Signature
f_x = x^(k/2 - 1) * exp(-x / 2) / (2^(k/2) * Gamma(k/2))
| Inputs | Definition | Unit |
|---|---|---|
x | Value of the chi-square variable, above zero, for example an observed test statistic | none |
k | Degrees of freedom, above zero; the number of squared standard normal terms in the sum | none |
f_x | Probability density of the chi-square distribution at x | per unit of x |
|---|
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).
Implementations
Excel
Chi-square density in one cell
With named cells Stat and DegFree, CHISQ.DIST with FALSE as its last argument returns the density. Excel truncates a degrees of freedom argument that is not a whole number.
=CHISQ.DIST(Stat,DegFree,FALSE)
Assumptions
Independent standard normal terms behind the chi-square density
The variable is a sum of k independent squared standard normal deviates. A test statistic follows the distribution only approximately, in large samples and under the null hypothesis, so the density describes the reference distribution rather than the exact behaviour of the statistic.
Worked examples
Chi-square density at 2 on two degrees of freedom
With k = 2 the gamma term is one and the density at 2 is one half of exp(minus 1), about 0.1839.
x = 2; k = 2; f_x = 0.1839
Chi-square density at 1 on one degree of freedom
Gamma(1/2) is the square root of pi, so the density at 1 is exp(minus 0.5) divided by the square root of 2 pi, about 0.2420.
x = 1; k = 1; f_x = 0.2420
Common errors
Reading the chi-square density as a p-value
The density at the observed statistic is not a probability. For the Weibull against exponential statistic of 9.0 on one degree of freedom the density is about 0.0015, whereas the p-value, the area of the right-hand tail, is about 0.0027.
Sources
Chi-square density, mean and variance in the R documentation
R Core Team. The (non-central) chi-squared distribution: Chisquare, package stats. R Documentation; accessed 2 October 2026. Details: the chi-squared distribution with n degrees of freedom has density 1/(2^(n/2) Gamma(n/2)) x^(n/2 minus 1) e^(minus x/2) for x above zero, with mean n and variance 2n.
Chi-square as a special case of the gamma distribution in SciPy
SciPy developers. scipy.stats.chi2. SciPy v1.18.0 Manual; accessed 2 October 2026. Notes: the chi-squared distribution is a special case of the gamma distribution, with shape df/2, location 0 and scale 2.
Canonical Identity
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