11.1 Facts About the Chi-Square Distribution

Chi-square distribution is the resulting distribution from squaring variables that are have a standard normal distribution. The random variable for a chi-square distribution with k degrees of freedom is the sum of k independent, squared standard normal variables. \[\chi_k^2=\left(Z_1\right)^2+\left(Z_2\right)^2+…+\left(Z_k\right)^2\]

• 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.
• The population mean, [latex]\mu = df[/latex]. Mean is located just to the right of the peak.
• The population standard deviation, [latex]\sigma = \sqrt{2(df)}[/latex].
• When [latex]df > 90[/latex], the chi-square curve approximates the normal distribution.

You can use Desmos to draw/visualize Chi-square distributions relatively easily.

An Introduction to the [latex]\chi^2[/latex] distribution