E1C04 09/14/2010
14:7:42 Page 131
The value for the t estimator provides a coverage factor that is a function of the probability, P,
and the degrees of freedom in the data set, n ¼ N À 1. These t values can be obtained from Table 4.4,
which is a tabulation from the Student’s t distribution as developed by William S. Gosset
5 (1876–
1937). Gossett recognized that the use of the z variable with s x in place of s did not yield accurate
estimates of the precision interval, particularly at small degrees of freedom. Careful inspection of
Table 4.4 shows that the t value inflates the size of the interval required to attain a percent
probability, P%. That is, it has the effect of increasing the magnitude of t n;P s x relative to z 1 s at a
desired probability. As the value of N increases, t approaches those values given by the z variable just
as the value of s x must approach s. It should be understood that for very small sample sizes (N 10),
sample statistics can be misleading. In that situation other information regarding the measurement
may be required, including additional measurements.
Table 4.4 Student’s t Distribution
n
t 50
t 90
t 95
t 99
1
1.000
6.314
12.706
63.657
2
0.816
2.920
4.303
9.925
3
0.765
2.353
3.182
5.841
4
0.741
2.132
2.770
4.604
5
0.727
2.015
2.571
4.032
6
0.718
1.943
2.447
3.707
7
0.711
1.895
2.365
3.499
8
0.706
1.860
2.306
3.355
9
0.703
1.833
2.262
3.250
10
0.700
1.812
2.228
3.169
11
0.697
1.796
2.201
3.106
12
0.695
1.782
2.179
3.055
13
0.694
1.771
2.160
3.012
14
0.692
1.761
2.145
2.977
15
0.691
1.753
2.131
2.947
16
0.690
1.746
2.120
2.921
17
0.689
1.740
2.110
2.898
18
0.688
1.734
2.101
2.878
19
0.688
1.729
2.093
2.861
20
0.687
1.725
2.086
2.845
21
0.686
1.721
2.080
2.831
30
0.683
1.697
2.042
2.750
40
0.681
1.684
2.021
2.704
50
0.680
1.679
2.010
2.679
60
0.679
1.671
2.000
2.660
1
0.674
1.645
1.960
2.576
5 At the time, Gosset was employed as a brewer and statistician by a well-known Irish brewery. You might pause to reflect on
his multifarious contributions.
4.4 Statistics of Finite-Sized Data Sets 131
14:7:42 Page 131
The value for the t estimator provides a coverage factor that is a function of the probability, P,
and the degrees of freedom in the data set, n ¼ N À 1. These t values can be obtained from Table 4.4,
which is a tabulation from the Student’s t distribution as developed by William S. Gosset
5 (1876–
1937). Gossett recognized that the use of the z variable with s x in place of s did not yield accurate
estimates of the precision interval, particularly at small degrees of freedom. Careful inspection of
Table 4.4 shows that the t value inflates the size of the interval required to attain a percent
probability, P%. That is, it has the effect of increasing the magnitude of t n;P s x relative to z 1 s at a
desired probability. As the value of N increases, t approaches those values given by the z variable just
as the value of s x must approach s. It should be understood that for very small sample sizes (N 10),
sample statistics can be misleading. In that situation other information regarding the measurement
may be required, including additional measurements.
Table 4.4 Student’s t Distribution
n
t 50
t 90
t 95
t 99
1
1.000
6.314
12.706
63.657
2
0.816
2.920
4.303
9.925
3
0.765
2.353
3.182
5.841
4
0.741
2.132
2.770
4.604
5
0.727
2.015
2.571
4.032
6
0.718
1.943
2.447
3.707
7
0.711
1.895
2.365
3.499
8
0.706
1.860
2.306
3.355
9
0.703
1.833
2.262
3.250
10
0.700
1.812
2.228
3.169
11
0.697
1.796
2.201
3.106
12
0.695
1.782
2.179
3.055
13
0.694
1.771
2.160
3.012
14
0.692
1.761
2.145
2.977
15
0.691
1.753
2.131
2.947
16
0.690
1.746
2.120
2.921
17
0.689
1.740
2.110
2.898
18
0.688
1.734
2.101
2.878
19
0.688
1.729
2.093
2.861
20
0.687
1.725
2.086
2.845
21
0.686
1.721
2.080
2.831
30
0.683
1.697
2.042
2.750
40
0.681
1.684
2.021
2.704
50
0.680
1.679
2.010
2.679
60
0.679
1.671
2.000
2.660
1
0.674
1.645
1.960
2.576
5 At the time, Gosset was employed as a brewer and statistician by a well-known Irish brewery. You might pause to reflect on
his multifarious contributions.
4.4 Statistics of Finite-Sized Data Sets 131