Mean, variance and skewness of a chi-square distribution

A statistic that follows chi-square on k degrees of freedom under the null hypothesis has expected value k, so only values well above k give small p-values. The skewness falls towards zero as k grows, and because the variable is a sum of k independent terms the central limit theorem makes it close to a normal distribution with the same mean and variance. The function sqrt is the square root.

Signature

E_X = k; V_X = 2 * k; g_X = sqrt(8 / k)
Inputs
InputsDefinitionUnit
kDegrees of freedom, above zeronone
Output
E_XExpected value of the chi-square variablenone
V_XVariance of the chi-square variablenone
g_XSkewness of the chi-square variable, equal to 2^1.5 divided by the square root of knone

Function

Chi-square distribution of a sum of squared standard normal deviates

Maps the degrees of freedom k to the distribution of the sum of k independent squared standard normal variables. Health economic models meet it as the reference for test statistics, each read from the right-hand tail: a likelihood ratio between nested survival models, Cochran's Q for heterogeneity in a meta-analysis and Pearson's statistic for calibration targets. It is the gamma distribution with shape k/2 and scale 2. The formulae for Cochran's Q, I-squared and the between-study variance are on the Aggregate Data Meta-Analysis page (HE-FM-ADMA-002 to HE-FM-ADMA-004). AIC and BIC, which TSD 14 prefers for choosing between survival curves, are on the Akaike Information Criterion page (HE-FM-AIC-001 and HE-FM-AIC-004) and the Bayesian Information Criterion page (HE-FM-BIC-001).

Try this function

Implementations

  • Excel

    Chi-square mean, variance and skewness in three cells

    With the degrees of freedom in a cell named DegFree, the formulas return the mean, the variance and the skewness.

    =DegFree; =2*DegFree; =SQRT(8/DegFree)

Assumptions

  • Chi-square moments describe the null reference distribution

    The moments describe the reference distribution that a test statistic follows when the null hypothesis holds and the large-sample approximation applies. Under an alternative the statistic tends to be larger.

Worked examples

  • Moments of chi-square on one degree of freedom

    With one degree of freedom the mean is 1, the variance 2 and the skewness about 2.83, a strongly right-skewed distribution.

    k = 1; E_X = 1; V_X = 2; g_X = 2.8284

  • Moments of chi-square on four degrees of freedom

    With four degrees of freedom the mean is 4, the variance 8 and the skewness about 1.41, already less skewed than with one degree of freedom.

    k = 4; E_X = 4; V_X = 8; g_X = 1.4142

  • Moments of chi-square on fifty degrees of freedom

    With fifty degrees of freedom the skewness has fallen to 0.4 and the distribution is close to a normal with mean 50 and variance 100.

    k = 50; E_X = 50; V_X = 100; g_X = 0.4

Common errors

  • Treating a chi-square statistic near its degrees of freedom as evidence of misfit

    A statistic close to k is what the null hypothesis predicts. The generalised gamma against Weibull statistic of 0.8 on one degree of freedom is below its mean of 1 and has a p-value of about 0.37, so it gives no evidence that the extra parameter improves the fit.

Sources

  • NIST common statistics of the chi-square distribution

    NIST/SEMATECH. e-Handbook of Statistical Methods. Section 1.3.6.6.6, Chi-Square Distribution. National Institute of Standards and Technology; accessed 2 October 2026. The distribution results when independent standard normal variables are squared and summed; common statistics give mean nu, standard deviation the square root of 2 nu and skewness 2^1.5 divided by the square root of nu.

    View source →

Canonical Identity

Persistent URI
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