Chi-Square Test

Testing whether observed counts match expected ones (goodness of fit) or whether two categorical variables are independent.

Chi-square distribution — right-tail p-value

df=4

χ²=6.0

p-value ≈ 0.1992 → fail to reject H₀

Definition

The chi-square test ( $\chi^2$ ) tests whether categorical data matches an expected distribution.

The test statistic is:

\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}

where $O_i$ are observed counts and $E_i$ are expected counts under $H_0$ .

Under $H_0$ , this follows a $\chi^2$ distribution with $k - 1$ degrees of freedom (for goodness of fit) or $(r-1)(c-1)$ for a contingency table with $r$ rows and $c$ columns.

Key properties

The $\chi^2$ statistic is always non-negative, and exactly 0 only when observed counts match expected counts perfectly
Larger deviations from expected counts contribute quadratically, so a few large discrepancies can dominate many small ones
The approximation to the $\chi^2$ distribution improves as sample size grows — it's asymptotic, not exact
A significant result only establishes association (for independence tests), never causation

Is a die fair?

Roll a die 60 times. Under $H_0$ (fair die), expect 10 of each face. Observed: 8, 14, 11, 9, 7, 11.

$\chi^2 = \frac{(8-10)^2}{10} + \frac{(14-10)^2}{10} + \ldots = 0.4+1.6+0.1+0.1+0.9+0.1 = 3.2$

With 5 df, the critical value at $\alpha=0.05$ is 11.07. Since $3.2 < 11.07$ , fail to reject — no evidence the die is unfair.

Try it

A genetics experiment expects a 9:3:3:1 ratio in a cross (Mendelian genetics). Observed counts: 315, 108, 101, 32 (total 556). Expected: $556 \times (9/16, 3/16, 3/16, 1/16)$ . Compute $\chi^2$ and use the fact that the critical value with 3 df at $\alpha=0.05$ is 7.81.

Solution

Expected: 312.75, 104.25, 104.25, 34.75.

$\chi^2 = (315-312.75)^2/312.75 + (108-104.25)^2/104.25 + (101-104.25)^2/104.25 + (32-34.75)^2/34.75$ $\approx 0.016 + 0.135 + 0.101 + 0.218 = 0.47$

Since $0.47 < 7.81$ , we fail to reject — the data fits Mendel's expected 9:3:3:1 ratio.

Related concepts

Needs first

Hypothesis Testing Probability

Multiple Testing

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