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 each calculated differently.)

For the χ² distribution, the population mean is μ = df and the population standard deviation is σ=2(df)−−−−−√

$�=\sqrt{2(��)}$.

The random variable is shown as χ².

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

χ² = (Z₁)² + (Z₂)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 ~ χ21,000
   ${�}_{\mathrm{1,000}}^2$ 

   the mean, μ = df = 1,000 and the standard deviation, σ = 2(1,000)−−−−−−−√ $\sqrt{2(\mathrm{1,000})}$ 

   = 44.7. Therefore, X ~ N(1,000, 44.7), approximately.

5. The mean, μ, is located just to the right of the peak.