Conceptually and mathematically, the F-test of the independent-samples single-factor ANOVA is an extension of the t-test of $\mu _ { 1 } - \mu _ { 2 }$ .

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The F-test of the independent-samples single-factor ANOVA compares the variance between groups to the variance within groups, while the t-test compares the means of two groups. However, mathematically, the F-value for a single-factor ANOVA can be calculated using the ratio of the variances of the groups, which is similar to how the t-value for a two-sample t-test is calculated using the ratio of the difference in means and the standard error of the difference. Therefore, in some sense, the F-test can be seen as an extension of the t-test.

Conceptually and Mathematically, the F-Test of the Independent-Samples Single-Factor ANOVA μ1−μ2\mu _ { 1 } - \mu _ { 2 }μ1​−μ2​

Conceptually and Mathematically, the F-Test of the Independent-Samples Single-Factor ANOVA $\mu _ { 1 } - \mu _ { 2 }$