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Derivation of the Chi-Square Distribution from Sum of Squared Standard Normal Variables

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The Formal Theorem

Let Z1,Z2,…,Zk Z_1, Z_2, \dots, Z_k be independent and identically distributed random variables such that Zi∼N(0,1) Z_i \sim N(0, 1) . The sum of their squares, defined as X=∑i=1kZi2 X = \sum_{i=1}^{k} Z_i^2 , follows a Chi-square distribution with k k degrees of freedom, denoted X∼χ2(k) X \sim \chi^2(k) , with probability density function:
fX(x;k)=12k/2Γ(k/2)x(k/2)−1e−x/2,x>0 f_X(x; k) = \frac{1}{2^{k/2} \Gamma(k/2)} x^{(k/2)-1} e^{-x/2}, \quad x > 0

Analytical Intuition.

Imagine k k independent scouts wandering through a featureless, multi-dimensional fog. Each scout's position along any single axis is dictated by the standard normal distribution N(0,1) N(0, 1) , centered stubbornly at the origin. We are interested in the 'energy' or 'squared distance' of this group from the center, defined by the Euclidean norm squared, X=∑Zi2 X = \sum Z_i^2 . As we add each dimension k k , we are essentially accumulating degrees of freedom. For k=1 k=1 , the density peaks at the origin; as k k increases, the geometry of the hypersphere pushes the probability mass away from zero, transforming the shape into the skewed, characteristic Chi-square curve. We derive this by utilizing the moment generating function of a single Zi2 Z_i^2 , which reveals itself to be a Gamma distribution. Through the convolution of these independent variables, the Chi-square distribution emerges as a special, elegant case of the Gamma family, perfectly capturing how random fluctuations aggregate into a quantifiable metric of dispersion.
CAUTION

Institutional Warning.

Students often struggle to differentiate between the individual standard normal variables Zi Z_i and the resulting χ2 \chi^2 variable. Remember: the Zi Z_i are symmetric about zero, but their squared sum χ2 \chi^2 is strictly non-negative, shifting the focus from location to magnitude and variability.

Academic Inquiries.

01

Why is the Chi-square distribution a special case of the Gamma distribution?

The Gamma distribution has the form f(x)∝xα−1e−βx f(x) \propto x^{\alpha-1} e^{-\beta x} . By setting the shape parameter α=k/2 \alpha = k/2 and the rate parameter β=1/2 \beta = 1/2 , we recover the Chi-square density exactly.

02

What happens to the shape of the distribution as k k approaches infinity?

By the Central Limit Theorem, as the degrees of freedom k k increase, the sum of these independent squared variables begins to approach a Normal distribution, specifically N(k,2k) N(k, 2k) .

Standardized References.

  • Definitive Institutional SourceCasella, G., & Berger, R. L., Statistical Inference.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Derivation of the Chi-Square Distribution from Sum of Squared Standard Normal Variables: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/applied-statistics/derivation-of-the-chi-square-distribution-from-sum-of-squared-standard-normal-variables

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