F Distribution Calculator: Online Statistical Table
The F distribution calculator makes it easy to find the cumulative
probability associated with an f value.
For help in using the calculator, read the
FrequentlyAsked Questions
or review the
Sample Problems.
To learn more about the F distribution, read Stat Trek's
tutorial on the F distribution.
Instructions: To find the answer to a frequentlyasked
question, simply click on the question. If you don't see the answer you need,
read Stat Trek's tutorial on the F
distribution or visit the
Statistics Glossary.
What are degrees of freedom?
Degrees of freedom can be described as the number of scores that
are free to vary. For example, suppose your friend tossed three dice, and the
total score added up to 12. If your friend told you that he rolled a 3 on the
first die and a 5 on the second, then you know that the third die must be a 4
(otherwise, the total would not add up to 12). In this example, 2 die are free
to vary while the third is not. Therefore, there are 2 degrees of freedom.
In many situations, the degrees of freedom are equal to the
number of observations minus one. Thus, if the sample size were 20, there would
be 20 observations; and the degrees of freedom would be 20 minus 1 or 19.
What are degrees of freedom (v_{1})
and degrees of freedom (v_{2}) ?
You can use the following equation to compute an f statistic:
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.
The degrees of freedom (v_{1}) refers to the
degrees of freedom associated with the sample standard deviation s_{1}
in the numerator; and the degrees of freedom (v_{2}) refers to
the degrees of freedom associated with the sample standard deviation s_{2}
in the denominator.
What is a cumulative probability?
A cumulative probability is a sum of probabilities. In connection
with the F distribution calculator,
cumulative probability refers to the probability
that an f statistic will be less than or equal to a specified value.
What is an f value?
An f value (also known as an f statistic) is a
random variable that has an
F distribution.
Here are the steps required to compute an f value:

Select a random sample of size n_{1} from a normal population,
having a standard deviation equal to σ_{1}.

Select an independent random sample of size n_{2} from a normal
population, having a standard deviation equal to σ_{2}.

The f value is the ratio of s_{1}^{2}/σ_{1}^{2}
and s_{2}^{2}/σ_{2}^{2}. Thus,
f = [ s_{1}^{2}/σ_{1}^{2}
] / [ s_{2}^{2}/σ_{2}^{2}]
What is a probability?
A probability is a number expressing the chances that a specific
event will occur. This number can take on any value from 0 to 1. A probability
of 0 means that there is zero chance that the event will occur; a probability
of 1 means that the event is certain to occur. Numbers between 0 and 1 quantify
the uncertainty associated with the event. For example, the probability of a
coin flip resulting in Heads (rather than Tails) would be 0.50. Fifty percent
of the time, the coin flip would result in Heads; and fifty percent of the
time, it would result in Tails.

Suppose we take independent random samples of size n_{1}
= 11 and n_{2} = 16 from normal populations. If the cumulative
probability of the f statistic is equal to 0.75, what is the value of
the f statistic?
Solution:
We know the following:
 Since the sample size n_{1} = 11, the degrees of freedom v_{1}
= n_{1}  1 = 10.
 And since the sample size n_{2} = 16, the degrees of freedom v_{2}
= n_{2}  1 = 15.
 The cumulative probability is equal to 0.75.
Now, we are ready to use the F Distribution
Calculator. We enter the degrees of freedom (v_{1} =
10), the degrees of freedom (v_{2} = 15), and the cumulative
probability (0.75) into the calculator ; and hit the Calculate button. The
calculator reports that the f value is 1.45.

Suppose we take independent random samples of size n_{1}
= 25 and n_{2} = 13 from normal populations. If the f statistic
(aka, f value) is equal to 2.51, what is the cumulative probability of
the f statistic?
Solution:
We know the following:
 Since the sample size n_{1} = 25, the degrees of freedom v_{1}
= n_{1}  1 = 24.
 And since the sample size n_{2} = 13, the degrees of freedom v_{2}
= n_{2}  1 = 12.
 The f statistic is equal to 2.51.
Now, we are ready to use the F Distribution
Calculator. We enter the degrees of freedom (v_{1} =
24), the degrees of freedom (v_{2} = 12), and the f value
(2.51) into the calculator; and hit the Calculate button. The calculator
reports that the cumulative probability is 0.95.