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 σ=2(df)σ=2(df).

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.
    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.
    Figure 11.2
  • 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 ~ χ1,0002χ1,000 , the mean, μ = df = 1,000 and the standard deviation, σ = 2(1,000)2(1,000) = 44.7. Therefore, X ~ N(1,000, 44.7), approximately.
  • The mean, μ, is located just to the right of the peak.
    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.
    Figure 11.3