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:

• 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 statistic is the ratio of s₁²/σ₁² and s₂²/σ₂².

The following equivalent equations are commonly used to compute an f statistic:

f = [ s₁²/σ₁² ] / [ s₂²/σ₂² ]

f = [ s₁² * σ₂² ] / [ s₂² * σ₁² ]

f = [ Χ²₁ / v₁ ] / [ Χ²₂ / v₂ ]

f = [ Χ²₁ * v₂ ] / [ Χ²₂ * v₁ ]

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, s₂ is the standard deviation of the sample drawn from population 2, Χ²₁ is the chi-square statistic for the sample drawn from population 1, v₁ is the degrees of freedom for Χ²₁, Χ²₂ is the chi-square statistic for the sample drawn from population 2, and v₂ is the degrees of freedom for Χ²₂ . Note that degrees of freedom v₁ = n₁ - 1, and degrees of freedom v₂ = n₂ - 1 .

The F Distribution

The distribution of all possible values of the f statistic is called an F distribution, with v₁ = n₁ - 1 and v₂ = n₂ - 1 degrees of freedom.

The curve of the F distribution depends on the degrees of freedom, v₁ and v₂. 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 v₁ = 5 and v₂ = 9 degrees of freedom; whereas f(9, 5) would refer to an F distribution with v₁ = 9 and v₂ = 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 mean of the distribution is equal to v₂ / ( v₂ - 2 ) for v₂ > 2.
• The variance is equal to [ 2 * v₂² * ( v₁ + v₁ - 2 ) ] / [ v₁ * ( v₂ - 2 )² * ( v₂ - 4 ) ] for v₂ > 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, v₁ and v₂. Notationally, the degrees of freedom appear in parentheses as follows: f_α(v₁,v₂). Thus, f_0.05(5, 7) refers to value of the f statistic having a cumulative probability of 0.95, v₁ = 5 degrees of freedom, and v₂ = 7 degrees of freedom.

How does one find the probability associated with a particular f statistic? Many statistics texts include a table of probabilities for the F distribution. And many graphing calculators can compute f statistics. On this website, we use Stat Trek's F Distribution Calculator, as illustrated below in Problem 2.

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 = [ s₁²/σ₁² ] / [ s₂²/σ₂² ]

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.

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 = ( 35² / 30² ) / ( 45² / 50² )

f = (1225 / 900) / (2025 / 2500)

f = 1.361 / 0.81 = 1.68

For this calculation, the numerator degrees of freedom v₁ are 7 - 1 or 6; and the denominator degrees of freedom v₂ 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 = ( 45² / 50² ) / ( 35² / 30² )

f = (2025 / 2500) / (1225 / 900)

f = 0.81 / 1.361 = 0.595

For this calculation, the numerator degrees of freedom v₁ are 12 - 1 or 11; and the denominator degrees of freedom v₂ are 7 - 1 or 6.

When you are trying to find the cumulative probability associated with an f statistic, you need to know v₁ and v₂. 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.

Here are the degrees of freedom for each sample.

• The degrees of freedom for the sample of women is equal to n - 1 = 7 - 1 = 6.
• 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 v₁ is equal to 6; and the denominator degrees of freedom v₂ 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 about 0.78.

When the men's data appear in the numerator, the numerator degrees of freedom v₁ is equal to 11; and the denominator degrees of freedom v₂ is equal to 6. And, based on the computations shown in the previous example, the f statistic is equal to 0.595. If you plug these values into the F Distribution Calculator, you will find that the cumulative probability is about 0.22.