# F Distribution Calculator

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

• Enter values for degrees of freedom.
• Enter a value for one, and only one, of the remaining text boxes.
• Click the Calculate button to compute a value for the blank text box.
 Degrees of freedom (v1) Degrees of freedom (v2) Cumulative prob:P(F < f) f  value

Instructions: To find the answer to a frequently-asked 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 (v1) and (v2)?

You can use the following equation to compute an f statistic:

f = [ s1212 ] / [ s2222 ]

where σ1 is the standard deviation of population 1, s1 is the standard deviation of the sample drawn from population 1, σ2 is the standard deviation of population 2, and s1 is the standard deviation of the sample drawn from population 2.

The degrees of freedom (v1) refers to the degrees of freedom associated with the sample standard deviation s1 in the numerator; and the degrees of freedom (v2) refers to the degrees of freedom associated with the sample standard deviation s2 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 n1 from a normal population, having a standard deviation equal to σ1.
• Select an independent random sample of size n2 from a normal population, having a standard deviation equal to σ2.
• The f value is the ratio of s1212 and s2222. Thus, f = [ s1212 ] / [ s2222]

### 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

1. Suppose we take independent random samples of size n1 = 11 and n2 = 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 n1 = 11, the degrees of freedom v1 = n1 - 1 = 10.
• And since the sample size n2 = 16, the degrees of freedom v2 = n2 - 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 (v1 = 10), the degrees of freedom (v2 = 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.

1. Suppose we take independent random samples of size n1 = 25 and n2 = 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 n1 = 25, the degrees of freedom v1 = n1 - 1 = 24.
• And since the sample size n2 = 13, the degrees of freedom v2 = n2 - 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 (v1 = 24), the degrees of freedom (v2 = 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.