Chi-square distribution

The chi-square distribution with k degrees of freedom is the distribution of a sum of k squared independent standard normals; it is right-skewed and non-negative.

\chi^2_k = Z_1^2 + \dots + Z_k^2, \qquad \E = k, \qquad \Var = 2k

Definition, mean and variance

$Z_i$: independent standard normal variables
$k$: degrees of freedom

\frac{(n-1)s^2}{\sigma^2} \sim \chi^2_{n-1}

Sampling distribution of a variance (normal population)

$s^2$: sample variance (divisor $n - 1$ )
$\sigma^2$: population variance

When to use

Reference distribution for goodness-of-fit and independence tests (always the upper tail), likelihood-ratio tests and the Kruskal–Wallis approximation.
Intervals for a variance from normal data.

How to interpret

Right-skewed; becomes more symmetric as df grows. Mean equals df, so a statistic near its df is unremarkable.
Chi-square tests of fit use only the upper tail: large values mean poor agreement.

Common mistakes

Doubling the tail for a chi-square test of fit; the p-value is $P(\chi^2 \ge X^2)$ .
Wrong df: goodness of fit uses $k - 1$ (minus estimated parameters); an $r \times c$ table uses $(r-1)(c-1)$ .
Using the variance interval for non-normal data: it is very sensitive to non-normality.

Example

With df $= 3$ , the 5% critical value is $\chi^2_{0.95,3} = 7.8147$ ; an observed $X^2 = 9.2$ has p-value $P(\chi^2_3 \ge 9.2) = 0.0267$ . For df $= 9$ , the central 95% lies between 2.7004 and 19.0228.