The F - Distribution

The F Distribution

Named in honor of R.A. Fisher who studied it in 1924.

Used for comparing the variances of two populations.

It is defined in terms of ratio of the variances of two normally distributed populations. So, it sometimes also called variance ratio.

F–distribution: where

22

22

21

21

s

s

Degrees of freedom: v1 = n1 – 1 , v2 = n2 – 1

For different values of v1 and v2 we will get different distributions, so v1 and v2 are parameters of F distribution.

If σ12 = σ2

2, then , the statistic F = s12/s2

2 follows F distribution with n1 – 1 and n2 – 1 degrees of freedom.

1

1

2

2

2222

1

2

1121

n

xxs

n

xxs

Probability density function

Probability density function of F-distribution:

where Y0 is a constant depending on the values of v1 and v2 such that the area under the curve is unity.

21 21

2

1

121

FYFf o

Properties of F-distribution

It is positively skewed and its skewness decreases with increase in v1 and v2.

Value of F must always be positive or zero, since variances are squares. So its value lies between 0 and ∞.

Mean and variance of F-distribution:

Mean = v2/(-v2-2), for v2 > 2

Variance = 2v22(v1+v2-2) , for v2 > 4

v1(v2-2)2(v2-4) Shape of F-distribution depends upon the

number of degrees of freedom.

Testing of hypothesis for equality of two variances

It is based on the variances in two independently selected random samples drawn from two normal populations. Null hypothesis Ho: σ1

2 = σ22

F = s12/σ1

2 , which reduces to F = s12

s22/σ2

2 s22

Degrees of freedom v1 and v2.

Find table value using v1 and v2. If calculated F value exceeds table F value, null hypothesis is rejected.

Problem 1

Two sources of raw materials are under consideration by a company. Both sources seem to have similar characteristics but the company is not sure about their respective uniformity.

A sample of 10 lots from source A yields a variance of 225 and a sample of 11 lots from source B yields a variance of 200. Is it likely that the variance of source A is significantly greater than the variance of source B at α = 0.01 ?

Solution

Null hypothesis Ho: σ12 = σ2

2 i.e. the variances of source A and that of source B are the same. The F statistic to be used here is

F = s12 / s2

2

where s12 = 225 and s2

2 = 200

F = 225/200 = 1.1

Value for v1=9 and v2=10 at 1% level of significance is 4.49. Since computed value of F is smaller than the table value of F, the null hypothesis is accepted.

Hence the variances of two populations are same.

Problem 2

In order to test the belief that incomes earned by college graduates show much greater variability than the earnings of those who did not attend college. A sample of 21 college graduates is taken and the sample std. deviation for the sample is s1 = Rs 17,000. For the second sample of 25 non-graduates and obtains a std dev. s2 = Rs 7,500.

s1 = Rs 17,000 n1 = 21

s1 = Rs 7,500 n2 = 25

Solution

Null hypothesis Ho: σ12 = σ2

2 or (σ12 / σ2

2 = 1)

Alternative hypothesis H1: σ12 > σ2

2

Level of significance α = 0.01

F = s12 / s2

2

F = (17,000)2 / (7,500)2

F = 5.14

Value for v1=20 and v2=24 at 1% level of significance is 2.74. Since computed value of F is greater than the table value of F, the null hypothesis is rejected.