Variance-Ratio Test and Analysis of Variance (ANOVA)

Variance has always been used as a measure of statistical analysis. Sir RA Fisher, the great statistician of the twentieth century, introduced the term “variance” in 1920 for the analysis of statistical data. The technique for the analysis of variance of two or more samples was developed by Fisher himself. The “variance-ratio test” is also known as “F-ratio test” or F-test. The F-test demonstrates that whether the variance of two populations from which the samples have been drawn is equal or not, whereas the “analysis of variance” ascertains the difference of variance among more than two samples. We know that the “square of standard deviation” (s 2) is termed as “variance.” F-test is defined as a “ratio of variances” of two samples. Mathematically: F = s 1 2 s 2 2 , where s 1 2 = ∑ x 1 − x ¯ 1 2 n 1 − 1 and s 2 2 = ∑ x 1 − x ¯ 2 2 n 2 − 1 $$ F=\frac{s_1^2}{s_2^2},\kern1em \mathrm{where}\kern0.5em {s}_1^2=\frac{\sum {\left({x}_1-{\overline{x}}_1\right)}^2}{n_1-1}\kern0.5em \mathrm{and}\kern0.5em {s}_2^2=\frac{\sum {\left({x}_1-{\overline{x}}_2\right)}^2}{n_2-1} $$ Important condition: The “numerator” should be “greater than the ‘denominator’” ( s 1 2 > s 2 2 $$ {s}_1^2>{s}_2^2 $$ ). If s 2 2 > $$ {s}_2^2> $$ s 1 2 $$ {s}_1^2 $$ , then F value should be calculated as F = s 2 2 s 1 2 $$ \frac{s_2^2}{s_1^2} $$ . Degrees of freedom would be v 1 = n 1 − 1 and v 2 = n 2 − 1.