Facts About the Chi-Square Distribution

The notation for the chi-square distribution is

where df = degrees of freedom, which depends on how chi-square is being used. If you want to practice calculating chi-square probabilities then use df = n – –1. The degrees of freedom for the three major uses are calculated differently.

For the χ² distribution, the population mean is μ = df, and the population standard deviation is $\sigma =\sqrt{2(df)}$.

The random variable is shown as χ², but it may be any uppercase letter.

The random variable for a chi-square distribution with k degrees of freedom is the sum of k independent, squared standard normal variables is

χ² = (Z₁)² + (Z₂)² + ... + (Z_k)², where the following are true:

• The curve is nonsymmetrical and skewed to the right.
• There is a different chi-square curve for each df.

  

  Figure 11.2
• The test statistic for any test is always greater than or equal to zero.
• When df > 90, the chi-square curve approximates the normal distribution. For X ~ ${\chi }_{\mathrm{1,000}}^2$, the mean, μ = df = 1,000 and the standard deviation, σ = $\sqrt{2(\mathrm{1,000})}$ = 44.7. Therefore, X ~ N(1,000, 44.7), approximately.
• The mean, μ, is located just to the right of the peak.

  

  Figure 11.3