The Chi-Square Distribution

Read this chapter, which introduces you to the three major uses of the chi-squared distribution: the goodness-of-fit test, the test of independence, and the test of a single variance. Attempt the practice problems and homework at the end of the chapter.

Facts About the Chi-Square Distribution

The notation for the chi-square distribution is:

$χ∼χ^2_{df}$

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 $χ^2$ distribution, the population mean is μ = df and the population standard deviation is $σ=\sqrt{2(df)}$ .

The random variable is shown as $χ^2$ .

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

$χ^2 = (Z_1)^2 + (Z_2)^2 + ... + (Z_k)^2$

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

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