Chi-Square Calculator

How it works

χ² = Σ (O − E)² ÷ E · Goodness-of-fit: df = k − 1 (E = total ÷ k if equal) · Independence: E_ij = (row total_i × column total_j) ÷ grand total, df = (r − 1)(c − 1) · p = P(χ² > statistic) = 1 − P(df/2, χ²/2)

The chi-square test compares counts you observed with the counts a hypothesis predicts. In a goodness-of-fit test you check whether one categorical variable follows an expected distribution — if no expected counts are supplied, equal frequencies (total ÷ k) are assumed. In a test of independence you check whether two categorical variables in a contingency table are related, deriving the expected count in each cell from the row and column totals. Both sum the squared, standardised gap (O − E)² ÷ E across every cell into a single χ² statistic, and its upper-tail p-value — computed here from the regularized incomplete gamma function — measures how surprising that total is under the null hypothesis. Each expected count should generally be at least 5 for the approximation to hold.

Worked example

A die is rolled 100 times giving observed counts 18, 22, 20, 25, 15 against an expected 20 in each of the 5 categories. The contributions are (18−20)²/20 + (22−20)²/20 + (20−20)²/20 + (25−20)²/20 + (15−20)²/20 = 0.2 + 0.2 + 0 + 1.25 + 1.25 = 2.9. With df = 5 − 1 = 4, the upper-tail p-value is about 0.5747 — not significant at α = 0.05, so the die looks fair.

Sources: Chi-squared test — Wikipedia

Frequently asked questions