Stats - Chi Squared

Frequency/Count

number of sample items in category

Goodness-of-Fit Test

if a particular distribution could reasonably model our sample data

Contingency Test

if 2 categorical variables are independent or associated

Test of Homogeneity

tests whether different populations have the same distribution of a categorical variable

Goodness-of-Fit Hypothesis

Null: distribution of x is a particular probability distribution

Alt: distribution is NOT a particular probability distribution

• Do counts observed in sample match our expectation from the particular probability distribution?

Contingency Hypothesis

Null: variable a is independent of variable B

Alt: variable a is NOT independent of variable B

• Is there evidence that 2 nominal variables are related/associated?

• Is there evidence of differences among 2 or more population proportions?

Homogeneity Hypothesis

Null: distributions are the same among all proportions

Alt: at least one population has a different distribution

Difference between goodness-of-fit and homogeneity

• Goodness-of-Fit → one group vs expected

• Homogeneity → multiple groups vs each other
  

Fast memory trick ⚡

• “Fit” → Does this group fit what I expected?

• “Homogeneity” → Are these groups homogeneous (the same)?

Observed Count

number of items in the sample with a given characteristic or set of characteristic

Expected Count

number of items in the sample that should have a given characteristic if the null statement is true

Chi Squared Test Statistic

A measure of how far observed counts differ from expected counts, calculated by summing (Observed − Expected)² / Expected across all categories.

• Small χ² → observed is close to expected (supports H₀) ✅

• Large χ² → observed is far from expected (evidence against H₀) ❌

Required Data Conditions

• SRS

• Data summarized as counts per category

• all expected counts ≥ 5

• n ≥ 30 (for goodness-of-fit)

DF Goodness-of-Fit

If parameter given: k-1

If parameter not given: k-m-1

• m = # of parameters estimated

DF Contingency

(r-1)(c-1)