The notation for the chi-square distribution is [latex]\chi \sim {\chi }_{df}^{2}[/latex], where [latex]df = \text{degrees of freedom}[/latex], which depends on how chi-square is being used. (If you want to practice calculating chi-square probabilities then use [latex]df = n - 1[/latex]. The degrees of freedom for the three major uses are each calculated differently.)

For the [latex]\chi^2[/latex] distribution, the population mean is [latex]\mu = df[/latex] and the population standard deviation is [latex]\sigma =\sqrt{2(df)}[/latex].

The random variable is shown as [latex]\chi^2[/latex], but may be any uppercase letter.

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

[latex]\chi^2 = (Z_1)^2+(Z_2)^2+...+(Z_k)^2[/latex]

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