### Introduction

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 *χ ^{2}* distribution, the population mean is

*μ*=

*df*, and the population standard deviation is $\sigma =\sqrt{2(df)}$.

The random variable is shown as *χ ^{2}*, 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

*χ*^{2} = (*Z*_{1})^{2} + (*Z*_{2})^{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*. - 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*~ ${\chi}_{\mathrm{1,000}\text{}}^{2}$, 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.