Are My Results Unexpected?

When counts surprise us

• Some questions ask whether observed counts 计数 differ from what we'd expect.
• "Do the colors in a candy bag match the advertised mix?"
• We compare the observed counts 观测频数 to the expected counts 期望频数 under a claim.
• Big gaps between observed and expected are the signal of something unusual.

The chi-square family

• The tool is the chi-square 卡方 family of tests, written $\chi^2$.
• It's built for categorical data — counts in categories, not measurements.
• Every chi-square test compares observed counts to expected counts.
• A larger total gap → a larger $\chi^2$ → stronger evidence against the claim.

Observed against expected counts: large gaps make χ² large

Which chi-square question?

• Goodness-of-fit 拟合优度: does one categorical variable match a claimed distribution?
• Two-way table questions: are two categorical variables related (across groups or within one)?
• One variable, one sample → goodness-of-fit; a table of two variables → homogeneity/independence.
• Spotting the type is the first decision.

Observed vs. expected

• Every chi-square test rests on one comparison: observed vs. expected counts.
• Expected = what the null hypothesis predicts each count should be.
• Observed = what the data actually show.
• The test measures how far, overall, observed strays from expected.

Write from an expected count of 20

• A category is expected to hold 20 observations if the model is right.
• The study sees 28. The lesson expected count 25 is another model.