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

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

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:
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 \(w ≥ 0\).
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(\(r_2\), \(r_1\)).

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:

An F-value, or
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−\(\alpha\):

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:

Find the column that corresponds to the relevant numerator degrees of freedom, \(r_1\).
Find the three rows that correspond to the relevant denominator degrees of freedom, \(r_2\).
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.
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, \(r_1\)
\(\alpha\)	\(P(F ≤ f)\)	Den. d.f. \(r_2\)	1	2	3	4	5	6	7	8
0.05 0.0025 0.01	0.95 0.975 0.99	1	161.40 647.74 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	2	18.51 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:

Find the column headed by \(r_1 = 4\).
Find the three rows that correspond to \(r_2 = 5\).
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, \(r_1\)
\(\alpha\)	\(P(F ≤ f)\)	Den. d.f. \(r_2\)	1	2	3	4	5	6	7	8
0.05 0.0025 0.01	0.95 0.975 0.99	1	161.40 647.74 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	2	18.51 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	3	10.13 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	4	7.71 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	5	6.61 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, \(r_1\)
\(\alpha\)	\(P(F ≤ f)\)	Den. d.f. \(r_2\)	1	2	3	4	5	6	7	8
0.05 0.0025 0.01	0.95 0.975 0.99	1	161.40 647.74 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	2	18.51 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	3	10.13 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	4	7.71 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	5	6.61 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:

Find the column headed by \(r_1 = 5\).
Find the three rows that correspond to \(r_2 = 4\).
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, \(r_1\)
\(\alpha\)	\(P(F ≤ f)\)	Den. d.f. \(r_2\)	1	2	3	4	5	6	7	8
0.05 0.0025 0.01	0.95 0.975 0.99	1	161.40 647.74 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	2	18.51 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	3	10.13 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	4	7.71 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	5	6.61 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, \(r_1\)
\(\alpha\)	\(P(F ≤ f)\)	Den. d.f. \(r_2\)	1	2	3	4	5	6	7	8
0.05 0.0025 0.01	0.95 0.975 0.99	1	161.40 647.74 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	2	18.51 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	3	10.13 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	4	7.71 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	5	6.61 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:

Find the column headed by \(r_1 = 4\).
Find the three rows that correspond to \(r_2 = 5\).
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.
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, \(r_1\)
\(\alpha\)	\(P(F ≤ f)\)	Den. d.f. \(r_2\)	1	2	3	4	5	6	7	8
0.05 0.0025 0.01	0.95 0.975 0.99	1	161.40 647.74 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	2	18.51 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	3	10.13 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	4	7.71 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	5	6.61 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, \(r_1\)
\(\alpha\)	\(P(F ≤ f)\)	Den. d.f. \(r_2\)	1	2	3	4	5	6	7	8
0.05 0.0025 0.01	0.95 0.975 0.99	1	161.40 647.74 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	2	18.51 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	3	10.13 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	4	7.71 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	5	6.61 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

What does the F-distribution tell you? ›

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 does F mean in probability distribution? ›

Definition. 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].

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Can F-distribution be greater than 1? ›

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

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What is the limit of the F-distribution? ›

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. The F distribution approaches, but never quite touches the horizontal axis.

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How do you interpret the 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 is the significance level of the F distribution? ›

If the F-test statistic is greater than or equal to 2.92, our results are statistically significant. The probability distribution plot below displays this graphically. The shaded area is the probability of F-values falling within the rejection region of the F-distribution when the null hypothesis is true.

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What does F value mean in statistics? ›

An F-value is the ratio of two variances, or technically, two mean squares. Mean squares are simply variances that account for the degrees of freedom (DF) used to estimate the variance. F-values are the test statistic for F-tests. Learn more about Test Statistics.

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What is F in normal distribution? ›

The normal distribution is defined as the probability density function f(x) for the continuous random variable, say x, in the system.

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What does a large value for the F ratio indicate? ›

A large F value indicates that the differences in group means are substantially greater than the variability within each group, suggesting that the observed differences are unlikely to be due to chance alone.

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What is the range of the F-distribution? ›

The F-distribution curve is positively skewed towards the right with a range of 0 and ∞. The value of F is always positive or zero. No negative values. The shape of the distribution depends on the degrees of freedom of numerator ϑ1 and denominator ϑ2.

What if the F-test is significant? ›

The F-test of overall significance is the hypothesis test for this relationship. If the overall F-test is significant, you can conclude that R-squared does not equal zero, and the correlation between the model and dependent variable is statistically significant.

What is F distribution in simple words? ›

The F-Distribution is a probability distribution that is commonly used in statistical analysis. It arises when comparing the variances of two normal populations.

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When should you use F distribution? ›

The F distribution is used in many cases for the critical regions for hypothesis tests and in determining confidence intervals. Two common examples are the analysis of variance and the F test to determine if the variances of two populations are equal.

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What is the conclusion of the F distribution? ›

If the F statistic is larger than the critical value from the F distribution, the null hypothesis is rejected and it can be concluded that there is a significant difference between the means of the groups.

How do you interpret the F-test results in Excel? ›

Decoding the F-Test Output in Excel

F Statistic: Think of it as the ratio of variance between two datasets. A higher value indicates a significant difference in variances, suggesting that not all samples come from populations with the same variance.

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What is the significance F in a regression? ›

F is a test for statistical significance of the regression equation as a whole. It is obtained by dividing the explained variance by the unexplained variance. By rule of thumb, an F-value of greater than 4.0 is usually statistically significant but you must consult an F-table to be sure.

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When interpreting F 2 27 8.80 p .05, what would you conclude? ›

Answer and Explanation:

We know that, If P-value is less than a level of significance then we reject the null hypothesis. So, We reject the null hypothesis and conclude that there is a significant difference between the groups.

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