The chi-square statistic measures how far a table of observed counts is from what H₀ predicts. Edit the table and watch each component, the shaded p-value, and the four-step write-up update. The biggest components show where the difference comes from.
Edit the observed counts, choose the test and α, and follow the calculation from expected counts to the χ² statistic, the p-value, and a conclusion in context.
Highlighted cells contribute the most to χ². Add every component to get the statistic.
χ² = Σ (O − E)² / E adds up how far each observed count is from its expected count, relative to the expected count. χ² = 0 means a perfect match.
Chi-square distributions take only nonnegative values and are skewed right. Their mean equals df, and they become more symmetric as df grows.
The p-value is P(χ² ≥ observed) using df = (r − 1)(c − 1). Large χ² values give small p-values.
Look at the largest components to see where the observed counts differ most from expected, and say whether each is above or below expected.
Homogeneity: convincing evidence the distributions differ among the populations. Independence: convincing evidence of an association between the variables.
A randomized experiment supports cause and effect. Random samples support generalizing to the populations. An association in observational data is not causation.
The four steps of a chi-square test.
| Step | What to write | Example |
|---|---|---|
| State | H₀ and Hₐ in words, α | H₀: no association between age group and phone type; α = 0.05 |
| Plan | Name the test; random, 10%, all expected ≥ 5 | Chi-square test for independence; smallest expected count 9.3 |
| Do | χ², df, p-value | χ² = 17.69, df = 4, p-value = 0.0014 |
| Conclude | Compare p to α; conclusion in context | Reject H₀. There is convincing evidence of an association… |
| Calculator | χ²-Test with matrix [A] of observed counts | Expected counts appear in matrix [B] |
| Never | "Accept H₀" or use a two-tailed p-value | — |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 3.15. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 3 terminology guide.