**Wilcoxon Two Independent Samples Test
{Adapted from the Institute of
Phonetic Sciences (IFA):
http://www.fon.hum.uva.nl/}**

Note: this test is identical to the Mann-Whitney *U*
test for two independent samples.

(EDL 7150 Class NOTE: Re-do the exercise with the
Mann-Whitney *U *test.)

*Characteristics: *A *most* useful test to see whether the values in two samples differ in
size. It resembles the
Median-Test in scope, but it is *much* more sensitive. In fact, for
large numbers it is almost as sensitive as the
Two Sample Student t-test. For small numbers with unknown distributions this
test is even *more* sensitive than the Student t-test.

Since it is only on rare occasions that we do know that values are Normal distributed, this test may be preferred over the Student t-test.

*H _{0}: *The populations from which the two samples are taken have identical

*Assumptions: *None, really.

*Scale: *Ordinal.

*Procedure: *Rank order all *N* = *m* + *n* values from both samples (*m*
and *n*) combined. Sum the ranks of the smallest sample (W_{smallest}). This
value is used to determine the level of significance.

*Level of Significance: *Look up the level of significance in a table using W_{smallest}, *m* and *n*.

Calculating the exact level of significance is based on calculating all possible
permutations of ranks over both samples. This is computationally demanding if *
n* and *m* are larger than 7.

*Approximation: *If *m*>10 and *n*>10,

Z = ( W

_{smallest}- 0.5 -m* (m+n+ 1 ) / 2 ) / sqrt(m*n* (m+n+ 1 ) / 12 )

is approximately Normal distributed.(Use

W_{smallest}-0.5 ifW>_{smallest}N*(N+1)/4, else useWsmallest+0.5)

*Remarks: *In this example, exact probabilities are calculated for *m* <= 10 or *n*
<= 10. If both are larger than 7 this can take more time than is available
within this system (the number of calculations grows as N!/(m!*n!) , with
N!=N*(N-1)*(N-2)*...*1). Therefore, if it is anticipated that the calculations
take too much time, the Normal approximation is used. However, the resulting
values are unreliable and this will be indicated with a *. You are advised to
check the level of significance in a table.

Note, the symbol, !, denotes
factorial. Hence, N! is read, "N factorial."

If N is 5, say, then N! is

5 × 4 × 3 × 2 × 1,

You can compute the Sign Test by clicking HERE.

Alternatively, you could try programming it in Excel.

For *m* > 10 and *n* > 10 the Normal approximation is used.