Module 11: The Chi Square Distribution
The notation for the chi-square distribution is , 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 . The degrees of freedom for the three major uses are each calculated differently.)
For the χ2 distribution, the population mean is μ = df and the population standard deviation is .
The random variable is shown as χ2, 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.
- 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.
- When df > 90, the chi-square curve approximates the normal distribution. For the mean, and the standard deviation, . Therefore, , approximately.
- The mean, μ, is located just to the right of the peak.
Data from Parade Magazine.
“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.
chi-square distribution random variable
chi-square distribution population mean
Chi-Square distribution population standard deviation