4.2 - The F-Distribution | STAT 415 (2024)

Table of Contents
Characteristics of the F-Distribution Section The F-Table Section Example 4-2 Section Answer Answer Answer FAQs

As we'll soon see, the confidence interval for the ratio of two variances requires the use of the probability distribution known as the F-distribution. So, let's spend a few minutes learning the definition and characteristics of the F-distribution.

F-distribution

If U and V are independent chi-square random variables with \(r_1\) and \(r_2\) degrees of freedom, respectively, then:

\(F=\dfrac{U/r_1}{V/r_2}\)

follows an F-distribution with \(r_1\) numerator degrees of freedom and \(r_2\) denominator degrees of freedom. We write F ~ F(\(r_1\), \(r_2\)).

Characteristics of the F-Distribution Section

  1. F-distributions are generally skewed. The shape of an F-distribution depends on the values of \(r_1\) and \(r_2\), the numerator and denominator degrees of freedom, respectively, as this picture pirated from your textbook illustrates:

  2. The probability density function of an F random variable with \(r_1\) numerator degrees of freedom and \(r_2\) denominator degrees of freedom is:

    \(f(w)=\dfrac{(r_1/r_2)^{r_1/2}\Gamma[(r_1+r_2)/2]w^{(r_1/2)-1}}{\Gamma[r_1/2]\Gamma[r_2/2][1+(r_1w/r_2)]^{(r_1+r_2)/2}}\)

    over the support \(w ≥ 0\).

  3. The definition of an F-random variable:

    \(F=\dfrac{U/r_1}{V/r_2}\)

    implies that if the distribution of W is F(\(r_1\), \(r_2\)), then the distribution of 1/W is F(\(r_2\), \(r_1\)).

See Also
F distribution | Properties, proofs, exercises5.11: The F Distribution13.3: The F Distribution and the F-Ratio13.4: Facts About the F Distribution

The F-Table Section

One of the primary ways that we will need to interact with an F-distribution is by needing to know either:

  1. An F-value, or
  2. The probabilities associated with an F-random variable, in order to complete a statistical analysis.

We could go ahead and try to work with the above probability density function to find the necessary values, but I think you'll agree before long that we should just turn to an F-table, and let it do the dirty work for us. For that reason, we'll now explore how to use a typical F-table to look up F-values and/or F-probabilities. Let's start with two definitions.

\(100 \alpha^{th}\) percentile

Let \(\alpha\) be some probability between 0 and 1 (most often, a small probability less than 0.10). The upper \(100 \alpha^{th}\) percentile of an F-distribution with \(r_1\) and \(r_2\) degrees of freedom is the value \(F_\alpha(r_1,r_2)\) such that the area under the curve and to the right of \(F_\alpha(r_1,r_2)\) is \(\alpha\):

The above definition is used in Table VII, the F-distribution table in the back of your textbook. While the next definition is not used directly in Table VII, you'll still find it necessary when looking for F-values (or F-probabilities) in the left tail of an F-distribution.

\(100 \alpha^{th}\) percentile

Let \(\alpha\) be some probability between 0 and 1 (most often, a small probability less than 0.10). The \(100 \alpha^{th}\) percentile of an F-distribution with \(r_1\) and \(r_2\) degrees of freedom is the value \(F_{1-\alpha}(r_1,r_2)\) such that the area under the curve and to the right of \(F_{1-\alpha}(r_1,r_2)\) is 1−\(\alpha\):

See Also
11.2 The F-Distribution – Introduction to Statistics

With the two definitions behind us, let's now take a look at the F-table in the back of your textbook.

In summary, here are the steps you should take in using the F>-table to find an F-value:

  1. Find the column that corresponds to the relevant numerator degrees of freedom, \(r_1\).
  2. Find the three rows that correspond to the relevant denominator degrees of freedom, \(r_2\).
  3. Find the one row, from the group of three rows identified in the second step, that is headed by the probability of interest... whether it's 0.01, 0.025, 0.05.
  4. Determine the F-value where the \(r_1\) column and the probability row identified in step 3 intersect.

Now, at least theoretically, you could also use the F-table to find the probability associated with a particular F-value. But, as you can see, the table is pretty (very!) limited in that direction. For example, if you have an F random variable with 6 numerator degrees of freedom and 2 denominator degrees of freedom, you could only find the probabilities associated with the F values of 19.33, 39.33, and 99.33:

\(P(F ≤ f)\) = \(\displaystyle \int^f_0\dfrac{\Gamma[(r_1+r_2)/2](r_1/r_2)^{r_1/2}w^{(r_1/2)-1}}{\Gamma[r_1/2]\Gamma[r_2/2][1+(r_1w/r_2)]^{(r_1+r_2)/2}}dw\)
\(\alpha\)\(P(F ≤ f)\)Den.
d.f.
\(r_2\)		Numerator Degrees of Freedom, \(r_1\)
12345678
0.05
0.0025
0.01		0.95
0.975
0.99		1161.40
647.74
4052.00		199.50
799.50
4999.50		215.70
864.16
5403.00		224.60
899.58
5625.00		230.20
921.85
5764.00		234.00
937.11
5859.00		236.80
948.22
5928.00		238.90
956.66
5981.00
0.05
0.0025
0.01		0.95
0.975
0.99		218.51
38.51
98.50		19.00
39.00
99.00		19.16
39.17
99.17		19.25
39.25
99.25		19.30
39.30
99.30		19.33
39.33
99.33		19.35
39.36
99.36		19.37
39.37
99.37

What would you do if you wanted to find the probability that an F random variable with 6 numerator degrees of freedom and 2 denominator degrees of freedom was less than 6.2, say? Well, the answer is, of course... statistical software, such as SAS or Minitab! For what we'll be doing, the F table will (mostly) serve our purpose. When it doesn't, we'll use Minitab. At any rate, let's get a bit more practice now using the F table.

Example 4-2 Section

Let X be an F random variable with 4 numerator degrees of freedom and 5 denominator degrees of freedom. What is the upper fifth percentile?

Answer

The upper fifth percentile is the F-value x such that the probability to the right of x is 0.05, and therefore the probability to the left of x is 0.95. To find x using the F-table, we:

  1. Find the column headed by \(r_1 = 4\).
  2. Find the three rows that correspond to \(r_2 = 5\).
  3. Find the one row, from the group of three rows identified in the above step, that is headed by \(\alpha = 0.05\) (and \(P(X ≤ x) = 0.95\).

Now, all we need to do is read the F-value where the \(r_1 = 4\) column and the identified \(\alpha = 0.05\) row intersect. What do you get?

\(P(F ≤ f)\) = \(\displaystyle \int^f_0\dfrac{\Gamma[(r_1+r_2)/2](r_1/r_2)^{r_1/2}w^{(r_1/2)-1}}{\Gamma[r_1/2]\Gamma[r_2/2][1+(r_1w/r_2)]^{(r_1+r_2)/2}}dw\)
\(\alpha\)\(P(F ≤ f)\)Den.
d.f.
\(r_2\)		Numerator Degrees of Freedom, \(r_1\)
12345678
0.05
0.0025
0.01		0.95
0.975
0.99		1161.40
647.74
4052.00		199.50
799.50
4999.50		215.70
864.16
5403.00		224.60
899.58
5625.00		230.20
921.85
5764.00		234.00
937.11
5859.00		236.80
948.22
5928.00		238.90
956.66
5981.00
0.05
0.0025
0.01		0.95
0.975
0.99		218.51
38.51
98.50		19.00
39.00
99.00		19.16
39.17
99.17		19.25
39.25
99.25		19.30
39.30
99.30		19.33
39.33
99.33		19.35
39.36
99.36		19.37
39.37
99.37
0.05
0.0025
0.01		0.95
0.975
0.99		310.13
17.44
34.12		9.55
16.04
30.82		9.28
15.44
29.46		9.12
15.10
28.71		9.01
14.88
28.24		8.94
14.73
27.91		8.89
14.62
27.67		8.85
14.54
27.49
0.05
0.0025
0.01		0.95
0.975
0.99		47.71
12.22
21.20		6.94
10.65
18.00		6.59
9.98
16.69		6.39
9.60
15.98		6.26
9.36
15.52		6.16
9.20
15.21		6.09
9.07
14.98		6.04
8.98
14.80
0.05
0.0025
0.01		0.95
0.975
0.99		56.61
10.01
16.26		5.79
8.43
13.27		5.41
7.76
12.06		5.19
7.39
11.39		5.05
7.15
10.97		4.95
6.98
10.67		4.88
6.85
10.46		4.82
6.76
10.29
0.05
0.0025
0.01		0.95
0.975
0.99		6

5.99
8.81
13.75

5.14
7.26
10.92		4.76
6.60
9.78		4.53
6.23
9.15		4.39
5.99
8.75		4.28
5.82
8.47		4.21
5.70
8.26		4.15
5.60
8.10
\(P(F ≤ f)\) = \(\displaystyle \int^f_0\dfrac{\Gamma[(r_1+r_2)/2](r_1/r_2)^{r_1/2}w^{(r_1/2)-1}}{\Gamma[r_1/2]\Gamma[r_2/2][1+(r_1w/r_2)]^{(r_1+r_2)/2}}dw\)
\(\alpha\)\(P(F ≤ f)\)Den.
d.f.
\(r_2\)		Numerator Degrees of Freedom, \(r_1\)
12345678
0.05
0.0025
0.01		0.95
0.975
0.99		1161.40
647.74
4052.00		199.50
799.50
4999.50		215.70
864.16
5403.00		224.60
899.58
5625.00		230.20
921.85
5764.00		234.00
937.11
5859.00		236.80
948.22
5928.00		238.90
956.66
5981.00
0.05
0.0025
0.01		0.95
0.975
0.99		218.51
38.51
98.50		19.00
39.00
99.00		19.16
39.17
99.17		19.25
39.25
99.25		19.30
39.30
99.30		19.33
39.33
99.33		19.35
39.36
99.36		19.37
39.37
99.37
0.05
0.0025
0.01		0.95
0.975
0.99		310.13
17.44
34.12		9.55
16.04
30.82		9.28
15.44
29.46		9.12
15.10
28.71		9.01
14.88
28.24		8.94
14.73
27.91		8.89
14.62
27.67		8.85
14.54
27.49
0.05
0.0025
0.01		0.95
0.975
0.99		47.71
12.22
21.20		6.94
10.65
18.00		6.59
9.98
16.69		6.39
9.60
15.98		6.26
9.36
15.52		6.16
9.20
15.21		6.09
9.07
14.98		6.04
8.98
14.80
0.05
0.0025
0.01		0.95
0.975
0.99		56.61
10.01
16.26		5.79
8.43
13.27		5.41
7.76
12.06		5.19
7.39
11.39		5.05
7.15
10.97		4.95
6.98
10.67		4.88
6.85
10.46		4.82
6.76
10.29
0.05
0.0025
0.01		0.95
0.975
0.99		6

5.99
8.81
13.75

5.14
7.26
10.92		4.76
6.60
9.78		4.53
6.23
9.15		4.39
5.99
8.75		4.28
5.82
8.47		4.21
5.70
8.26		4.15
5.60
8.10

The table tells us that the upper fifth percentile of an F random variable with 4 numerator degrees of freedom and 5 denominator degrees of freedom is 5.19.

Let X be an F random variable with 4 numerator degrees of freedom and 5 denominator degrees of freedom. What is the first percentile?

Answer

The first percentile is the F-value x such that the probability to the left of x is 0.01 (and hence the probability to the right of x is 0.99). Since such an F-value isn't directly readable from the F-table, we need to do a little finagling to find x using the F-table. That is, we need to recognize that the F-value we are looking for, namely \(F_{0.99}(4,5)\), is related to \(F_{0.01}(5,4)\), a value we can read off of the table by way of this relationship:

\(F_{0.99}(4,5)=\dfrac{1}{F_{0.01}(5,4)}\)

That said, to find x using the F-table, we:

  1. Find the column headed by \(r_1 = 5\).
  2. Find the three rows that correspond to \(r_2 = 4\).
  3. Find the one row, from the group of three rows identified in (2), that is headed by \(\alpha = 0.01\) (and \(P(X ≤ x) = 0.99\).

Now, all we need to do is read the F-value where the \(r_1 = 5\) column and the identified \(\alpha = 0.01\) row intersect, and take the inverse. What do you get?

\(P(F ≤ f)\) = \(\displaystyle \int^f_0\dfrac{\Gamma[(r_1+r_2)/2](r_1/r_2)^{r_1/2}w^{(r_1/2)-1}}{\Gamma[r_1/2]\Gamma[r_2/2][1+(r_1w/r_2)]^{(r_1+r_2)/2}}dw\)
\(\alpha\)\(P(F ≤ f)\)Den.
d.f.
\(r_2\)		Numerator Degrees of Freedom, \(r_1\)
12345678
0.05
0.0025
0.01		0.95
0.975
0.99		1161.40
647.74
4052.00		199.50
799.50
4999.50		215.70
864.16
5403.00		224.60
899.58
5625.00		230.20
921.85
5764.00		234.00
937.11
5859.00		236.80
948.22
5928.00		238.90
956.66
5981.00
0.05
0.0025
0.01		0.95
0.975
0.99		218.51
38.51
98.50		19.00
39.00
99.00		19.16
39.17
99.17		19.25
39.25
99.25		19.30
39.30
99.30		19.33
39.33
99.33		19.35
39.36
99.36		19.37
39.37
99.37
0.05
0.0025
0.01		0.95
0.975
0.99		310.13
17.44
34.12		9.55
16.04
30.82		9.28
15.44
29.46		9.12
15.10
28.71		9.01
14.88
28.24		8.94
14.73
27.91		8.89
14.62
27.67		8.85
14.54
27.49
0.05
0.0025
0.01		0.95
0.975
0.99		47.71
12.22
21.20		6.94
10.65
18.00		6.59
9.98
16.69		6.39
9.60
15.98		6.26
9.36
15.52		6.16
9.20
15.21		6.09
9.07
14.98		6.04
8.98
14.80
0.05
0.0025
0.01		0.95
0.975
0.99		56.61
10.01
16.26		5.79
8.43
13.27		5.41
7.76
12.06		5.19
7.39
11.39		5.05
7.15
10.97		4.95
6.98
10.67		4.88
6.85
10.46		4.82
6.76
10.29
0.05
0.0025
0.01		0.95
0.975
0.99		6

5.99
8.81
13.75

5.14
7.26
10.92		4.76
6.60
9.78		4.53
6.23
9.15		4.39
5.99
8.75		4.28
5.82
8.47		4.21
5.70
8.26		4.15
5.60
8.10
\(P(F ≤ f)\) = \(\displaystyle \int^f_0\dfrac{\Gamma[(r_1+r_2)/2](r_1/r_2)^{r_1/2}w^{(r_1/2)-1}}{\Gamma[r_1/2]\Gamma[r_2/2][1+(r_1w/r_2)]^{(r_1+r_2)/2}}dw\)
\(\alpha\)\(P(F ≤ f)\)Den.
d.f.
\(r_2\)		Numerator Degrees of Freedom, \(r_1\)
12345678
0.05
0.0025
0.01		0.95
0.975
0.99		1161.40
647.74
4052.00		199.50
799.50
4999.50		215.70
864.16
5403.00		224.60
899.58
5625.00		230.20
921.85
5764.00		234.00
937.11
5859.00		236.80
948.22
5928.00		238.90
956.66
5981.00
0.05
0.0025
0.01		0.95
0.975
0.99		218.51
38.51
98.50		19.00
39.00
99.00		19.16
39.17
99.17		19.25
39.25
99.25		19.30
39.30
99.30		19.33
39.33
99.33		19.35
39.36
99.36		19.37
39.37
99.37
0.05
0.0025
0.01		0.95
0.975
0.99		310.13
17.44
34.12		9.55
16.04
30.82		9.28
15.44
29.46		9.12
15.10
28.71		9.01
14.88
28.24		8.94
14.73
27.91		8.89
14.62
27.67		8.85
14.54
27.49
0.05
0.0025
0.01		0.95
0.975
0.99		47.71
12.22
21.20		6.94
10.65
18.00		6.59
9.98
16.69		6.39
9.60
15.98		6.26
9.36
15.52		6.16
9.20
15.21		6.09
9.07
14.98		6.04
8.98
14.80
0.05
0.0025
0.01		0.95
0.975
0.99		56.61
10.01
16.26		5.79
8.43
13.27		5.41
7.76
12.06		5.19
7.39
11.39		5.05
7.15
10.97		4.95
6.98
10.67		4.88
6.85
10.46		4.82
6.76
10.29
0.05
0.0025
0.01		0.95
0.975
0.99		6

5.99
8.81
13.75

5.14
7.26
10.92		4.76
6.60
9.78		4.53
6.23
9.15		4.39
5.99
8.75		4.28
5.82
8.47		4.21
5.70
8.26		4.15
5.60
8.10

The table, along with a minor calculation, tells us that the first percentile of an F random variable with 4 numerator degrees of freedom and 5 denominator degrees of freedom is 1/15.52 = 0.064.

What is the probability that an F random variable with 4 numerator degrees of freedom and 5 denominator degrees of freedom is greater than 7.39?

Answer

There I go... just a minute ago, I said that the F-table isn't very helpful in finding probabilities, then I turn around and ask you to use the table to find a probability! Doing it at least once helps us make sure that we fully understand the table. In this case, we are going to need to read the table "backwards." To find the probability, we:

  1. Find the column headed by \(r_1 = 4\).
  2. Find the three rows that correspond to \(r_2 = 5\).
  3. Find the one row, from the group of three rows identified in the second point above, that contains the value 7.39 in the \(r_1 = 4\) column.
  4. Read the value of \(\alpha\) that heads the row in which the 7.39 falls.

What do you get?

\(P(F ≤ f)\) = \(\displaystyle \int^f_0\dfrac{\Gamma[(r_1+r_2)/2](r_1/r_2)^{r_1/2}w^{(r_1/2)-1}}{\Gamma[r_1/2]\Gamma[r_2/2][1+(r_1w/r_2)]^{(r_1+r_2)/2}}dw\)
\(\alpha\)\(P(F ≤ f)\)Den.
d.f.
\(r_2\)		Numerator Degrees of Freedom, \(r_1\)
12345678
0.05
0.0025
0.01		0.95
0.975
0.99		1161.40
647.74
4052.00		199.50
799.50
4999.50		215.70
864.16
5403.00		224.60
899.58
5625.00		230.20
921.85
5764.00		234.00
937.11
5859.00		236.80
948.22
5928.00		238.90
956.66
5981.00
0.05
0.0025
0.01		0.95
0.975
0.99		218.51
38.51
98.50		19.00
39.00
99.00		19.16
39.17
99.17		19.25
39.25
99.25		19.30
39.30
99.30		19.33
39.33
99.33		19.35
39.36
99.36		19.37
39.37
99.37
0.05
0.0025
0.01		0.95
0.975
0.99		310.13
17.44
34.12		9.55
16.04
30.82		9.28
15.44
29.46		9.12
15.10
28.71		9.01
14.88
28.24		8.94
14.73
27.91		8.89
14.62
27.67		8.85
14.54
27.49
0.05
0.0025
0.01		0.95
0.975
0.99		47.71
12.22
21.20		6.94
10.65
18.00		6.59
9.98
16.69		6.39
9.60
15.98		6.26
9.36
15.52		6.16
9.20
15.21		6.09
9.07
14.98		6.04
8.98
14.80
0.05
0.0025
0.01		0.95
0.975
0.99		56.61
10.01
16.26		5.79
8.43
13.27		5.41
7.76
12.06		5.19
7.39
11.39		5.05
7.15
10.97		4.95
6.98
10.67		4.88
6.85
10.46		4.82
6.76
10.29
0.05
0.0025
0.01		0.95
0.975
0.99		6

5.99
8.81
13.75

5.14
7.26
10.92		4.76
6.60
9.78		4.53
6.23
9.15		4.39
5.99
8.75		4.28
5.82
8.47		4.21
5.70
8.26		4.15
5.60
8.10
\(P(F ≤ f)\) = \(\displaystyle \int^f_0\dfrac{\Gamma[(r_1+r_2)/2](r_1/r_2)^{r_1/2}w^{(r_1/2)-1}}{\Gamma[r_1/2]\Gamma[r_2/2][1+(r_1w/r_2)]^{(r_1+r_2)/2}}dw\)
\(\alpha\)\(P(F ≤ f)\)Den.
d.f.
\(r_2\)		Numerator Degrees of Freedom, \(r_1\)
12345678
0.05
0.0025
0.01		0.95
0.975
0.99		1161.40
647.74
4052.00		199.50
799.50
4999.50		215.70
864.16
5403.00		224.60
899.58
5625.00		230.20
921.85
5764.00		234.00
937.11
5859.00		236.80
948.22
5928.00		238.90
956.66
5981.00
0.05
0.0025
0.01		0.95
0.975
0.99		218.51
38.51
98.50		19.00
39.00
99.00		19.16
39.17
99.17		19.25
39.25
99.25		19.30
39.30
99.30		19.33
39.33
99.33		19.35
39.36
99.36		19.37
39.37
99.37
0.05
0.0025
0.01		0.95
0.975
0.99		310.13
17.44
34.12		9.55
16.04
30.82		9.28
15.44
29.46		9.12
15.10
28.71		9.01
14.88
28.24		8.94
14.73
27.91		8.89
14.62
27.67		8.85
14.54
27.49
0.05
0.0025
0.01		0.95
0.975
0.99		47.71
12.22
21.20		6.94
10.65
18.00		6.59
9.98
16.69		6.39
9.60
15.98		6.26
9.36
15.52		6.16
9.20
15.21		6.09
9.07
14.98		6.04
8.98
14.80
0.05
0.0025
0.01		0.95
0.975
0.99		56.61
10.01
16.26		5.79
8.43
13.27		5.41
7.76
12.06		5.19
7.39
11.39		5.05
7.15
10.97		4.95
6.98
10.67		4.88
6.85
10.46		4.82
6.76
10.29
0.05
0.0025
0.01		0.95
0.975
0.99		6

5.99
8.81
13.75

5.14
7.26
10.92		4.76
6.60
9.78		4.53
6.23
9.15		4.39
5.99
8.75		4.28
5.82
8.47		4.21
5.70
8.26		4.15
5.60
8.10

The table tells us that the probability that an F random variable with 4 numerator degrees of freedom and 5 denominator degrees of freedom is greater than 7.39 is 0.025.

4.2 - The F-Distribution | STAT 415 (2024)

FAQs

How do you solve for F-distribution? ›

The selection of one or two-tailed tests depends upon the problem. In the F-sampling distribution, F is calculated by dividing the variance of one sample by the other sample's variance. For the right-tailed and two-tailed tests, keep the highest variance as the numerator and the lowest variance as the denominator.

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Can F-distribution be greater than 1? ›

Since variances are always positive, if the null hypothesis is false, MSbetween will generally be larger than MSwithin. Then the F-ratio will be larger than one. However, if the population effect is small, it is not unlikely that MSwithin will be larger in a given sample.

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What is the F-distribution level? ›

The F distribution is an asymmetric distribution that has a minimum value of 0, but no maximum value. The curve reaches a peak not far to the right of 0, and then gradually approaches the horizontal axis the larger the F value is.

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How do you find F in a probability distribution? ›

The formulas to find the probability distribution function are as follows:
  1. Discrete distributions: F(x) = ∑xi≤xp(xi) ∑ x i ≤ x p ( x i ) . Here p(x) is the probability mass function.
  2. Continuous distributions: F(x) = ∫x−∞f(u)du ∫ − ∞ x f ( u ) d u . Here f(u) is the probability density function.

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What is F distribution calculator? ›

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.

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What is F distribution used for best answer? ›

The F-distribution can be used for several types of applications, including testing hypotheses about the equality of two population variances and testing the validity of a multiple regression equation.

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How to calculate F value? ›

The formula to calculate the F-statistic, or F-value, is: F = σ 1 σ 2 , or Variance 1/Variance 2. In order to accommodate the skewed right shape of the F-distribution, the larger variance is placed in the numerator and the smaller variance is used in the denominator.

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What is a good F value in ANOVA? ›

If the null hypothesis is true, you expect F to have a value close to 1.0 most of the time. A large F ratio means that the variation among group means is more than you'd expect to see by chance.

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How to interpret F-test results? ›

Result of the F Test (Decided Using F Directly)

If the F value is smaller than the critical value in the F table, then the model is not significant. If the F value is larger, then the model is significant. Remember that the statistical meaning of significant is slightly different from its everyday usage.

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What does F distribution look like? ›

The graph of the F distribution is always positive and skewed right, though the shape can be mounded or exponential depending on the combination of numerator and denominator degrees of freedom.

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When should F distribution be used? ›

The F-distribution has numerous real-world applications. For example, it is used in finance to test whether the variances of stock returns are equal across two or more portfolios. It is also used in engineering to test the effectiveness of different manufacturing processes by comparing the variances of the outcomes.

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What is F distribution called? ›

In probability theory and statistics, the F-distribution or F-ratio, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W.

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How to calculate mean of F-distribution? ›

The F distribution has the following properties:
  1. The mean of the distribution is equal to v2 / ( v2 - 2 ) for v2 > 2.
  2. The variance is equal to [ 2 * v22 * ( v1 + v1 - 2 ) ] / [ v1 * ( v2 - 2 )2 * ( v2 - 4 ) ] for v2 > 4.

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Is the F-distribution positive or negative? ›

The F-distribution cannot take negative values, because it is a ratio of variances and variances are always non-negative numbers. The distribution represents the ratio between the variance between groups and the variance within groups.

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What is the formula for the F-distribution test? ›

The f test statistic formula is given below: F statistic for large samples: F = σ21σ22 σ 1 2 σ 2 2 , where σ21 σ 1 2 is the variance of the first population and σ22 σ 2 2 is the variance of the second population.

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What is the formula for the F test distribution? ›

F-Distribution Formula

The formula to calculate the F-statistic, or F-value, is: F = σ 1 σ 2 , or Variance 1/Variance 2. In order to accommodate the skewed right shape of the F-distribution, the larger variance is placed in the numerator and the smaller variance is used in the denominator.

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How do you calculate the F value? ›

State the null hypothesis and the alternate hypothesis. Calculate the F value. The F Value is calculated using the formula F = (SSE1 – SSE2 / m) / SSE2 / n-k, where SSE = residual sum of squares, m = number of restrictions and k = number of independent variables.

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What is the formula for the F ratio? ›

We calculate the F-ratio by dividing the Mean of Squares Between (MSB) by the Mean of Squares Within (MSW).

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How do you find the distribution function? ›

The cumulative distribution function (cdf) gives the probability that the random variable X is less than or equal to x and is usually denoted F(x) . The cumulative distribution function of a random variable X is the function given by F(x)=P[X≤x]. F ( x ) = P [ X ≤ x ] .

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