Signature
X_1 = z^2
| Inputs | Definition | Unit |
|---|---|---|
z | Test statistic that is approximately standard normal under the null hypothesis, such as a Wald statistic | none |
X_1 | Test statistic referred to chi-square on one degree of freedom | none |
|---|
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).
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Implementations
Excel
One-degree-of-freedom chi-square from a normal statistic in one cell
With the normal statistic in a cell named Z, the formula returns its square.
=Z^2
Assumptions
Standard normal statistic under the null for a one-degree-of-freedom chi-square
z is approximately standard normal when the null hypothesis holds, as for a Wald statistic of one parameter in a large sample.
Worked examples
Squared 5% normal critical value on one degree of freedom
The squared normal critical value, 1.96 squared, is 3.8416, matching the 95th percentile of chi-square on one degree of freedom, 3.841.
z = 1.96; X_1 = 3.8416
Likelihood ratio of 9.0 as a squared normal deviate of 3
The Weibull against exponential statistic of 9.0 is the square of 3, so its p-value equals the two-sided normal tail beyond 3, about 0.0027, the article's step 1.
z = 3; X_1 = 9
Common errors
One-sided normal tail used for a one-degree-of-freedom chi-square p-value
The chi-square p-value on one degree of freedom equals the two-sided normal tail. Taking the one-sided tail beyond 3 gives about 0.00135, half the correct 0.0027.
Sources
Chi-square as a sum of squared standard normal variables in the NIST handbook
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 chi-square distribution results when nu independent variables with standard normal distributions are squared and summed; with nu equal to 1 it is the distribution of one squared standard normal variable.
Canonical Identity
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