<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Skip Navigation

Test of Independence

Construct a table of values that you would expect to see if the variables were independent.

Atoms Practice
Estimated10 minsto complete
Practice Test of Independence
This indicates how strong in your memory this concept is
Estimated10 minsto complete
Practice Now
Turn In
Chi-Square Test

We use the chi-square test to examine patterns between categorical variables, such as genders, political candidates, locations, or preferences.

There are two types of chi-square tests: the goodness-of-fit test and the test for independence. We use the goodness-of-fit test to estimate how closely a sample matches the expected distribution.  We use the test for independence to determine whether there is a significant association between two categorical variables in a single population.

To test for significance, it helps to make a table containing the observed and expected frequencies of the data sample. If you have two different categorical variables, this is called a contingency table.

The Chi-Square Statistic

The value that indicates the comparison between the observed and expected frequency is called the chi-square statistic . The idea is that if the observed frequency is close to the expected frequency, then the chi-square statistic will be small. On the other hand, if there is a substantial difference between the two frequencies, then we would expect the chi-square statistic to be large.

To calculate the chi-square statistic,  , we use the following formula:


 is the chi-square test statistic.

 is the observed frequency value for each event.

 is the expected frequency value for each event.

The number of degrees of freedom associated with a goodness-of-fit chi-square test is df = c - 1 where c is the number of categories.  The number of degrees of freedom associated with a chi-square test of independence is, df = (r-1) * (c-1) where where r is the number of levels for one catagorical variable, and c is the number of levels for the other categorical variable.

We use the chi-square test statistic and the degrees of freedom to determine the p-value on a chi-square probability table.

Using the p-value and the level of significance, we are able to determine whether to reject or fail to reject the null hypothesis and write a summary statement based on these results.

Test of Single Variance

We can use the chi-square test if we want to test two samples to determine if they belong to the same population.  We are testing the hypothesis that the sample comes from a population with a variance greater than the observed variance.

Here is the formula for the chi-square statistic:


 is the chi-square statistical value.

 , where  is the size of the sample.

 is the sample variance.

 is the population variance.

Once we have the chi-square statistic, find the p-value and complete the test as usual.

Explore More

Sign in to explore more, including practice questions and solutions for Chi-Square Test.
Please wait...
Please wait...