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F-Distribution

Definition: The F-Distribution is also called as Variance Ratio Distribution as it usually defines the ratio of the variances of the two normally distributed populations. The F-distribution got its name after the name of R.A. Fisher, who studied this test for the first time in 1924.

Symbolically, the quantity is distributed as F-distribution with ν1 =n1-1 and ν2 = n2-1 degrees of freedom and is represented as:

F-Distribution-1Where,
S12 is the unbiased estimator of σ12 and is calculated as:

F-Distribution-2S22 is the unbiased estimator of σ22 and is calculated as:

F-Distribution-3The parameters of the F-distribution are degrees of freedom ν1 for the numerator and degrees of freedom ν2 for the denominator. Thus, with the change in the values of these parameters the distribution also changes. The F distribution probability density function is given by:

F-Distribution-4Y0 = constant depending on the values of ν1 and ν2.

For testing the hypothesis of the equality of two population variances, the following statistic is used:

F-Distribution-5Here, the Null hypothesis = σ12 = σ22 which follows the F-distribution with degrees of freedom ν1 and ν2. Often, the larger sample variance is placed in the numerator for the convenient computations. In doing so, we get the ratio of sample variance equals to or greater than one.

If the computed value of F exceeds the table value of F, then the null hypothesis is rejected and the alternative hypothesis gets accepted. On the other hand, if the computed value of F is less than the table value, the null hypothesis is accepted.

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