F Distribution

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F Distribution


References: Dr. Mikel J. Harry, Juran's Quality Control Handbook

Purpose

The F Distribution is a continuous random variable distribution used in the F test in variance analysis, the F test verifies if the groups tested are from same mean populations.

In regression analysis, it verifies if there is a connection between the independent and the dependent variables.


The F distribution is a positively-skewed distribution used most commonly in Analysis of Variance. A distribution is positively-skewed if the mean is greater than the median.

The mean is the average value of a distribution, and the median is the midpoint; half of the values in the distribution are below the median, and half are above. 

The F-distribution can be used for several other types of applications, including testing a hypotheses about the equality of two population variances and testing the validity of a multiple regression equation.


Anatomy

how to use the f distributionF Distribution

Terminology

A. F statistic - A continuous random variable equal to the ratio of two independent chi-square random variables divided by their respective degrees of freedom.

B. Test parameter - To test the similarity of the variance of two populations based on two independent random samples of variance we use the F test (see tool F test). If we assume that the variances of the two populations are equal, then the following ratio follows an F probability distribution:

the f distribution formulaF Distribution Formula

C. Vertical axis Y(F) - Scale to measure the value of the F-statistic function.

D. Probability Density function - Curves representing the distribution for three different sets of values for the degrees of freedom of the numerator (df num) and the degrees of freedom of the denominator (df den). The total area under each of the curves is equal to one (1).

E. Horizontal axis - Scale of measure of the F-statistic.


Major Considerations

That samples are taken from populations that follow a normal distribution. Conduct normality testing. The standard deviation of the populations can be estimated by the sample standard deviations.

Application Cookbook

1. Use Excel functions to calculate the F distribution and the inverse of the F probability distribution.

2. Alternatively, use the tables printed at the end of most statistic books. For example, for df numerator = 2 and df denominator = 10 and a 95% confidence level we can say that the probability is 0.05 that the F-value will be 4.1028 or greater.

the f distribution tableF Distribution Table

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