t-statistics

Prob. is the two-sided probability of the absolute value of the reported t with the reported d.f., so that a statistically significant finding for a single two-sided t-test is Prob.<.05, and a highly significant finding is Prob.<.01. Please interpret this conservatively, i.e., "barely significant is probably not significant". If you wish to make a Bonferroni multiple-comparison correction, compare this Prob. with your chosen significance level, e.g., p<.05, divided by the number of DIF tests in this Table. This is approximate because of dependencies between the statistics underlying the computation.

 

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)

 

A calculator for the probability of any t value and d.f. is at http://www.danielsoper.com/statcalc3/calc.aspx?id=8

 


 

Welch's refinement of Student's t-test for possibly unequal variances:

 

For sample 1,

M1 = mean of the sample

SS1 = sum of squares of observations from the individual sample means

N1 = sample size (or number of observations)

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 measure)

 

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 http://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

 

Example: Gender subtotals for Example0.txt Table 28:

M1 = 1.62, M2 = .76, SE1 = .38, SE2 = .16, N1 = 18, N2 = 57

Welch: t = 2.08, d.f. = 23, p = .049


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