Chi-Square Tests Goodness-of-Fit, Independence, Homogeneity Cheat Sheet

A printable reference covering chi-square goodness-of-fit, independence, homogeneity, expected counts, degrees of freedom, test statistic, and p-values for grades 11-12.

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Chi-square tests help students analyze categorical data and decide whether observed counts differ from what a model or comparison predicts. This cheat sheet covers the goodness-of-fit test, the test of independence, and the test of homogeneity. These tests are important because many real data sets involve counts in categories rather than measurements like height or time. A clear reference helps students choose the correct test, compute expected counts, and interpret results correctly. All three tests use the chi-square statistic $\chi^2 = \sum \frac{(O - E)^2}{E}$ , where $O$ is an observed count and $E$ is an expected count. A larger $\chi^2$ value means the observed counts are farther from the expected counts. The p-value is found from a chi-square distribution using the correct degrees of freedom. Students should always check that data are counts, observations are independent, and expected counts are large enough before using the test.

Key Facts

The chi-square test statistic for all three tests is $\chi^2 = \sum \frac{(O - E)^2}{E}$ .
For a goodness-of-fit test with $k$ categories and no estimated parameters, the degrees of freedom are $df = k - 1$ .
For a test of independence or homogeneity in an $r \times c$ table, the degrees of freedom are $df = (r - 1)(c - 1)$ .
For a two-way table, the expected count in a cell is $E = \frac{(\text{row total})(\text{column total})}{\text{grand total}}$ .
A goodness-of-fit test compares one categorical variable to a claimed distribution, such as $p_1 = 0.25$ , $p_2 = 0.50$ , and $p_3 = 0.25$ .
A test of independence checks whether two categorical variables from one population are associated or independent.
A test of homogeneity compares the distribution of one categorical variable across two or more populations or treatments.
A small p-value, usually $p < \alpha$ , gives evidence against the null hypothesis and supports the alternative hypothesis.

Vocabulary

Observed count: An observed count is the actual number of data values recorded in a category or table cell.
Expected count: An expected count is the count predicted by the null hypothesis for a category or table cell.
Chi-square statistic: The chi-square statistic $\chi^2$ measures the total squared difference between observed and expected counts, scaled by expected counts.
Degrees of freedom: Degrees of freedom describe which chi-square distribution to use when finding the p-value.
Goodness-of-fit test: A goodness-of-fit test determines whether the distribution of one categorical variable matches a claimed distribution.
Test of independence: A test of independence determines whether two categorical variables in one population are associated.

Common Mistakes to Avoid

Using proportions instead of counts in the formula is wrong because $\chi^2 = \sum \frac{(O - E)^2}{E}$ is based on counts, not percentages.
Confusing independence and homogeneity is wrong because independence uses one random sample from one population, while homogeneity compares separate samples or groups.
Forgetting to check expected counts is wrong because the chi-square approximation may not be reliable when expected counts are too small.
Using $df = k - 1$ for a two-way table is wrong because independence and homogeneity tests use $df = (r - 1)(c - 1)$ .
Concluding that the null hypothesis is true is wrong because a large p-value means there is not enough evidence to reject the null, not proof that it is true.

Practice Questions

1 A six-sided die is rolled $120$ times, and the observed counts are $18$ , $22$ , $17$ , $19$ , $24$ , and $20$ . Find the expected count for each side and compute the chi-square contribution for the side rolled $24$ times.
2 A survey table has $3$ rows and $4$ columns. Find the degrees of freedom for a chi-square test of independence.
3 In a two-way table, one cell has row total $80$ , column total $50$ , and grand total $200$ . Find the expected count for that cell using $E = \frac{(\text{row total})(\text{column total})}{\text{grand total}}$ .
4 A researcher compares favorite music genre across students from three different schools. Explain whether this situation calls for a goodness-of-fit test, a test of independence, or a test of homogeneity.

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