MATH 2441
Probability and Statistics for Biological Sciences
The F-Distribution
The so-called F-distribution is the fourth (and last) major continuous probability distribution that arises in the
statistical inference methods that we study in this course. It joins the normal distribution, t-distribution, and
2-distribution in our list of standard probability distributions that describe commonly arising situations.
Almost all of the physical, chemical and biological quantities we measure directly are random variables.
Many of these random variables are found to be approximately normally distributed (or, when sample sizes
are small, for example, the similarly shaped t-distribution applies).
Mathematical expressions involving random variables are themselves random variables, and thus are
associated with probability distributions of their own. Linear combinations of approximately normallydistributed random variables (for example, sums and differences) also tend to be approximately normally
distributed under appropriate conditions. Products or squares of approximately normally-distributed random
variables tend to have the 2-distribution. Hence, to construct confidence interval estimates for population
variances, or to test hypotheses involving population variances, we had to use the 2-distribution because
variances are essentially sums of squares of the original data values. We will find when we look at
goodness-of-fit and related issues later in the course that the 2-distribution is again involved, because those
analyses involve the calculation of squares of proportions.
Ratios of 2-distributed random variables (such
as variances) have another type of probability
distribution  the F-distribution. Like the 2distribution, the F-distribution is not symmetric,
and is defined only for F > 0. In fact, like both
the t-distribution and the 2-distribution, there is
a whole family of F-distributions, distinguished
by two parameters, 1 (called the numerator
degrees of freedom) and 2 (called the
denominator degrees of freedom). These two
F
parameters have whole number values greater
than or equal to 1. The shape of a typical Fdistribution is shown in the figure to the right. It has a peak very near 1, and then a long tail to the right. The
mean value of an F-distributed random variable is

2
2  2
 2
 2
and so the location of this peak will tend to stay in the vicinity of F = 1, regardless of the values of 1 and 2.
In this course, the F-distribution arises in connection with test statistics that are quotients of two variances,
hence the terminology numerator degrees of freedom and denominator degrees of freedom in identifying the
parameters 1 and 2. In the days when statisticians had to rely on printed probability tables, it was much
less usual to have to calculate F-probabilities than it was to require critical values of the F-distribution. As a
result, F-distribution tables were limited to tables for critical values for a specific right-hand tail area.
Because of the two parameters, it takes a whole page to give critical values for each value of , the right
hand tail area, and so what was one line in a t-table or 2-table now becomes one page. As a result, it was
common practice to give critical F-values only for  = 0.05 and 0.01. If the symbol F(1, 2) denotes the
value of F cutting off a right-hand tail of area , then you can use the relation
F1    1, 2  
1
F  2 , 1 
to get critical values for left-hand tails of area  (though in this course, all of the methods we consider use
just right-hand tails of the F-distribution).
© David W. Sabo (1999)
The F-Distribution
Page 1 of 4
F-distribution probabilities can be obtained using the FDIST() function in Excel/97 and critical values of the
F-distribution can be obtained using the FINV() function in Excel/97. You would use the FDIST() function,
for example, to calculate p-values for hypothesis tests that involve F-distributed test statistics.
The next two pages of this document contain traditional-style F-tables, one giving critical values of F for
 = 0.05 (and various combinations of 1 and 2) and the other giving critical values of F for  = 0.01. In
these tables, the symbol "N" stands for the numerator degrees of freedom, and the symbol "D" stands for
the denominator degrees of freedom.
To understand what these numbers mean, note the
following example. In the table for  = 0.05, we find
the entry 3.22 in the row labeled D = 2 = 10 and
the column labeled N = 1 = 6. This means that if
F is a random variable with the F-distribution having
1 = 6 and 2 = 10, then
area to the right of
this line is 1 - 
, so
the area to the left
must be 
.
Pr(F > 3.22) = 0.05
area in right-hand
tail is 
F
F1-(1, 2)
= 1/F(2, 1)
F(1, 2)
(Since the row labeled 6 and the column labeled 10 gives the entry 4.06, we know also that
1


Pr  F 
 0.246   1  0.05  0.95
4.06


and hence, that
Pr(F < 0. 246) = 0.05
for 1 = 6 and 2 = 10.)
Page 2 of 4
The F-Distribution
© David W. Sabo (1999)
Critical Values for the F-Distribution: Right-Hand Tail Area = 0.05
vN:
vD:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
120
250
inf.
1
2
3
4
5
6
7
8
9
10
12
15
20
25
30
40
60
120
inf.
161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 241.9 243.9 245.9 248.0 249.3 250.1 251.1 252.2 253.3 254.3
18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.43 19.45 19.46 19.46 19.47 19.48 19.49 19.50
10.13
9.55
9.28
9.12
9.01
8.94
8.89
8.85
8.81
8.79
8.74
8.70
8.66
8.63
8.62
8.59
8.57
8.55
8.53
7.71
6.94
6.59
6.39
6.26
6.16
6.09
6.04
6.00
5.96
5.91
5.86
5.80
5.77
5.75
5.72
5.69
5.66
5.63
6.61
5.79
5.41
5.19
5.05
4.95
4.88
4.82
4.77
4.74
4.68
4.62
4.56
4.52
4.50
4.46
4.43
4.40
4.36
5.99
5.14
4.76
4.53
4.39
4.28
4.21
4.15
4.10
4.06
4.00
3.94
3.87
3.83
3.81
3.77
3.74
3.70
3.67
5.59
4.74
4.35
4.12
3.97
3.87
3.79
3.73
3.68
3.64
3.57
3.51
3.44
3.40
3.38
3.34
3.30
3.27
3.23
5.32
4.46
4.07
3.84
3.69
3.58
3.50
3.44
3.39
3.35
3.28
3.22
3.15
3.11
3.08
3.04
3.01
2.97
2.93
5.12
4.26
3.86
3.63
3.48
3.37
3.29
3.23
3.18
3.14
3.07
3.01
2.94
2.89
2.86
2.83
2.79
2.75
2.71
4.96
4.10
3.71
3.48
3.33
3.22
3.14
3.07
3.02
2.98
2.91
2.85
2.77
2.73
2.70
2.66
2.62
2.58
2.54
4.84
3.98
3.59
3.36
3.20
3.09
3.01
2.95
2.90
2.85
2.79
2.72
2.65
2.60
2.57
2.53
2.49
2.45
2.40
4.75
3.89
3.49
3.26
3.11
3.00
2.91
2.85
2.80
2.75
2.69
2.62
2.54
2.50
2.47
2.43
2.38
2.34
2.30
4.67
3.81
3.41
3.18
3.03
2.92
2.83
2.77
2.71
2.67
2.60
2.53
2.46
2.41
2.38
2.34
2.30
2.25
2.21
4.60
3.74
3.34
3.11
2.96
2.85
2.76
2.70
2.65
2.60
2.53
2.46
2.39
2.34
2.31
2.27
2.22
2.18
2.13
4.54
3.68
3.29
3.06
2.90
2.79
2.71
2.64
2.59
2.54
2.48
2.40
2.33
2.28
2.25
2.20
2.16
2.11
2.07
4.49
3.63
3.24
3.01
2.85
2.74
2.66
2.59
2.54
2.49
2.42
2.35
2.28
2.23
2.19
2.15
2.11
2.06
2.01
4.45
3.59
3.20
2.96
2.81
2.70
2.61
2.55
2.49
2.45
2.38
2.31
2.23
2.18
2.15
2.10
2.06
2.01
1.96
4.41
3.55
3.16
2.93
2.77
2.66
2.58
2.51
2.46
2.41
2.34
2.27
2.19
2.14
2.11
2.06
2.02
1.97
1.92
4.38
3.52
3.13
2.90
2.74
2.63
2.54
2.48
2.42
2.38
2.31
2.23
2.16
2.11
2.07
2.03
1.98
1.93
1.88
4.35
3.49
3.10
2.87
2.71
2.60
2.51
2.45
2.39
2.35
2.28
2.20
2.12
2.07
2.04
1.99
1.95
1.90
1.84
4.32
3.47
3.07
2.84
2.68
2.57
2.49
2.42
2.37
2.32
2.25
2.18
2.10
2.05
2.01
1.96
1.92
1.87
1.81
4.30
3.44
3.05
2.82
2.66
2.55
2.46
2.40
2.34
2.30
2.23
2.15
2.07
2.02
1.98
1.94
1.89
1.84
1.78
4.28
3.42
3.03
2.80
2.64
2.53
2.44
2.37
2.32
2.27
2.20
2.13
2.05
2.00
1.96
1.91
1.86
1.81
1.76
4.26
3.40
3.01
2.78
2.62
2.51
2.42
2.36
2.30
2.25
2.18
2.11
2.03
1.97
1.94
1.89
1.84
1.79
1.73
4.24
3.39
2.99
2.76
2.60
2.49
2.40
2.34
2.28
2.24
2.16
2.09
2.01
1.96
1.92
1.87
1.82
1.77
1.71
4.23
3.37
2.98
2.74
2.59
2.47
2.39
2.32
2.27
2.22
2.15
2.07
1.99
1.94
1.90
1.85
1.80
1.75
1.69
4.21
3.35
2.96
2.73
2.57
2.46
2.37
2.31
2.25
2.20
2.13
2.06
1.97
1.92
1.88
1.84
1.79
1.73
1.67
4.20
3.34
2.95
2.71
2.56
2.45
2.36
2.29
2.24
2.19
2.12
2.04
1.96
1.91
1.87
1.82
1.77
1.71
1.65
4.18
3.33
2.93
2.70
2.55
2.43
2.35
2.28
2.22
2.18
2.10
2.03
1.94
1.89
1.85
1.81
1.75
1.70
1.64
4.17
3.32
2.92
2.69
2.53
2.42
2.33
2.27
2.21
2.16
2.09
2.01
1.93
1.88
1.84
1.79
1.74
1.68
1.62
4.08
3.23
2.84
2.61
2.45
2.34
2.25
2.18
2.12
2.08
2.00
1.92
1.84
1.78
1.74
1.69
1.64
1.58
1.51
4.03
3.18
2.79
2.56
2.40
2.29
2.20
2.13
2.07
2.03
1.95
1.87
1.78
1.73
1.69
1.63
1.58
1.51
1.44
4.00
3.15
2.76
2.53
2.37
2.25
2.17
2.10
2.04
1.99
1.92
1.84
1.75
1.69
1.65
1.59
1.53
1.47
1.39
3.92
3.07
2.68
2.45
2.29
2.18
2.09
2.02
1.96
1.91
1.83
1.75
1.66
1.60
1.55
1.50
1.43
1.35
1.25
3.88
3.03
2.64
2.41
2.25
2.13
2.05
1.98
1.92
1.87
1.79
1.71
1.61
1.55
1.50
1.44
1.37
1.29
1.17
3.84
3.00
2.60
2.37
2.21
2.10
2.01
1.94
1.88
1.83
1.75
1.67
1.57
1.51
1.46
1.39
1.32
1.22
1.00
© David W. Sabo (1999)
The F-Distribution
Page 3 of 4
Critical Values for the F-Distribution: Right-Hand Tail Area = 0.01
vN:
vD:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
120
250
inf.
Page 4 of 4
1
2
3
4
5
6
7
8
9
10
12
15
20
25
30
40
60
120
inf.
4052.2
98.50
34.12
21.20
16.26
13.75
12.25
11.26
10.56
10.04
9.65
9.33
9.07
8.86
8.68
8.53
8.40
8.29
8.18
8.10
8.02
7.95
7.88
7.82
7.77
7.72
7.68
7.64
7.60
7.56
7.31
7.17
7.08
6.85
6.74
6.63
4999.3
99.00
30.82
18.00
13.27
10.92
9.55
8.65
8.02
7.56
7.21
6.93
6.70
6.51
6.36
6.23
6.11
6.01
5.93
5.85
5.78
5.72
5.66
5.61
5.57
5.53
5.49
5.45
5.42
5.39
5.18
5.06
4.98
4.79
4.69
4.61
5403.5
99.16
29.46
16.69
12.06
9.78
8.45
7.59
6.99
6.55
6.22
5.95
5.74
5.56
5.42
5.29
5.19
5.09
5.01
4.94
4.87
4.82
4.76
4.72
4.68
4.64
4.60
4.57
4.54
4.51
4.31
4.20
4.13
3.95
3.86
3.78
5624.3
99.25
28.71
15.98
11.39
9.15
7.85
7.01
6.42
5.99
5.67
5.41
5.21
5.04
4.89
4.77
4.67
4.58
4.50
4.43
4.37
4.31
4.26
4.22
4.18
4.14
4.11
4.07
4.04
4.02
3.83
3.72
3.65
3.48
3.40
3.32
5764.0
99.30
28.24
15.52
10.97
8.75
7.46
6.63
6.06
5.64
5.32
5.06
4.86
4.69
4.56
4.44
4.34
4.25
4.17
4.10
4.04
3.99
3.94
3.90
3.85
3.82
3.78
3.75
3.73
3.70
3.51
3.41
3.34
3.17
3.09
3.02
5859.0
99.33
27.91
15.21
10.67
8.47
7.19
6.37
5.80
5.39
5.07
4.82
4.62
4.46
4.32
4.20
4.10
4.01
3.94
3.87
3.81
3.76
3.71
3.67
3.63
3.59
3.56
3.53
3.50
3.47
3.29
3.19
3.12
2.96
2.87
2.80
5928.3
99.36
27.67
14.98
10.46
8.26
6.99
6.18
5.61
5.20
4.89
4.64
4.44
4.28
4.14
4.03
3.93
3.84
3.77
3.70
3.64
3.59
3.54
3.50
3.46
3.42
3.39
3.36
3.33
3.30
3.12
3.02
2.95
2.79
2.71
2.64
5981.0
99.38
27.49
14.80
10.29
8.10
6.84
6.03
5.47
5.06
4.74
4.50
4.30
4.14
4.00
3.89
3.79
3.71
3.63
3.56
3.51
3.45
3.41
3.36
3.32
3.29
3.26
3.23
3.20
3.17
2.99
2.89
2.82
2.66
2.58
2.51
6022.4
99.39
27.34
14.66
10.16
7.98
6.72
5.91
5.35
4.94
4.63
4.39
4.19
4.03
3.89
3.78
3.68
3.60
3.52
3.46
3.40
3.35
3.30
3.26
3.22
3.18
3.15
3.12
3.09
3.07
2.89
2.78
2.72
2.56
2.48
2.41
6055.9
99.40
27.23
14.55
10.05
7.87
6.62
5.81
5.26
4.85
4.54
4.30
4.10
3.94
3.80
3.69
3.59
3.51
3.43
3.37
3.31
3.26
3.21
3.17
3.13
3.09
3.06
3.03
3.00
2.98
2.80
2.70
2.63
2.47
2.39
2.32
6106.7
99.42
27.05
14.37
9.89
7.72
6.47
5.67
5.11
4.71
4.40
4.16
3.96
3.80
3.67
3.55
3.46
3.37
3.30
3.23
3.17
3.12
3.07
3.03
2.99
2.96
2.93
2.90
2.87
2.84
2.66
2.56
2.50
2.34
2.26
2.18
6157.0
99.43
26.87
14.20
9.72
7.56
6.31
5.52
4.96
4.56
4.25
4.01
3.82
3.66
3.52
3.41
3.31
3.23
3.15
3.09
3.03
2.98
2.93
2.89
2.85
2.81
2.78
2.75
2.73
2.70
2.52
2.42
2.35
2.19
2.11
2.04
6208.7
99.45
26.69
14.02
9.55
7.40
6.16
5.36
4.81
4.41
4.10
3.86
3.66
3.51
3.37
3.26
3.16
3.08
3.00
2.94
2.88
2.83
2.78
2.74
2.70
2.66
2.63
2.60
2.57
2.55
2.37
2.27
2.20
2.03
1.95
1.88
6239.9
99.46
26.58
13.91
9.45
7.30
6.06
5.26
4.71
4.31
4.01
3.76
3.57
3.41
3.28
3.16
3.07
2.98
2.91
2.84
2.79
2.73
2.69
2.64
2.60
2.57
2.54
2.51
2.48
2.45
2.27
2.17
2.10
1.93
1.85
1.77
6260.4
99.47
26.50
13.84
9.38
7.23
5.99
5.20
4.65
4.25
3.94
3.70
3.51
3.35
3.21
3.10
3.00
2.92
2.84
2.78
2.72
2.67
2.62
2.58
2.54
2.50
2.47
2.44
2.41
2.39
2.20
2.10
2.03
1.86
1.77
1.70
6286.4
99.48
26.41
13.75
9.29
7.14
5.91
5.12
4.57
4.17
3.86
3.62
3.43
3.27
3.13
3.02
2.92
2.84
2.76
2.69
2.64
2.58
2.54
2.49
2.45
2.42
2.38
2.35
2.33
2.30
2.11
2.01
1.94
1.76
1.67
1.59
6313.0
99.48
26.32
13.65
9.20
7.06
5.82
5.03
4.48
4.08
3.78
3.54
3.34
3.18
3.05
2.93
2.83
2.75
2.67
2.61
2.55
2.50
2.45
2.40
2.36
2.33
2.29
2.26
2.23
2.21
2.02
1.91
1.84
1.66
1.56
1.47
6339.5
99.49
26.22
13.56
9.11
6.97
5.74
4.95
4.40
4.00
3.69
3.45
3.25
3.09
2.96
2.84
2.75
2.66
2.58
2.52
2.46
2.40
2.35
2.31
2.27
2.23
2.20
2.17
2.14
2.11
1.92
1.80
1.73
1.53
1.43
1.32
6366.0
99.50
26.13
13.46
9.02
6.88
5.65
4.86
4.31
3.91
3.60
3.36
3.17
3.00
2.87
2.75
2.65
2.57
2.49
2.42
2.36
2.31
2.26
2.21
2.17
2.13
2.10
2.06
2.03
2.01
1.80
1.68
1.60
1.38
1.24
1.00
The F-Distribution
© David W. Sabo (1999)