*Example:*

**t = -2.489, DF = 8, p <= 0.04061 **

This is the standard test for matched-pairs. It is also the yard-stick for calculating the relative efficiency of other tests. The Student-

*H0:*

The mean value of the difference is *zero*.

*Assumptions:*

The difference is Normal distributed. If there is any reason to doubt this
assumption, use another, distribution-free, test (e.g.,
Wilcoxon Matched-Pairs Signed-Ranks Test).

*Scale:*

Interval

*Procedure:*

Calculate the Mean value and standard deviation (*SD*) of the differences,
determine the number of sample pairs, *N*, and *Degrees of Freedom*
(= *N*-1, in this case).

The test parameter is *t* = ( *Mean* / *SD* ) * sqrt( *N* ).

If necessary, *Mean* can be replaced by (*Mean* - expected value).

*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. This can lead to illusory high sensitivities.
Although it is the most powerfull test when the experimental data *are*
indeed Normal distributed, do *not* hesitate to use a distribution-free
test instead whenever there is some doubt about Normality (the
Wilcoxon Matched-Pairs Signed-Ranks 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)

Return to Statistics