F DISTRIBUTION

Sir Ronald Fisher was interested in extending our knowledge of testing mean differences to an analysis of the variability of scores, i.e., variance. He was specifically interested in comparing the variance of two random samples of data. For example, if a random sample of data was drawn from one population and a second random sample of data was drawn from a second population, then the two sample variances could be compared by computing an F ratio: F=S21 / S22. The F ratio is equal to one if the variances of the two random samples are the same. The F-distribution in the appendix reveals this for F values with df=∞, ∞ in the numerator and denominator. The F ratio could be less than one, depending upon which sample variances were in the numerator and denominator, but F values less than one are not considered, so we always place the larger sample variance in the numerator. If several random samples of data were drawn from each population and the F ratio computed on the variances for each pair of samples, a sampling distribution of the F’s would create the F distribution. Sir Ronald Fisher determined that like the t distribution and chisquare distribution, the F distribution was a function of sample size; specifically the sizes of the two random samples. Consequently, a family of F curves can be formed based on the degrees of freedom in the numerator and denominator. An F curve is positively skewed with F ratio values ranging from zero to infinity (∞). If the degrees of freedom for both samples are large, then the F distribution approaches symmetry (bell-shaped). Example F curves for certain degree of freedom pairs can be illustrated as:

Because there are two degrees of freedom associated with an F ratio, F tables were constructed to list the F values expected by chance with the degrees of freedom for the numerator across the top (column values) and the degrees of freedom for the denominator along the side (row values). The corresponding intersection of a column and row degrees of freedom would indicate the tabled F value. If the computed F value is greater than the tabled F value, we conclude that the two sample variances are statistically different at a specified level of probability, e.g., .05 level of significance.