Friday, December 10, 2010

Statistics Tutorial: F Distribution

Statistics Tutorial: F Distribution
The f statistic, also known as an f value, is a random variable that has an F distribution. (We discuss the F distribution in the next section.)
Here are the steps required to compute an f statistic:
  • Select a random sample of size n1 from a normal population, having a standard deviation equal to σ1.
  • Select an independent random sample of size n2 from a normal population, having a standard deviation equal to σ2.
  • The f statistic is the ratio of s1212 and s2222.
The following equivalent equations are commonly used to compute an f statistic:
f = [ s1212 ] / [ s2222 ]
f = [ s12 * σ22 ] / [ s22 * σ12 ]
f = [ Χ21 / v1 ] / [ Χ22 / v2 ]
f = [ Χ21 * v2 ] / [ Χ22 * v1 ]
where σ1 is the standard deviation of population 1, s1 is the standard deviation of the sample drawn from population 1, σ2 is the standard deviation of population 2, s2 is the standard deviation of the sample drawn from population 2, Χ21 is the chi-square statistic for the sample drawn from population 1, v1 is the degrees of freedom for Χ21, Χ22 is the chi-square statistic for the sample drawn from population 2, and v2 is the degrees of freedom for Χ22 . Note that degrees of freedom v1 = n1 - 1, and degrees of freedom v2 = n2 - 1 .
The F Distribution
The distribution of all possible values of the f statistic is called an F distribution, with v1 = n1 - 1 and v2 = n2 - 1 degrees of freedom.
The curve of the F distribution depends on the degrees of freedom, v1 and v2. When describing an F distribution, the number of degrees of freedom associated with the standard deviation in the numerator of the f statistic is always stated first. Thus, f(5, 9) would refer to an F distribution with v1 = 5 and v2 = 9 degrees of freedom; whereas f(9, 5) would refer to an F distribution with v1 = 9 and v2 = 5 degrees of freedom. Note that the curve represented by f(5, 9) would differ from the curve represented by f(9, 5).
The F distribution has the following properties:
  • The mean of the distribution is equal to v2 / ( v2 - 2 ) for v2 > 2.
  • The variance is equal to [ 2 * v22 * ( v1 + v1 - 2 ) ] / [ v1 * ( v2 - 2 )2 * ( v2 - 4 ) ] for v2 > 4.
Cumulative Probability and the F Distribution
Every f statistic can be associated with a unique cumulative probability. This cumulative probability represents the likelihood that the f statistic is less than or equal to a specified value.
Statisticians use fα to represent the value of an f statistic having a cumulative probability of (1 - α). For example, suppose we were interested in the f statistic having a cumulative probability of 0.95. We would refer to that f statistic as f0.05, since (1 - 0.95) = 0.05.
Of course, to find the value of fα, we would need to know the degrees of freedom, v1 and v2. Notationally, the degrees of freedom appear in parentheses as follows: fα(v1,v2). Thus, f0.05(5, 7) refers to value of the f statistic having a cumulative probability of 0.95, v1 = 5 degrees of freedom, and v2 = 7 degrees of freedom.
The easiest way to find the value of a particular f statistic is to use the F Distribution Calculator, a free tool provided by Stat Trek. For example, the value of f0.05(5, 7) is 3.97. The use of the F Distribution Calculator is illustrated in the examples below.

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