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# F -Distribution and Its Applications

F-statistic is the ratio of two sums of the squares of deviations of observations from respective means. The sampling distribution of the statistic is F-distribution.

F -DISTRIBUTION AND ITS APPLICATIONS

F-statistic is the ratio of two sums of the squares of deviations of observations from respective means. The sampling distribution of the statistic is F-distribution.

## Definition: F -Distribution

Let X and Y be two independent χ2 random variates with m and n degrees of freedom respectively.

Then F = is said to follow F-distribution with (m, n) degrees of freedom. This F-distribution is named after the famous statistician R.A. Fisher (1890 to 1962).

## Definition: F-Statistic

Let (X1, X2 , …, Xm) and (Y1, Y2, …, Yn) be two independent random samples drawn from N(μX, σX2) and N(μY, σY2) populations respectively.

Then, are independent

(1) Hence, F-Statistic is defined as (2) F-Statistic is also defined as the ratio of two mean square errors.

## Applications of F-distribution

The following are some of the important applications where the sampling distribution of the respective statistic under H0 is F–distribution.

(i) Testing the equality of variances of two normal populations. [Using (1)]

(ii) Testing the equality of means of k (>2) normal populations. [Using (2)]

(iii) Carrying out analysis of variance for two-way classified data. [Using (2)]

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12th Statistics : Chapter 3 : Tests Based on Sampling Distributions II : F -Distribution and Its Applications | Properties, Definition | Statistics