Chi-Square Distribution

Statistics · Axiom Academy

LESSON Chi-Square Distribution Understanding the distribution formed by summing squared standard normal random variables 1. Definition: Sum of Squared Standard Normals The chi-square distribution arises when we square independent standard normal random variables and sum them together. Each Z i represents a standard normal variable, and when we square it, we always get a positive value. The sum of these squared values creates the chi-square distribution. 2. Degrees of Freedom and Shape The degrees of freedom (df) parameter determines the shape of the chi-square distribution. As df increases, the distribution's shape changes dramatically. Low df (1-2): Distribution is highly right-skewed with mode at or near zero Medium df (3-10): Right-skewed but more balanced, mode moves rightward High df (>30): Distribution becomes more symmetric and approaches normal distribution The degrees of freedom equals the number of independent squared standard normal variables being summed. In practice, df often equals n - 1 for sample variance estimates. 3. Key Properties: Mean and Variance The chi-square distribution has elegant mathematical properties directly related to its degrees of freedom. Mean: E[χ²] = df (the mean equals the degrees of freedom) Variance: Var(χ²) = 2 × df (variance is twice the degrees of freedom) Mode: max(df - 2, 0) for df ≥ 2 Range: (0, ∞) - only positive values Additive Property: If X ~ χ²(df₁) and Y ~ χ²(df₂) are independent, then X + Y ~ χ²(df₁ + df₂)

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