Statistics · CED Unit 8: Chi-Squared Tests · 16 min read · Updated 2026-05-11
Chi-Squared Tests — AP Statistics
AP Statistics · CED Unit 8: Chi-Squared Tests · 16 min read
1. Overview of Chi-Squared Tests★☆☆☆☆⏱ 3 min
Chi-squared ($\chi^2$) tests are non-parametric hypothesis tests for categorical (count) data, used to compare observed sample counts to expected counts under a null hypothesis. The test uses the right-skewed chi-squared probability distribution, defined by its degrees of freedom. This topic makes up 2-5% of your total AP Statistics exam score.
2. Goodness-of-Fit Test★★☆☆☆⏱ 4 min
A chi-squared goodness-of-fit test tests whether the distribution of a single categorical variable from one sample matches a pre-specified hypothesized distribution.
$H_0$: The distribution of the variable matches the hypothesized distribution
$H_a$: The distribution of the variable does not match the hypothesized distribution
Test statistic formula:
\chi^2 = \sum \frac{(O - E)^2}{E}
Where $O$ = observed count, $E$ = expected count ($E = n \times p_i$ for sample size $n$ and hypothesized proportion $p_i$), and degrees of freedom are $df = k - 1$ for $k$ categories.
Exam tip: Always label expected counts clearly in free response answers; examiners regularly dock marks for unlabeled calculations.
3. Tests of Independence and Homogeneity★★★☆☆⏱ 5 min
These two tests use identical calculations but differ in study design and hypothesis framing, so it is critical to distinguish them for full AP exam marks.
Both tests use two-way contingency tables. Expected count for each cell, and degrees of freedom, are calculated as:
Exam tip: Always name the correct test type in your free response answer to earn full points for your conclusion.
4. Validity Conditions for Inference★★☆☆☆⏱ 3 min
All three chi-squared tests require the same three conditions to be met for valid inference. You must explicitly check all three on every free response question to earn full credit.
**Random**: Data comes from a random sample or randomized experiment. For homogeneity, each sample must be independently randomly selected.
**Independent**: Individual observations are independent. For sampling without replacement, the population must be at least 10 times the sample size (the 10% condition).
**Large Counts**: All expected counts are at least 5. If any expected count is <5, combine adjacent logically related categories to meet this condition.
5. Interpreting Test Statistics and P-Values★★★☆☆⏱ 4 min
Correct interpretation in context is required for full marks on AP Statistics free response. Generic interpretations will not earn credit.
The $\chi^2$ test statistic is always non-negative, since it is a sum of squared differences. Larger $\chi^2$ values indicate larger gaps between observed and expected counts, meaning stronger evidence against the null hypothesis.
For the die example, a correct interpretation is: *The p-value of 0.699 means that if the die is fair, there is a 69.9% chance of observing a $\chi^2$ test statistic of 3 or larger purely by random chance. Since this is greater than our 0.05 significance level, we do not have sufficient evidence to conclude the die is unfair.*
Exam tip: Always tie your interpretation back to the specific context of the problem. Generic interpretations without reference to the variables being studied will not earn full marks.
Common Pitfalls
Why: Confusion between hypothesized proportions and expected counts when setting up calculations.
Why: Mixing up the degrees of freedom formulas for goodness-of-fit versus two-way chi-squared tests.
Why: Confusing association and causation, a common AP Statistics theme.
Why: Carrying over one-sided alternative habits from z-tests or t-tests for means and proportions.
Why: Assuming conditions are met and forgetting to document checks, which are explicitly graded on the AP exam.