Chi-Square Distribution — Sum of Squared Normals
Foundations of Statistics
The Foundation of Variance-Based Inference
The chi-square distribution arises from summing squared normal variables, making it essential for testing variances and independence. Its applications span from quality control to genetics, wherever squared deviations matter.
- Genetics — Testing Hardy-Weinberg equilibrium in population studies
- Manufacturing — Quality control through variance testing and goodness-of-fit
- Market Research — Analyzing survey response patterns against expected distributions
The chi-square distribution connects normal theory to categorical data analysis.
Core Concepts
The chi-square distribution arises as the sum of squared independent standard normal random variables. It is fundamental to tests of variance and independence.
Interactive Visualization
Mean, Variance, and Higher Moments
Relationship to Normal and Gamma
Derivation: Sample Variance and Chi-Square
Worked Example
A quality engineer tests whether a filling machine has variance mL. From bottles, mL.
Step 1. Compute the chi-square test statistic:
Step 2. Under , with and , so .
Step 3. The observed value 37.8 is standard deviations above the mean.
Step 4. For a two-sided test at , the critical values are and . Since , we fail to reject .
Step 5. The chi-square distribution is asymmetric, so the test is inherently two-sided in a different way than z/t tests. The p-value is .