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Unit 28: F-distribution

 This distribution has two parameters n (= n - 1) and n (= n - 1). Notes
1 1 2 2
n
 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
( 1
2
ç ÷ .
è n - 2ø n n - ) 4
1
2
( 2
We note that the mean will exist if n > 2 and standard error will exist if n > 4. Further, the
2 2
mean > 1.
 The random variate F can take only positive values from 0 to . The curve is positively
skewed, as shown in Fig. 20.3
 For large values of n and n , the distribution approaches normal distribution. This
1 2
behaviour is shown in the following figure.
 If a random variate follows t-distribution with n degrees of freedom, then its square
2
follows F-distribution with 1 and n d.f. i.e. t = F
n 1,n
28.3 Keywords
The random variate: The random variate F can take only positive values from 0 to . The curve
is positively skewed.
F-distribution: If a random variate follows t-distribution with n degrees of freedom, then its
2
square follows F-distribution with 1 and n d.f. i.e. t = F
n 1,n
28.4 Self Assessment

1. Fill in the Blanks:
(1) The mean and standard error of a F-variate depend upon its ...... parameters.
(ii) The sum of squares of standard normal variates is a ...... variate.

(iii) If N = 8 and n = 3, the number of samples without replacement is equal to ...... .
(iv) The ratio of two sample variances follows F - distribution when the variances of
their parent population are ...... .

(v) In sampling without replacement if N = n, the standard error of X is equal to ...... .
2
(vi) Both c and F-distributions are ...... skewed distributions.

28.5 Review Questions

1. Define F distribution and discuss feature of F distribution.

2. Two random samples of sizes 100 and 150 are drawn from two different normal populations.
Find mean and standard error of the statistic F.
3. Suppose you randomly select 8 women from a population of women, and 10 men from a
population of men. The table below shows the standard deviation in each sample and in
each population.

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