Chi-Square Goodness-of-Fit

You can't just eyeball whether a bag of candy matches its advertised color mix. Real bags vary by chance even when the company's claim is true, so you need a way to measure how far observed counts drift from expected counts, and whether that drift is bigger than random sampling would explain. The chi-square goodness-of-fit test gives you one number that captures the total mismatch across every category at once.

Start with one categorical variable and a claimed distribution, like a company's stated 1/3 red, 1/3 blue, 1/3 green. Multiply the claimed proportions by your sample size to get expected counts for each category. Then for every category, subtract expected from observed, square that difference, and divide by the expected count. Squaring keeps every term positive so deviations don't cancel out, and dividing by expected scales each gap relative to how many you expected there, so a miss of 10 matters more when you expected 20 than when you expected 200. Add up these terms across all categories to get the chi-square statistic. A small value means observed counts stayed close to the claim; a large value means they drifted far enough that you'd question the claim.

This test sits next to the chi-square test for independence, and the two get confused constantly because they use the same formula and the same chi-square distribution for the p-value. The difference is what question each one answers. Goodness-of-fit takes one variable and asks whether its distribution matches a specific claimed set of proportions. Independence takes two variables, arranged in a two-way table, and asks whether knowing one tells you anything about the other. Same arithmetic, different setup and different hypotheses.