Facts About the Chi-Square Distribution

Learning objectives

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 df = n – –1. The degrees of freedom for the three major uses are calculated differently.

For the χ² distribution, the population mean is μ = df, and the population standard deviation is $\sigma =\sqrt{2\left(df\right)}$.

The random variable is shown as χ², but it may be any uppercase letter.

The random variable for a chi-square distribution with k degrees of freedom is the sum of k independent, squared standard normal variables is

χ² = (Z₁)² + (Z₂)² + ... + (Z_k)², where the following are true:

• The curve is nonsymmetrical and skewed to the right.
• There is a different chi-square curve for each df.

  View original figure (external; image not reproduced in this text edition)

  Original figure description: 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.

• 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 X ~ ${\chi }_{\mathrm{1,000}}^2$, the mean, μ = df = 1,000 and the standard deviation, σ = $\sqrt{2\left(\mathrm{1,000}\right)}$ = 44.7. Therefore, X ~ N(1,000, 44.7), approximately.
• The mean, μ, is located just to the right of the peak.

  View original figure (external; image not reproduced in this text edition)

  Original figure description: 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.

References

Parade Magazine. (n.d.). Retrieved from https://parade.com/

Santa Clara County Public Health Department. (2011, May). HIV/AIDS epidemiology Santa Clara County. Retrieved from http://sccgov.iqm2.com/Citizens/FileOpen.aspx?Type=4&ID=32762

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

χ² = (Z₁)² + (Z₂)² + . . . (Z_df)² chi-square distribution random variable

μ_χ² = df chi-square distribution population mean

${\sigma }_{{\chi }^2}\text{=}\sqrt{2\left(df\right)}$ chi-square distribution population standard deviation

Homework

Decide whether the following statements are true or false.