40 Facts About the Chi-Square Distribution

The notation for the chi-square distribution is:

$$\chi \sim \chi_{df}^2$$

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 each calculated differently.)

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

The random variable is shown as χ², 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.

χ² = (Z₁)² + (Z₂)² + … + (Z_k)²

1. The curve is non-symmetrical and skewed to the right.
2. There is a different chi-square curve for each df.

   
   
   

   Figure 11.2
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 $\chi \sim \chi_{1000}^2$ the mean, μ = df = 1,000 and the standard deviation, $\sigma = \sqrt{2(1000)}=44.7$. Therefore, $X\sim N(1,000, 44.7)$, approximately.
5. The mean, μ, is located just to the right of the peak.

   
   
   

   Figure 11.3