# The Chi-Square Distribution

## Facts About 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 df = n – 1. The degrees of freedom for the three major uses are calculated differently.
For the χ distribution, the population mean is μ = df, and the population standard deviation is 
 .
The random variable is shown as χ, 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)2, where the following are true:
1. The curve is nonsymmetrical and skewed to the right.
2. There is a different chi-square curve for each df.

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 X ~ 
 , the mean, μ = df = 1,000 and the standard deviation, σ = 
  = 44.7. Therefore, X ~ N(1,000, 44.7), approximately.
5. The mean, μ, is located just to the right of the 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)2
chi-square distribution random variable
μ = df chi-square distribution population mean

  chi-square distribution population standard deviation

### Homework

Decide whether the following statements are true or false.