# The Student-t Test for Two Samples

Example:
#A = 15, #B = 15, t = 1.378, DF = 28, p <= 0.1793

The two samples (raw data)
A B

### Summary

Sample A:
Mean Standard Dev Sample Size
Sample B:
Mean Standard Dev Sample Size

Characteristics:
This is the standard test for two samples. It is also the yard-stick for calculating the relative efficiency of other tests. The Student-t test is the most sensitive test for interval data, but it also requires the most stringent assumptions.

H0:
Both populations have identical mean values. That is, the difference between the means is zero.

Assumptions:
Both distributions are Normal distributed with identical variances. If there is any reason to doubt these assumptions, use another, distribution-free, test (e.g. the Wilcoxon Test).

Scale:
Interval

Procedure:
Calculate the Mean values (M1, M2) and standard deviations (SD1 and SD2) of both samples. Determine SDg = sqrt(((n1-1)*SD1**2 + (n2-1)*SD2**2)/(n1+n2-2))
The test parameter is t = ( M1 - M2 ) / ( SDg * sqrt( 1 / n1 + 1 / n2 ) ).
The number of Degrees of Freedom = n1 + n2 - 2.

Level of Significance:
The significance levels of t for different Degrees of Freedom are tabulated.

Approximation:
If the Degrees of Freedom > 30, the distribution of t can be approximated by a Standard Normal Distribution.

Remarks:
Because of its popularity, this test is very often applied indiscriminately, when the underlying assumptions are invalid, i.e., when the observations are not Normal distributed with equal variance. This can lead to illusory high sensitivities. Although it is the most powerfull test when the experimental data are indeed Normal distributed with equal variance, do not hesitate to use a distribution-free test instead whenever there is some doubt about Normality (the Wilcoxon Test is almost as sensitive).
The name Student-t test is derived from the pen-name of the man who developed the test. It has nothing to do with the popularity of this test in introductory courses.
WARNING: the level of significance given here is only an approximation, take care when using it! (use a table if necessary)