Algorithms (Two Sample Test for Variance)

The F-test calculates the ratio of two sample variance to test whether or not the two data samples come from populations with equal variances. And the hypotheses take the form:

H_0:\frac{\sigma_1^2}{\sigma_2^2}=1 vs H_1:\frac{\sigma_1^2}{\sigma_2^2}\ne 1 Two Tailed

H_0:\frac{\sigma_1^2}{\sigma_2^2} \le 1 vs H_1:\frac{\sigma_1^2}{\sigma_2^2} > 1 Upper tailed

H_0:\frac{\sigma_1^2}{\sigma_2^2} \ge 1 vs H_1:\frac{\sigma_1^2}{\sigma_2^2} < 1 Lower tailed

Test Statistics

And the F-test statistic is calculated as: F=\frac{s_1^2}{s_2^2}

where s_1^2\,\! and s_2^2\,\! are observed sample variances. A ratio of 1 implies equal sample variances, while ratios that deviate from 1 indicate unequal population variances. The hypothesis that the variances of the two samples are equal is rejected if p < \sigma\,\!, where p is the calculated probability and \sigma\,\! is the chosen significance level.

Confidence Intervals

The upper and lower confidence limit values for F-test statistic is:

Null Hypothesis Confidence Interval
H_0:\frac{\sigma_1^2}{\sigma_2^2}=1 \left[\frac{F}{F_{1-\alpha/2}},\frac{F}{F_{\alpha/2}}\right]
H_0:\frac{\sigma_1^2}{\sigma_2^2} \le 1 \left[\frac{F}{F_{1-\alpha}},\infty\right]
H_0:\frac{\sigma_1^2}{\sigma_2^2} \ge 1 \left[0,\frac{F}{F_{\alpha}}\right]

where F_{1-\sigma/2}\,\! and F_{\sigma/2}\,\! represents the lower and upper critical values for an F-distribution with n_1-1\,\! and n_2-1\,\! degrees of freedom, and \sigma\,\! level of significance.