Many statistical tests are reported as Student's t statistics. This table shows the significance-level values for different degrees of freedom (d.f.). Often the reported t-statistics have effectively infinite degrees of freedom and so approximate a unit normal distribution. t-statistics with infinite degrees of freedom are also called z-statistics, paralleling the use of "z" in z-scores.
Table of the two-sided t distribution:
|
d.f.
|
p=.05
|
p=.01
|
1
|
12.71
|
63.66
|
2
|
4.30
|
9.93
|
3
|
3.18
|
5.84
|
4
|
2.78
|
4.60
|
5
|
2.57
|
4.03
|
6
|
2.45
|
3.71
|
7
|
2.37
|
3.50
|
8
|
2.31
|
3.36
|
9
|
2.26
|
3.25
|
10
|
2.23
|
3.17
|
|
d.f.
|
p=.05
|
p=.01
|
11
|
2.20
|
3.11
|
12
|
2.18
|
3.06
|
13
|
2.16
|
3.01
|
14
|
2.15
|
2.98
|
15
|
2.13
|
2.95
|
16
|
2.12
|
2.92
|
17
|
2.11
|
2.90
|
18
|
2.10
|
2.88
|
19
|
2.09
|
2.86
|
20
|
2.09
|
2.85
|
|
d.f.
|
p=.05
|
p=.01
|
21
|
2.08
|
2.83
|
22
|
2.07
|
2.82
|
23
|
2.07
|
2.81
|
24
|
2.06
|
2.80
|
25
|
2.06
|
2.79
|
30
|
2.04
|
2.75
|
100
|
1.98
|
2.63
|
1000
|
1.96
|
2.58
|
Infinite
|
1.96
|
2.58
|
(z-statistic)
|
|
Welch's refinement of Student's t-test for possibly unequal variances:
For sample 1 (or estimate 1),
M1 = mean of the sample (or the estimate)
SS1 = sum of squares of observations from the individual sample means
N1 = sample size (or number of observations)
N1 - 1 = degrees of freedom
SS1 / (N1 - 1) = sample variance around the mean (or the measure)
SS1 / ((N1 - 1)(N1)) = standard error variance = EV1 = SE1²
SE1 = Sqrt(EV1) = standard error of the mean (or the standard error of the measure estimate)
(M1 - M2) = difference between means (or estimates)
sqrt (SE1² + SE2²) = joint standard error of the means (or of the estimates
Similarly for sample 2, then
t = (M1 - M2) / sqrt (EV1 + EV2) = (M1 - M2) / sqrt (SE1² + SE2²)
with Welch-Satterthwaite d.f. = (EV1 + EV2)² / (EV1²/ (N1-1) + EV2² /(N2-1))
which is the same as d.f = (SE1² + SE2²)² / (SE14 / (N1-1) + SE24 / (N2-1))
A calculator for this is at https://www.graphpad.com/quickcalcs/ttest1.cfm?Format=SEM
Satterthwaite, F. E. (1946), "An Approximate Distribution of Estimates of Variance Components.", Biometrics Bulletin 2: 110-114
Welch, B. L. (1947), "The generalization of "Student's" problem when several different population variances are involved.", Biometrika 34: 28-35
