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
11.2 The F-Distribution – Introduction to StatisticsF distribution | Properties, proofs, exercises5.11: The F Distribution13.4: 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
F-Distribution Definition & Examples - Quickonomics

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? ›

F Distribution
  1. Select a random sample of size n1 from a normal population, having a standard deviation equal to σ1.
  2. Select an independent random sample of size n2 from a normal population, having a standard deviation equal to σ2.
  3. The f statistic is the ratio of s12/σ12 and s22/σ22.

View Details ›
How to solve F-test? ›

In order to carry out the F-test, the statistics for the mean, standard deviation, and variance of the two populations must be calculated. The F-statistic is then derived by taking the ratio of the variances for the two populations. The variance is calculated by squaring the standard deviation for each sample.

Learn More Now ›
How to 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. Find the F Statistic (the critical value for this test).

Read On ›
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 F distribution calculator? ›

F Distribution Calculator is a free online tool that displays the f value for the given f-distribution. BYJU'S online F distribution calculator tool makes the calculation faster and it displays the f value in a fraction of seconds.

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What is the F distribution? ›

The F-distribution was developed by Fisher to study the behavior of two variances from random samples taken from two independent normal populations. In applied problems we may be interested in knowing whether the population variances are equal or not, based on the response of the random samples.

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What is an F-test for dummies? ›

F test is a statistical test that is used in hypothesis testing to check whether the variances of two populations or two samples are equal or not. In an f test, the data follows an f distribution. This test uses the f statistic to compare two variances by dividing them.

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What is a good F value? ›

A general rule of thumb that is often used in regression analysis is that if F > 2.5 then we can reject the null hypothesis. We would conclude that there is a least one parameter value that is nonzero.

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What is the F score in statistics? ›

The F-statistic is simply a ratio of two variances. Variances are a measure of dispersion, or how far the data are scattered from the mean. Larger values represent greater dispersion. Variance is the square of the standard deviation.

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How to interpret a F-test? ›

Interpreting F-test Results

Interpreting the results of an F-test is integral to comprehending the analysis' implications. The outcome hinges on comparing the calculated F-value with the critical F-value derived from tables or statistical software, based on a chosen significance level, usually 0.05 or 5%.

Tell Me More ›

When should you use F-distribution? ›

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 does a low F value mean? ›

Low F-value graph: The group means cluster together more tightly than the within-group variability. The distance between the means is small relative to the random error within each group. You can't conclude that these groups are truly different at the population level.

Keep Reading ›
What is the F ratio and how to calculate it? ›

We calculate the F-ratio by dividing the Mean of Squares Between (MSB) by the Mean of Squares Within (MSW). The calculated F-ratio is then compared to the F-value obtained from an F-table with the corresponding alpha.

Discover More Details ›
What is the formula for the F test distribution? ›

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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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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How to calculate F-distribution table? ›

The F-distribution results from the quotient of two chi-square distributions which are divided by the respective degrees of freedom. Here you can either calculate the critical F-value or the p-value with given degrees of freedom. You can also read the critical F-value for a given alpha level in the tables below.

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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).

Read More ›
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