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11.2: Facts About the Chi-Square Distribution

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    The notation for the chi-square distribution is:

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

    where \(d f=\) degrees of freedom which depends on how chi-square is being used. (If you want to practice calculating chi-square probabilities then use \(d f=n-1\). The degrees of freedom for the three major uses are each calculated differently.)

    For the \(\chi^2\) distribution, the population mean is \(\mu=d f\) 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.

    \[X^2=\left(Z_1\right)^2+\left(Z_2\right)^2+\ldots+\left(Z_k\right)^2\]

    1. The curve is nonsymmetrical and skewed to the right.
    2. There is a different chi-square curve for each \(d f\).

    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 \(\PageIndex{1}\)

    3. The test statistic for any test is always greater than or equal to zero.
    4. When \(d f>90\), the chi-square curve approximates the normal distribution. For \(X \sim \chi_{1,000}^2\) the mean, \(\mu=d f=\) 1,000 and the standard deviation, \(\sigma=\sqrt{2(1,000)}=44.7\). Therefore, \(X \sim N(1,000,44.7)\), approximately.
    5. The mean, \(\mu\), is located just to the right of the peak.

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