Module 11: The Chi Square Distribution

# Facts About the Chi-Square Distribution

Barbara Illowsky & OpenStax et al.

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 *χ*^{2} distribution, the population mean is μ = *df* and the population standard deviation is .

The random variable is shown as *χ*^{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.

- 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.

## References

Data from *Parade Magazine*.

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

## Concept 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-square distribution random variable

chi-square distribution population mean

Chi-Square distribution population standard deviation