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 [latex]\sigma =\sqrt{2\left(df\right)}[/latex].

The random variable is shown as χ², but 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.

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.
   
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 ~ [latex]{\chi }_{1,000}^{2}[/latex] the mean, μ = df = 1,000 and the standard deviation, σ = [latex]\sqrt{2\left(1,000\right)}[/latex] = 44.7. Therefore, X ~ N(1,000, 44.7), approximately.
5. The mean, μ, is located just to the right of the peak.