Page 387 - DMTH404_STATISTICS

Unit 28: F-distribution

Features of F- distribution Notes

1. This distribution has two parameters n (= n - 1) and n (= n - 1).
1 1 2 2
n
2. The mean of F - variate with n and n degrees of freedom is 2 and standard error is
1 2
n - 2
2
æ n 2 ö 2 n + n - ) 2
2
( 1
ç ÷ .
è n - 2ø n n - ) 4
2
( 2
1
We note that the mean will exist if n > 2 and standard error will exist if n > 4. Further, the
2 2
mean > 1.
3. The random variate F can take only positive values from 0 to . The curve is positively
skewed, as shown in Fig. 20.3
4. For large values of n and n , the distribution approaches normal distribution. This
1 2
behaviour is shown in the following figure.
5. If a random variate follows t-distribution with n degrees of freedom, then its square
follows F-distribution with 1 and n d.f. i.e. t = F
2
n 1,n
2
(c n )
2
6. F and c are also related as F = 1 as n  
n , n 2
1 2 n
1
Example 1: Suppose you randomly select 7 women from a population of women, and 12
men from a population of men. The table below shows the standard deviation in each sample
and in each population.
Population Population standard deviation Sample standard deviation

Women 30 35
Men 50 45

Compute the f statistic.
Solution A: The f statistic can be computed from the population and sample standard deviations,
using the following equation:
2
2
2
f = [ s /s ] / [ s /s ]
2
1 1 2 2
where s is the standard deviation of population 1, s is the standard deviation of the sample
1 1
drawn from population 1, s is the standard deviation of population 2, and s is the standard
2 1
deviation of the sample drawn from population 2.
As you can see from the equation, there are actually two ways to compute an f statistic from
these data. If the women’s data appears in the numerator, we can calculate an f statistic as
follows:
2
2
2
2
f = ( 35 / 30 ) / ( 45 / 50 ) = (1225 / 900) / (2025 / 2500) = 1.361 / 0.81 = 1.68
For this calculation, the numerator degrees of freedom v are 7 - 1 or 6; and the denominator
1
degrees of freedom v are 12 - 1 or 11.
2
On the other hand, if the men’s data appears in the numerator, we can calculate an f statistic as
follows:
2
2
f = ( 45 / 50 ) / ( 35 / 30 ) = (2025 / 2500) / (1225 / 900) = 0.81 / 1.361 = 0.595
2
2
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