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 (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