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 .
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 + ... + (Zk)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.
- 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 ~ , the mean, μ = df = 1,000 and the standard deviation, σ = = 44.7. Therefore, X ~ N(1,000, 44.7), approximately.
- The mean, μ, is located just to the right of the peak.