a relatively large sample size for better results.

Additionally, a small difference will appear to be statistically significant if the sample size is too large (~500).

Another limitation is that the expected frequencies should be 5 or more for 80% of the cells. SPSS will generate a warning message if input cells are fewer than 5. It is therefore recommended to use categorized variables with a specific number of groupings. Furthermore, all the variables must be independent and the subjects in each group must be selected randomly from the total population.

Statistics in APA

Whenever statistics is included in written text, it is important that the researcher includes sufficient information so that the reader can understand your study. APA has its own guidelines on statistical information:

  1. Do not provide formulas for common statistics such as the t-test and mean.
  2. Do not provide references for common statistics that have been used conventionally.
  3. To illustrate the relationship between series and numbers, it is important that you use terms such as in order and
  4. Use Boldface for matrices and vectors
  5. For statistical symbols, use italics.
  6. Use the italicized, uppercase, N, when referring to the total population. For example, N = 450
  7. When referring to a sample population, use an italicized lower case. For example, n = 29
  8. Use brackets to enclose the limits in confidence intervals. For example [3.44, 2.3]

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Chi-Square Test Sample Questions and Answers

Chi-Square question and answer

Chi-Square question and answer

Chi-Square question and answer

Chi-Square question and answer

Chi-Square question and answer

Chi-Square question and answer

Chi-Square question and answer

Chi-Square question and answer


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