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11.2: Facts About the Chi-Square Distribution

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 \(df\). 

    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 11.2.1.

  3. The test statistic for any test is always greater than or equal to zero.
  4. 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.
  5. The mean, \(\mu\), is located just to the right of the peak. 

    This is a nonsymmetrical chi-square curve which is skewed to the right. The mean, m, is labeled on the horizontal axis and is located to the right of the curve's peak.

    Figure 11.2.2.


Data from Parade Magazine.

“HIV/AIDS Epidemiology Santa Clara County.”Santa Clara County Public Health Department, May 2011.

Chapter Review

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 11.2.1

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 11.2.2

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

Exercise 11.2.3

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


when the number of degrees of freedom is greater than 90

Exercise 11.2.4

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


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

This is a nonsymmetrical  chi-square curve which slopes downward continually.

Figure 11.2.3.


\(df = 2\)