The F distribution calculator makes it easy to find the cumulative probability associated with a specified *f* statistic. Or you can find the *f* statistic associated with a specified cumulative probability. For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems.

To learn more about the F distribution, read Stat Trek's tutorial on the F distribution.

## Frequently-Asked Questions

Calculator | Sample Problems

**Instructions:** To find the answer to a frequently-asked question, simply click on the question.

### 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 *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. The F distribution calculator computes two cumulative probabilities:

- Probability (F≤f). This is the cumulative probability that an F statistic is less than or equal to the value
*f*. - Probability (F≥f). This is the cumulative probability that an F statistic is greater than or equal to the value
*f*.

### What is an f statistic?

An *f* statistic (also known as an *f* value) is a random variable that has an F distribution. Here are the steps required to compute an *f* statistic:

- 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*statistic 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.

## Sample Problems

Calculator | Frequently-Asked Questions

- Suppose we take independent random samples of size
*n*_{1}= 11 and*n*_{2}= 16 from normal populations. Find the*f*statistic that has the following cumulative probability: P(F≤f)=0.75.**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. - P(F≤f)=0.75.

Now, we are ready to use the F Distribution Calculator. We enter the degrees of freedom for

*v*_{1}(10), the degrees of freedom for*v*_{2}(15), and the cumulative probability (0.75) into the calculator; and hit the Calculate button.The calculator reports that the

*f*statistic is 1.44907. - Since the sample size

- Suppose we take independent random samples of size
*n*_{1}= 25 and*n*_{2}= 13 from normal populations. What is the probability that the*f*statistic is less than or equal to 2.51?**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 of interest is equal to 2.51.

Now, we are ready to use the F Distribution Calculator. We enter the degrees of freedom for

*v*_{1}(24), the degrees of freedom for*v*_{2}(12), and the*f*statistic (2.51) into the calculator; and hit the Calculate button.The calculator reports that the probability that

*f*is less than or equal to 2.51 is 0.95032. - Since the sample size