F Distribution
The F distribution is the probability distribution associated with the f statistic. In this
lesson, we show how to compute an f statistic and how to find probabilities
associated with specific f statistic values.
The f Statistic
The f statistic, also known as an f value,
is a random variable
that has an F distribution. (We discuss the F distribution in the next
section.)
Here are the steps required to compute an f statistic:

The f statistic is the ratio of s_{1}^{2}/σ_{1}^{2}
and s_{2}^{2}/σ_{2}^{2}.
The following equivalent equations are commonly used to compute an f statistic:
f = [ s_{1}^{2}/σ_{1}^{2}
] / [ s_{2}^{2}/σ_{2}^{2}
]
f = [ s_{1}^{2} * σ_{2}^{2}
] / [ s_{2}^{2} * σ_{1}^{2}
]
f = [ Χ^{2}_{1}
/ v_{1} ] / [ Χ^{2}_{2}
/ v_{2} ]
f = [ Χ^{2}_{1}
* v_{2} ] / [ Χ^{2}_{2}
* v_{1} ]
where σ_{1} is the standard deviation of
population 1, s_{1} is the standard deviation of the sample
drawn from population 1, σ_{2} is the
standard deviation of population 2, s_{2} is the standard
deviation of the sample drawn from population 2, Χ^{2}_{1} is the
chisquare statistic for the sample drawn from population 1, v_{1}
is the degrees of
freedom for Χ^{2}_{1},
Χ^{2}_{2} is
the chisquare statistic for the sample drawn from population 2, and v_{2}
is the degrees of freedom for Χ^{2}_{2}
. Note that degrees of freedom v_{1} = n_{1}  1,
and degrees of freedom v_{2} = n_{2}  1 .
The F Distribution
The distribution of all possible values of the f statistic is called an
F distribution, with v_{1} = n_{1} 
1 and v_{2} = n_{2}  1 degrees of freedom.
The curve of the F distribution depends on the degrees of freedom, v_{1}
and v_{2}. When describing an F distribution, the number of
degrees of freedom associated with the standard deviation in the numerator of
the f statistic is always stated first. Thus, f(5, 9) would refer
to an F distribution with v_{1} = 5 and v_{2} = 9
degrees of freedom; whereas f(9, 5) would refer to an F distribution
with v_{1} = 9 and v_{2} = 5 degrees of freedom.
Note that the curve represented by f(5, 9) would differ from the curve
represented by f(9, 5).
The F distribution has the following properties:

The variance is equal to
[ 2 * v_{2}^{2} *
( v_{1} + v_{1}  2 ) ] /
[ v_{1} * ( v_{2}  2 )^{2} * ( v_{2}  4 ) ]
for v_{2} > 4.
Cumulative Probability and the F Distribution
Every f statistic can be associated with a unique
cumulative probability. This cumulative probability represents the
likelihood that the f statistic is less than or equal to a specified
value.
Statisticians use f_{α} to
represent the value of an f statistic having a cumulative
probability of (1  α).
For example, suppose we were interested in the f statistic having
a cumulative probability of 0.95.
We would refer to that f statistic as
f_{0.05}, since (1  0.95) = 0.05.
Of course, to find the value of f_{α},
we would need to know the degrees of freedom,
v_{1} and v_{2}.
Notationally, the degrees of freedom appear in parentheses as
follows:
f_{α}(v_{1},v_{2}).
Thus, f_{0.05}(5, 7) refers to value of the f statistic having
a cumulative probability of 0.95, v_{1} = 5
degrees of freedom, and v_{2} = 7 degrees of freedom.
The easiest way to find the value of a
particular f statistic is to use the F
Distribution Calculator.
F Distribution Calculator
The F Distribution Calculator solves common statistics problems, based on the F
distribution. The calculator computes cumulative probabilities, based on simple
inputs. Clear instructions guide you to an accurate solution, quickly and
easily. If anything is unclear, frequentlyasked questions and sample problems
provide straightforward explanations. The
calculator is free. It can found in the Stat Trek
main menu under the Stat Tools tab. Or you can tap the button below.
F Distribution Calculator
The use of the F Distribution Calculator is illustrated in Example 2 below.
Test Your Understanding
Problem 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:
f = [ s_{1}^{2}/σ_{1}^{2}
] / [ s_{2}^{2}/σ_{2}^{2}
]
where σ_{1} is the standard deviation of
population 1, s_{1} is the standard deviation of the sample
drawn from population 1, σ_{2} is the
standard deviation of population 2, and s_{1} is the standard
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:
f = ( 35^{2} / 30^{2} ) / ( 45^{2} / 50^{2} )
f = (1225 / 900) / (2025 / 2500)
f = 1.361 / 0.81 = 1.68
For this calculation, the numerator degrees of freedom
v_{1} are 7  1 or 6;
and the denominator degrees of freedom
v_{2} are 12  1 or 11.
On the other hand, if the men's data appears in the numerator, we can calculate
an f statistic as follows:
f = ( 45^{2} / 50^{2} ) / ( 35^{2} / 30^{2} )
f = (2025 / 2500) / (1225 / 900)
f = 0.81 / 1.361 = 0.595
For this calculation, the numerator degrees of freedom
v_{1} are 12  1 or 11;
and the denominator degrees of freedom
v_{2} are 7  1 or 6.
When you are trying to find the cumulative probability associated with
an f statistic, you need to know v_{1} and
v_{2}.
This point is illustrated in the next example.
Problem 2
Find the cumulative probability associated with each of the
f statistics from Example 1, above.
Solution: To solve this problem, we need to find the degrees of
freedom for each sample. Then, we will use the
F Distribution Calculator
to find the probabilities.
 The degrees of freedom for the sample of men is equal to
n  1 = 12  1 = 11.
Therefore, when the women's data appear in the numerator, the
numerator degrees of freedom v_{1} is equal to 6;
and the denominator degrees of freedom v_{2} is equal
to 11.
And, based on the computations shown in the previous example,
the f statistic is equal to 1.68. We plug these
values into the F Distribution Calculator and find that the cumulative
probability is 0.78.
On the other hand, when the men's data appear in the numerator,
the numerator degrees of freedom v_{1} is equal to 11;
and the denominator degrees of freedom v_{2}
is equal to 6.
And, based on the computations shown in the previous example,
the f statistic is equal to 0.595.
We plug these values into the F Distribution Calculator and find
that the cumulative probability is 0.22.