Chi-Square Test of Independence
Test whether two categorical variables are related, from a two-way table (χ²-Test).
Results
Reject H₀: χ² = 10.7458, p-value = 0.0046
The p-value (0.0046) is less than or equal to α = 0.05, so the data give enough evidence to conclude that the two variables are related.
Results
| χ² test statistic | 10.7458 |
|---|
| df = (rows − 1)(columns − 1) | 2 |
|---|
| p-value | 0.0046 |
|---|
| Critical value | 5.991 |
|---|
| Decision at α = 0.05 | Reject H₀ |
|---|
On the TI-84
χ²-Test [A], [B]
Show steps
- Hypotheses. H₀: the two variables are independent Ha: they are related (not independent)
- Expected counts. E = (row total × column total) ÷ grand total. For example, the top-left cell: 60 × 50 ÷ 125 = 24.
- Test statistic. χ² = Σ (O − E)² ÷ E over all 6 cells = 10.7458, df = (2 − 1)(3 − 1) = 2.
- p-value. Right-tailed test: area to the right of the test statistic: p-value = 0.0046. Since 0.0046 ≤ 0.05, reject H₀.
- Critical-value method. At α = 0.05, the rejection region is χ² ≥ 5.991. The test statistic χ² = 10.7458 is in the rejection region, so we reject H₀. (Same decision as the p-value method.)
- Conclusion. The p-value (0.0046) is less than or equal to α = 0.05, so the data give enough evidence to conclude that the two variables are related.
About this calculator
When to use this
Use a test of independence when each person (or item) is classified by two categorical variables, such as course format and pass/fail, and the counts are arranged in a two-way table. The question is whether the variables are related.
Expected counts
If the variables were independent, each cell's expected count would be (row total × column total) ÷ grand total. The farther the observed counts are from those, the larger χ² gets. The test is right-tailed with df = (rows − 1)(columns − 1).
More chi-square & anova calculators: χ² Goodness of Fit · One-Way ANOVA