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.

Let *X* and *Y* be two independent χ^{2} 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).

Let (*X*_{1}, *X*_{2} , …, *X _{m}*)
and (

Then,

are
independent

(1)
Hence, F-Statistic is defined as

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

The following are some of the important applications where the
sampling distribution of the respective statistic under *H*_{0} 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)]

Tags : Properties, Definition | Statistics , 12th Statistics : Chapter 3 : Tests Based on Sampling Distributions II

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

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