http://www.ck12.org Chapter 11. Analysis of Variance and the F-Distribution
In this lesson we will examine the difference between theF- and Student’st-distributions, calculate the test statistic
and test hypotheses about multiple population variances. In addition, we will look a bit more closely at the limitations
of this test.
Differences between the F- and Student’s t-Distributions
As review, we use the Student’st-distribution when we are conducting hypotheses tests where the variance of the
population is unknown. Usually, the variance of the population isnotknown and it is necessary to estimate it by
using the variance of the sample. Using the variance of a sample to estimate population variance can be inappropriate
- especially if we have a small sample size. For estimating the population variance from a small sample we use a
statistical tool called theStudent’st-distribution.
The Student’st-distribution is a family of distributions that, like the normal distribution, are symmetrical, bell-
shaped and centered on the mean. The shape of these distributions changes as the sample sizes changes (see below)
and eacht-distribution is associated with a unique number of Degrees of Freedom (number of observations in the
sample minus one). As the number of observations (shown bykin the figure) increases, the difference between the
t-distribution and the normal distribution (in pink) decreases.
TheF-distribution is quite a bit different. When we test the hypothesis that two variances in the populations from
which random samples were selected are equal(H 0 :σ 12 =σ 22 )(or in other words that the ratio of the variances
(σ 12 )/(σ 22 )equals 1.00), we call this test theF-Max test.
Since we are testing ratios, theF-distribution looks quite different from the Student’st-distribution (see below). Like
the Student’st-distribution, theF-distribution is a family of distributions. The specificF-distribution for testing two
population variancesH 0 :σ 12 =σ 22 is based on two Degrees of Freedom (one for each of the populations). Unlike the
normal and thet-distributions, theF-distributions are not symmetrical and span only non-negative numbers (unlike
others that are symmetric and have both positive and negative values.) In addition, the shapes of theF-distribution
vary drastically, especially when the degrees of freedom values are small. These characteristics make determining
the critical values for theF-distribution more complicated than for the normal and Student’st-distributions.