Show why the Sum of Squared Standard Normals Follows a Chi-Square Distribution

Why Squared Normals Form a Chi-Square Distribution

Submit Homework

The chi-square distribution is one of the most important distributions in statistics tutoring, especially in hypothesis testing, variance estimation, and likelihood theory. Students often memorize the definition without understanding why squaring standard normal variables produces a chi-square distribution. This page explains the construction, intuition, and exam‑relevant logic behind the chi-square distribution.

If \(Z_1, \dots, Z_k \sim N(0,1)\), then \(\chi^2_k = Z_1^2 + \cdots + Z_k^2\).

Why does this happen? Because squaring a standard normal produces a gamma‑distributed variable with shape \(1/2\) and scale \(2\). The sum of independent gamma variables with the same scale parameter is also gamma. A chi-square distribution is simply a special case of the gamma distribution with integer or half‑integer shape. This mathematical structure explains why chi-square distributions appear naturally in variance estimation and likelihood theory.

1. Start with a standard normal variable. \(Z \sim N(0,1)\).
2. Square it. \(Z^2\) follows a chi-square distribution with 1 degree of freedom.
3. Use the gamma connection. \(Z^2 \sim \Gamma\left(\frac{1}{2}, 2\right)\).
4. Add independent squared normals. If \(Z_1, \dots, Z_k\) are independent, \(Z_1^2 + \cdots + Z_k^2\) is the sum of \(k\) gamma variables.
5. Apply the gamma summation rule. The sum of gamma variables with the same scale is gamma with shape equal to the sum of shapes.
6. Conclude the chi-square form. The resulting distribution is \(\chi^2_k\).

Suppose you square three independent standard normal variables:

\(Z_1^2 = 1.21,\quad Z_2^2 = 0.49,\quad Z_3^2 = 2.25\)

Then:

\(\chi^2_3 = 1.21 + 0.49 + 2.25 = 3.95\)

This value would be compared to a chi-square distribution with 3 degrees of freedom in hypothesis testing.

• Thinking chi-square variables must come from sample variances only.
• Forgetting that independence of normals is required.
• Confusing chi-square with t or F distributions.
• Ignoring the gamma distribution connection.

The chi-square distribution is essential for variance estimation, confidence intervals, likelihood ratio tests, and ANOVA. It appears in nearly every graduate statistics exam and underlies the t and F distributions through simple transformations.

This idea connects directly to:

• Math Statistics & Probability Theory (parent spoke)
• Tutoring Services
• Question Hub
• Statistics Post Hub