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 *Ï‡ ^{2}* distribution, the population mean is

*Î¼*=

*df*and the population standard deviation is $\mathrm{\xcf\u0192}=\sqrt{2(df)}$.

The random variable is shown as *Ï‡ ^{2}*.

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

*Ï‡*^{2} = (*Z*_{1})^{2} + (*Z*_{2})^{2} + ... + (*Z*_{k})^{2}

- 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*~ ${\mathrm{\xcf\u2021}}_{\mathrm{1,000}}^{2}$ the mean,*Î¼*=*df*= 1,000 and the standard deviation,*Ïƒ*= $\sqrt{2(\mathrm{1,000})}$ = 44.7. Therefore,*X*~*N*(1,000, 44.7), approximately. - The mean,
*Î¼*, is located just to the right of the peak.