11.2 Chi-Square Distribution

Just as z-tests are based on the normal distribution and t-tests are based on the t-distribution, chi-square tests are based on the chi-square distribution.

Key Facts: Chi-Square Distribution

  1. If [latex]Z \sim N(0, 1)[/latex], then [latex]Z^2 \sim \chi^2[/latex] distribution with degrees of freedom [latex]df=1[/latex].
  2. If [latex]Z_1, Z_2, \dots, Z_n[/latex] are independent and follow a standard normal distribution [latex]N(0, 1)[/latex], then [latex]Z_1^2 + Z_2^2 + \dots + Z_n^2 \sim \chi^2[/latex] distribution with [latex]df = 1 + 1 + \dots + 1 = n[/latex].
  3. If [latex]W \sim \chi^2[/latex] with [latex]df = p[/latex], [latex]V \sim \chi^2[/latex] with [latex]df = q[/latex] and they are independent, then [latex]W + V \sim \chi^2[/latex] with [latex]df = p + q[/latex].

Like the t distribution, the chi-square distribution is determined by one parameter, the degrees of freedom. The figure below shows the density curves of chi-square distributions with [latex]df = 1, 3, 5, 9, 15[/latex].

Several chi-square density curves are shown with different degrees of freedom. Image description available.
Figure 11.1: Chi-Square Density Curves. [Image Description (See Appendix D Figure 11.1)]

The properties of the chi-square density curve are as follows:

Key Facts: Properties of Chi-Square Density Curve

  • The total area under the curve is 1.
  • It is right skewed.
  • As the degrees of freedom increase, the chi-square curves appear more symmetric.
  • Every chi-square random variable is non-negative with possible values between 0 and [latex]\infty[/latex].
  • The mean of a chi-square is equal to its degrees of freedom and the standard deviation is the square root of twice the degrees of freedom. Suppose the degrees of freedom of a chi-square distribution is [latex]\gamma[/latex], then [latex]\mu=\gamma, \sigma=\sqrt{2\gamma}[/latex].

Like the t score table, the [latex]\chi^2[/latex] table (Table V) gives critical values [latex]\chi_{\alpha}^2[/latex]. Each critical value [latex]\chi_{\alpha}^2[/latex] has an area of [latex]\alpha[/latex] to its right under the curve of the chi-square distribution with a certain degrees of freedom df.

Table 11.1: Part of Chi-Square Table  (Table V)

Part of the chi-square table is shown. Image description available.
[Image Description (See Appendix D Table 11.1)]



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