The Chi-Squared Distribution

Sum of squared independent standard normals; foundation of chi-squared tests and ANOVA

Introduction

The chi-squared distribution arises as the sum of squared independent standard normal random variables. It is the sampling distribution of the sample variance under normality, the reference distribution for goodness-of-fit and contingency tests, and a key ingredient in the F distribution used throughout ANOVA.

Prerequisites

Normal distribution, basic calculus.

Theory

If \(Z_1, \ldots, Z_k\) are iid standard normal, then

\[Q = \sum_{i=1}^k Z_i^2 \sim \chi^2_k.\]

The parameter \(k\) is the degrees of freedom. The PDF is

\[f(x) = \frac{1}{2^{k/2} \Gamma(k/2)} x^{k/2 - 1} e^{-x/2}, \qquad x > 0.\]

Moments: \(E[Q] = k\), \(\mathrm{Var}(Q) = 2k\).

Sum property: if \(Q_1 \sim \chi^2_{k_1}\) and \(Q_2 \sim \chi^2_{k_2}\) are independent, then \(Q_1 + Q_2 \sim \chi^2_{k_1 + k_2}\).

Sample variance: for iid normal \(X_1, \ldots, X_n\) with variance \(\sigma^2\),

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

This is the basis of confidence intervals for a normal variance.

Large-df behaviour: \(\chi^2_k\) is approximately normal for large \(k\) (by the CLT applied to the sum of squared normals).

Assumptions

For the sample-variance result: iid normal observations. For the goodness-of-fit approximation in chi-squared tests: independent observations and large enough expected cell counts.

R Implementation

k <- 5

# PDF and CDF
x <- seq(0, 20, length.out = 400)
plot(x, dchisq(x, df = k), type = "l", col = "#2A9D8F", lwd = 2,
     main = paste("Chi-squared,", k, "df"), ylab = "f(x)")

# Moments
c(theoretical_mean = k, theoretical_var = 2 * k,
  empirical_mean   = mean(rchisq(1e5, k)),
  empirical_var    = var(rchisq(1e5, k)))

# Verify: sum of k squared standard normals
n <- 1e5
Q_direct <- rchisq(n, df = k)
Q_from_normals <- rowSums(matrix(rnorm(n * k), n, k)^2)
c(mean_direct = mean(Q_direct), mean_from_normals = mean(Q_from_normals))

# Confidence interval for a variance
set.seed(2026)
X <- rnorm(30, mean = 0, sd = 2)
s2 <- var(X)
n <- length(X)
chi_l <- qchisq(0.025, n - 1); chi_u <- qchisq(0.975, n - 1)
c((n - 1) * s2 / chi_u, (n - 1) * s2 / chi_l)

Output & Results

theoretical_mean  theoretical_var  empirical_mean  empirical_var
            5.0             10.0           5.003           9.98

mean_direct mean_from_normals
      5.005             5.004

[1] 2.542 7.206

A 95% CI for the true variance (which is 4) is (2.54, 7.21), which captures the true value. The sum-of-squared-normals construction produces the expected mean of 5.

Interpretation

Chi-squared tests of independence, goodness-of-fit, and variance homogeneity all reference the chi-squared distribution. Reporting a test statistic value like “\(\chi^2_3 = 9.5\), \(p = 0.023\)” is reporting a chi-squared-distributed statistic with its p-value.

Practical Tips

• Chi-squared confidence intervals for variance are highly sensitive to the normality assumption.
• For large df, use the normal approximation: \(\chi^2_k \approx \mathcal{N}(k, 2k)\), or more accurately \(\sqrt{2 \chi^2_k} \approx \mathcal{N}(\sqrt{2k - 1}, 1)\).
• Chi-squared tests with small expected counts are poorly approximated; use exact tests or Monte Carlo.
• Non-central chi-squared (with non-centrality parameter \(\lambda\)) governs power calculations for chi-squared tests.
• The square of a \(t_\nu\) is \(F_{1, \nu}\), not chi-squared; do not conflate.