10.1: Chi-Square Distribution

A $\chi^{2}$ -distribution (chi-square, pronounced “ki-square”) is another special type of distribution for a continuous random variable. The sampling distribution for a variance and standard deviation follows a chi-square distribution.

Properties of the $\chi^{2}$ -distribution density curve:

1. Right skewed starting at zero.
2. The center and spread of a $\chi^{2}$ -distribution are determined by the degrees of freedom with a mean = df and standard deviation = \(\sqrt{2df}\).
3. Chi-square variables cannot be negative.
4. As the degrees of freedom increase, the $\chi^{2}$ -distribution becomes normally distributed for df > 50. Figure 10-1 shows $\chi^{2}$ -distributions for df of 2, 4, 10, and 30.
5. The total area under the curve is equal to 1, or 100%.

Figure 10-1

We will use the $\chi^{2}$ -distribution for hypothesis testing later in this chapter. For now, we are just learning how to find a critical value $\chi_{\alpha}^{2}$.

The symbol $\chi_{\alpha}^{2}$ is the critical value on the $\chi^{2}$ -distribution curve with area 1 – $\alpha$ below the critical value and area $\alpha$ above the critical value, as shown below in Figure 10-2.

Figure 10-2

Use technology to compute the critical value for the $\chi^{2}$ -distribution.

TI-84: Use the INVCHI2 program downloaded at Rachel Webb’s website: http://MostlyHarmlessStatistics.com. Start the program and enter the area $\alpha$ and the df when prompted.

TI-89: Go to the [Apps] Stat/List Editor, then select F5 [DISTR]. This will get you a menu of probability distributions. Arrow down to Inverse > Inverse Chi-Square and press [ENTER]. Enter the area 1 – $\alpha$ to the left of the $\chi$ value and the df into each cell. Press [ENTER].

Excel: =CHISQ.INV(1 – $\alpha$, df) or =CHISQ.INV.RT($\alpha$, df)

Alternatively, use the following online calculator: https://homepage.divms.uiowa.edu/~mbognar/applets/chisq.html.

Compute the critical value $\chi_{\alpha}^{2}$ for a $\alpha$ = 0.05 and df = 6.

Solution

Start by drawing the curve and determining the area in the right-tail as shown in Figure 10-3. Then use technology to find the critical value.

Figure 10-3

In Excel there are two options. Use =CHISQ.INV(area in left-tail,df) or right-tail =CHISQ.INV.RT(area in righttail,df). For this example, then we would have $\chi_{\alpha}^{2}$ =CHISQ.INV(0.95,6) or =CHISQ.INV.RT(0.05,6) = 12.5916.

TI-89 use Distr > Inverse Chi-square with area 0.95 and df = 6