F-values, named after mathematician Sir Ronald Fisher who originally developed the test in the 1920s, provides a reliable means of determining whether the variance of a sample is significantly different than that of the population to which it belongs. While the mathematics required to calculate the critical value of F, the point at which variances are significantly different, the calculations to find the F-value of a sample and population is fairly simple.
Find the Total Sum of Squares
Calculate the sum of squares between. Square each value of each set. Add together each value of each set to find the sum of the set. Add together the squared values to find the sum of squares. For example, if a sample includes 11, 14, 12 and 14 as one set and 13, 18, 10 and 11 as another then the sum of the sets is 103. The squared values equal 121, 196, 144 and 196 for the first set and 169, 324, 100 and 121 for the second with a total sum of 1,371.
Square the sum of the set; in the example the sum of the sets equaled 103, its square is 10,609. Divide that value by the number of values in the set -- 10,609 divided by 8 equals 1,326.125.
Subtract the value just determined from the sum of the squared values. For example, the sum of the squared values in the example was 1,371. The difference between the two -- 44.875 in this example -- is the total sum of squares.
Find the Sum of Squares Between and Within Groups
Square the sum of the values of each set. Divide each square by the number of values in each set. For example, the square of the sum for the first set is 2,601 and 2,704 for the second. Dividing each by four equals 650.25 and 676, respectively.
Add those values together. For example, the sum of those values from the previous step is 1,326.25.
Divide the square of the total sum of the sets by the number of values in the sets. For example, the square of the total sum was 103, which when squared and divided by 8 equals 1,326.125. Subtract that value from the sum of the values from step two (1,326.25 minus 1,326.125 equals .125). The difference between the two is the sum of squares between.
Subtract the sum of squares between from the sum of squares total to find the sum of squares within. For example, 44.875 minus .125 equals 44.75.
Calculate F
Find the degrees of freedom between. Subtract one from the total number of sets. This example has two sets. Two minus one equals one, which is the degrees of freedom between.
Subtract the number of groups from the total number of values. For example, eight values minus two groups equals six, which is the degrees of freedom within.
Divide the sum of squares between (.125) by the degrees of freedom between (1). The result, .125, is the mean square between.
Divide the sum of squares within (44.75) by the degrees of freedom within (6). The result, 7.458, is the mean square within.
Divide the mean square between by the mean square within. The ratio between the two equals F. For example, .125 divided by 7.458 equals .0168.
References
About the Author
Sean Butner has been writing news articles, blog entries and feature pieces since 2005. His articles have appeared on the cover of "The Richland Sandstorm" and "The Palimpsest Files." He is completing graduate coursework in accounting through Texas A&M University-Commerce. He currently advises families on their insurance and financial planning needs.
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