Chi-Square Calculator

Goodness-of-fit — compare observed counts in c categories against a theoretical distribution. df = c − 1.

Test of independence — observed counts in an r × c contingency table. Tests H₀: rows and columns are independent. Expected count for cell (i, j) = (row total · column total) / grand total. df = (r − 1)(c − 1).

Both compute χ² = Σ (O − E)² / E and compare against a chi-square distribution. Larger χ² = bigger gap between observed and expected.

Independent observations: each subject counted once, no repeated measures on the same individual within the table.

Random sampling from the population of interest.

Expected counts ≥ 5 in at least 80% of cells, and no cell with expected < 1. If violated, combine adjacent categories or use Fisher's exact test for small 2×2 tables.

Categories must be mutually exclusive and exhaustive — every observation belongs to exactly one cell.

Roll a die 60 times, observe (12, 8, 11, 9, 10, 10). Is it fair? Under H₀, each face is expected 10 times.

χ² = (12−10)²/10 + (8−10)²/10 + (11−10)²/10 + (9−10)²/10 + 0 + 0 = 0.4 + 0.4 + 0.1 + 0.1 = 1.0. df = 5.

p = P(χ²₅ ≥ 1.0) ≈ 0.96 — totally consistent with a fair die.

Survey 349 people about coffee preference (regular/decaf) vs age group (18-34, 35-54, 55+). Observed counts form a 2×3 table.

Compute expected counts from row and column marginals. If all expected counts are large and observed counts close to expected, χ² will be small and you'll fail to reject 'no association.'

If you reject, examine the (O − E)² / E contributions per cell to see which cells drive the association — those are the 'where it matters' story behind the test.

Using chi-square on percentages or proportions. The test needs raw counts; convert back to counts before testing.

Comparing a tiny table with a tiny sample. Small expected counts inflate (O − E)² / E and break the chi-square approximation.

Treating significance as effect size. A huge sample can make trivial differences 'significant'. Report Cramér's V or odds ratios for effect magnitude.

Drawing causal conclusions. A significant independence test means 'they're related,' not 'one causes the other.'

Fisher's exact test is the classic alternative for 2×2 tables with small expected counts. It computes the exact probability of the observed table (and more extreme ones) under the hypergeometric distribution, avoiding the chi-square approximation entirely.

G-test (likelihood-ratio test) is asymptotically equivalent to chi-square but handles sparse tables better and has nicer theoretical properties for log-linear models. Most software reports both χ² and G for large tables.

McNemar's test is for paired categorical data — before/after studies on the same subjects, or matched case-control designs. It looks only at the discordant pairs (changed responses) and ignores the concordant ones.

For ordered categories (mild, moderate, severe), the chi-square test of independence wastes information by treating order as nominal. Use the Cochran-Armitage trend test or ordinal logistic regression instead.