Chi-Square Tests – AP Statistics

Learning Objectives

Categorical Data: Tests used for categorical bins (e.g., colors of M&Ms) rather than quantitative measurements like means.

Observed vs Expected: Does our real-world Observed count ($O$) match the theoretical Expected count ($E$)?

The $\chi^2$ Statistic: We calculate $(O-E)^2 / E$ for each category and sum them. If $\chi^2$ is very large, the skew is too immense to be random chance... we reject $H_0$!

Tags

Chi-SquareGoodness of FitCategoricalDegrees of Freedom

       CHI-SQUARE STATISTIC

       $\chi^2 = \sum \frac{(Observed - Expected)^2}{Expected}$

Observed Bins

Expected Bins

Error Penalty $(O-E)^2/E$

Null Hypothesis (Uniform)

Expected Candy Distribution

Assume a candy company claims exactly $20\%$ per color (Red, Blue, Green, Yellow, Orange). Total samples: $N = 100$.

Observed Deviation ($O$)

Red Candies = 20

Blue Candies = 20

TEST METRICS ($df = 4$)

Total Error $\chi^2$: 0.00

P-Value: 1.000

VERDICT ($a = 0.05$)

FAIL TO REJECT H₀

Random Chance: When all observed counts perfectly match expectations ($O=20$), the $\chi^2$ value is exactly 0. Try sliding the observed counts away from 20 to see the massive squared penalties stack up!