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was provided by George W. Snedecor. Snedecor's
Statistical Methods,
first
published in 1937, was probably the most influential statistics textbook of
the times. Snedecor taught at Iowa State University from 1913 to 1958, and
for much of that time he served as the director of its statistics laboratory.
Under his leadership, Iowa State became one of the leading statistics
centers in the world. R. A. Fisher was a somewhat regular summer visitor
to Iowa State during the 1930s (Salsburg, 2001). Snedecor held Fisher
in high regard, and Snedecor's prestige within the statistical community
led to the acceptance of the name he suggested in 1934 for the ratio (see
Snedecor, 1934).
As for the name of this ratio, Snedecor (1946, p. 219) tells us that
“Fisher and Yates [1938-1943] designate
F
as the
variance ratio,
while
Mahalanobis [1932], who first calculated it, called it
x.
”Butthemore
complete story as told by Snedecor (1946, p. 219) is as follows:
One null hypothesis that can be tested is that all of the [sets of scores] are
random samples from the same normal population....It is necessary to next
learn the magnitude of variation ordinarily encountered in such ratios. The
ratioofthetwoestimatesofvariance...hasadistributiondiscoveredbyFisher.
Inamedit
F
in his honor [Snedecor, 1934].
4.2 THE SAMPLING DISTRIBUTION OF
F
4.2.1 OBTAINING THE SAMPLING DISTRIBUTION OF
F
The
F
ratio is computed as the ratio of two variance estimates. Let's
build a hypothetical illustration to show the sampling distribution of
this ratio, not worrying, for the moment, about distinguishing between-
groups variance from within-groups variance. Instead, using the null
hypothesis as our base, we can decide on a sample size and randomly
draw two samples of values from the same population. We then compute
the variances of each sample and divide the variance of the first sample
by the variance of the second sample. We repeat this process an infinite
number of times. We then plot these ratios as a frequency distribution.
This plot from our hypothetical procedure is the
sampling distribution
of the
F
ratio for samples of this given size. Although the distribution is
slightly different for different sample sizes, the plot shown in Figure 4.1 is
representative of what the general shape looks like.
4.2.2 THE SHAPE OF THE SAMPLING DISTRIBUTION OF
F
As shown in Figure 4.1, the sampling distribution of
F
is positively skewed -
the bulk of scores are relatively lower values that are represented toward the
left portion of the
x
axis with the tail “pointing” to the right. Here is why it
is positively skewed. If the larger variance is in the denominator, then the
value that results from the division will be less than one. Although there
may be a theoretically infinite number of such values, with increasingly
