t-statistics

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.

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

Sqrt(EV1) = standard error of the mean (or the measure)

Similarly for sample 2, then

t = (M1 - M2) / sqrt (EV1 + EV2)

with  Welch-Satterthwaite d.f. = (EV1 + EV2)2 (N1-1)(N2-1) / (EV12(N2-1) + EV22(N1-1))

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