Definition: A non-parametric
test (distribution-free) used to compare two independent groups of sampled data.
Assumptions: Unlike the
parametric t-test, this non-parametric makes no assumptions about the
distribution of the data (e.g., normality).
Characteristics: This test
is an alternative to the independent group t-test, when the assumption of
normality or equality of variance is not met. This, like many non-parametric
tests, uses the ranks of the data rather than their raw values to calculate the
statistic. Since this test does not make a distribution assumption, it is not as
powerful as the t-test.
Test: The hypotheses for the
comparison of two independent groups are:
Ho: The two samples
come from identical populations
Ha: The two samples
come from different populations
Notice that the hypothesis makes no
assumptions about the distribution of the populations. These hypotheses are also
sometimes written as testing the equality of the central tendency of the
The test statistic for the
Mann-Whitney test is U. This value is compared to a table of critical values for
U based on the sample size of each group. If U exceeds the critical value for U
at some significance level (usually 0.05) it means that there is evidence to
reject the null hypothesis in favor of the alternative hypothesis. (See the Zar
reference for details.)
Note: Actually, there are
two versions of the U statistic calculated, where U' = n1n2
- U where n1 and n2 are the sample sizes of the
two groups. The largest of U or U' is compared to the critical value for the
purpose of the test.
Note: For sample sizes
greater than 8, a z-value can be used to approximate the significance level for
the test. In this case, the calculated z is compared to the standard normal
Note: The U test is usually
perform as a two-tailed test, however some text will have tabled one-tailed
significance levels for this purpose. If the sample size if large, the z-test
can be used for a one-sided test.
Graphical comparison: The
graphical comparison allows you to visually see the distribution of the two
groups. If the p-value is low, chances are there will be little overlap between
the two distributions. If the p-value is not low, there will be a fair amount of
overlap between the two groups. There are a number of options available in the
comparison graph to allow you to examine the two groups. These include box
plots, means, medians, and error bars.
Location in KWIKSTAT and WINKS:
The Mann-Whitney U test (independent group comparison test) is located in
the Analyze/Non-parametric comparisons menu. When there are more than two groups
in this comparison, the test becomes a Kruskal-Wallis test.
See Also: The independent
Example: The Mann-Whitney U test in WINKS
The FERTILIZ.DBF database contains
information on the heights of plants that were grown using two different
fertilizers. The Mann-Whitney test can be used to determine if there is evidence
that one fertilizer that causes the plants to grow taller than the other.
Step 1: Open the FERTILIZ.SDA data set. (or create it as described in the t-test example.)
Step 2: Select Analyze, Non-Parametric Comparisons/Ind. Grp
Step 3: Select GROUP as the group field and OBS (Height) as the data field and choose Ok. The results will be displayed in the viewer.
Non-Parametric Independent Group Comparison
Non- Parametric analysis:
variable = GROUP Observation variable = OBS
Whitney U' = 24. U = 18.
group 1 = 46. N = 7 Mean Rank = 6.57
group 2 = 45. N = 6 Mean Rank = 7.5
Significance estimated using the z statistic.
Z = .357 p
This Z calculation uses a correction for continuity.)
The program reports the p-value
based on the z approximation. Since the sample sizes for both groups are less
than 8, you should can look up the critical value for n1 = 6 and n2
= 7 in the table contained in the KWIKSTAT or WINKS manual (or in a textbook) to
see if the critical value is 34 for a test at the 0.05 level. Thus, if U' were
34 or greater, you could claim statistical significance at the 0.05 level. In
this case, the conclusion is that there is no difference in the two groups. Note
how this result agrees with the t-test for this same data set.
More about WINKS >>
Exercise: Mann-Whitney U test
Professor Testum wondered if
students tended to make better scores on his test depending if the test were
taken in the morning or afternoon. From a group of 19 similarly talented
students, he randomly selected some to take a test in the morning and some to
take it in the afternoon. The scores by groups were:
1. Perform a Mann-Whitney U test on
this data. Remember that the program expect two fields, a Group field and an
2. What was the result?
3. From this evidence, does it
appear that time of day makes a difference in performance on a test?
4. Change the value 83.9 in the
second group to 11 and rerun the test. Does this change the statistic calculated
or the conclusion? Also perform an independent group t-test on the original and
changed data. Does it effect this statistic? Why?
5. Display a graphical comparison
of the original and changed data.
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