11.1: Facts About the Chi-Square Distribution

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The notation for the chi-square distribution is:

\[\chi \sim \chi_{d f}^{2}\nonumber\]

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 \(\chi^2\) distribution, the population mean is \(\mu = df\) and the population standard deviation is \(\sigma=\sqrt{2(d f)}\).

The random variable is shown as \(\chi^2\).

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

\[\chi^{2}=\left(Z_{1}\right)^{2}+\left(Z_{2}\right)^{2}+\ldots+\left(Z_{k}\right)^{2}\nonumber\]

The curve is non-symmetrical and skewed to the right.
There is a different chi-square curve for each \(df\) (\(\PageIndex{1}\)).
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 \(\chi \sim \chi_{1,000}^{2}\) the mean, \(\mu = df = 1,000\) and the standard deviation, \(\sigma=\sqrt{2(1,000)}=44.7\). Therefore, \(\chi \sim N(1,000,44.7)\), approximately.
The mean, \(\mu\), is located just to the right of the peak.

Part (a) shows a chi-square curve with 2 degrees of freedom. It is nonsymmetrical and slopes downward continually. Part (b) shows a chi-square curve with 24 df. This nonsymmetrical curve does have a peak and is skewed to the right. The graphs illustrate that different degrees of freedom produce different chi-square curves. — Figure \(\PageIndex{1}\)