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8.2: Facts About the Chi-Square Distribution

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    The notation for the chi-square distribution is:

    \[\chi \sim \chi^{2}_{df}\]

    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

    \[\sigma = \sqrt{2(df)}.\]

    The random variable is shown as \(\chi^{2}\), but may be any upper case letter. The random variable for a chi-square distribution with \(k\) degrees of freedom is the sum of \(k\) independent, squared standard normal variables.

    \[\chi^{2} = (Z_{1})^{2} + ... + (Z_{k})^{2}\]

    1. The curve is nonsymmetrical and skewed to the right.
    2. There is a different chi-square curve for each \(df\).
    Figure \(\PageIndex{1}\)
    1. The test statistic for any test is always greater than or equal to zero.
    2. When \(df > 90\), the chi-square curve approximates the normal distribution. For \(\chi \sim \chi^{2}_{1,000}\) the mean, \(\mu = df = 1,000\) and the standard deviation, \(\mu = \sqrt{2(1,000)}\). Therefore, \(X \sim N(1,000, 44.7)\), approximately.
    3. The mean, \(\mu\), is located just to the right of the peak.
    Figure \(\PageIndex{2}\)


    1. Data from Parade Magazine.
    2. “HIV/AIDS Epidemiology Santa Clara County.”Santa Clara County Public Health Department, May 2011.


    The chi-square distribution is a useful tool for assessment in a series of problem categories. These problem categories include primarily (i) whether a data set fits a particular distribution, (ii) whether the distributions of two populations are the same, (iii) whether two events might be independent, and (iv) whether there is a different variability than expected within a population.

    An important parameter in a chi-square distribution is the degrees of freedom \(df\) in a given problem. The random variable in the chi-square distribution is the sum of squares of df standard normal variables, which must be independent. The key characteristics of the chi-square distribution also depend directly on the degrees of freedom.

    The chi-square distribution curve is skewed to the right, and its shape depends on the degrees of freedom \(df\). For \(df > 90\), the curve approximates the normal distribution. Test statistics based on the chi-square distribution are always greater than or equal to zero. Such application tests are almost always right-tailed tests.

    Formula Review

    \[\chi^{2} = (Z_{1})^{2} + (Z_{2})^{2} + ... + (Z_{df})^{2}\] chi-square distribution random variable

    \(\mu_{\chi^{2}} = df\) chi-square distribution population mean

    \(\sigma_{\chi^{2}} = \sqrt{2(df)}\) Chi-Square distribution population standard deviation

    Exercise \(\PageIndex{1}\)

    If the number of degrees of freedom for a chi-square distribution is 25, what is the population mean and standard deviation?


    mean \(= 25\) and standard deviation \(= 7.0711\)

    Exercise \(\PageIndex{2}\)

    If \(df > 90\), the distribution is _____________. If \(df = 15\), the distribution is ________________.

    Exercise \(\PageIndex{3}\)

    When does the chi-square curve approximate a normal distribution?


    when the number of degrees of freedom is greater than 90

    Exercise \(\PageIndex{4}\)

    Where is \(\mu\) located on a chi-square curve?

    Exercise \(\PageIndex{5}\)

    Is it more likely the df is 90, 20, or two in the graph?

    Figure \(\PageIndex{3}\).


    \(df = 2\)

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