The Chi-Square Distribution

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 each calculated differently).

For the \(χ^2\) distribution, the population mean is μ = df and the population standard deviation is \(σ=\sqrt{2(df)}\).

The random variable is shown as \(χ^2\).

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

\(χ^2 = (Z_1)^2 + (Z_2)^2 + ... + (Z_k)^2\)

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