Statistics. F-Ratio and Other Technical Terms

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F-ratio is a statistical concept in experimental research design, which helps to assess whether variance in two different samples is equal. It is also used to determine the variance within groups of sample data as well as the variance between two groups of data. The F-ration is mostly applied in ANOVA calculations and is calculated using the following formula:

F-ratio = MSM / MSR
Where MS=ss/df and
SS= sum of squares
Df= degrees of freedom.

The subscripted M is used to indicate ‘model’ and shows the systematic variance expected. This is often measured as a between-group variance. On the other hand, R stands for ‘residual’ and represents the unsystematic and random variance. This is often measured as a within-group variance. Occasionally the superscript w is also used very often.

Similarly, F can also be calculated using the Pearson correlation coefficient, that is, the r, as shown below:

F = r2 / (1 – r2)(n – 2)

In some cases, the f-ratio may not bear any significance statistically. Consequently, in such situations, it is assumed that there is homogeneity of variance. As a result, it will be prudent to employ the standard statistical T-test to show the differences of means. In cases where the F-ratio is significant statistically, alternative computations for the T-test such as the Cochran and Cox methodology may be used. Computation of T-tests involves dividing the ratios of intended and systematic variance with those of unsystematic and unexpected variance. The formula for the T-tests can be simplified as shown below;
T-tests= systematic variance/unsystematic variance.

Furthermore, the T-tests are often measured as the total value or sum of SS (squares), hence:

Test statistic = SSM / SSR

However, in situations of multiple variables, the F-ratio is often derived from the mean of the available multiple variables.