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Statistics LibreTexts

10.1: Chi-Square Distribution

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A χ2 -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 χ2 -distribution density curve:

  1. Right skewed starting at zero.
  2. The center and spread of a χ2 -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 χ2 -distribution becomes normally distributed for df > 50. Figure 10-1 shows χ2 -distributions for df of 2, 4, 10, and 30.
  5. The total area under the curve is equal to 1, or 100%.
clipboard_e4bc6fcabad9b911d3a268aeaf5de1162.png
Figure 10-1

We will use the χ2 -distribution for hypothesis testing later in this chapter. For now, we are just learning how to find a critical value χ2α.

The symbol χ2α is the critical value on the χ2 -distribution curve with area 1 – α below the critical value and area α above the critical value, as shown below in Figure 10-2.

clipboard_edff4463e62adbaa4742b5fa29a80fb4d.png
Figure 10-2

Use technology to compute the critical value for the χ2 -distribution.

TI-84: Use the INVCHI2 program downloaded at Rachel Webb’s website: http://MostlyHarmlessStatistics.com. Start the program and enter the area α 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 – α to the left of the χ value and the df into each cell. Press [ENTER].

Excel: =CHISQ.INV(1 – α, df) or =CHISQ.INV.RT(α, df)

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

Compute the critical value χ2α for a α = 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.

 clipboard_e7be50b99403a71daef1d9a363b5f5d48.png
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 χ2α =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

clipboard_ea1a1baed623f4f3224c7d4d4dfa2fe6f.png


This page titled 10.1: Chi-Square Distribution is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Rachel Webb via source content that was edited to the style and standards of the LibreTexts platform.

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