﻿ t-statistics

# 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.

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 (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 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

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