The Kruskal-Wallis test is a non-parametric method used to determine if samples originate from the same distribution. It compares the medians of three or more independent groups. The null hypothesis is that all groups have the same median. If significant, at least one group differs from the others, though post-hoc tests are needed to determine which ones. It assumes identically shaped distributions across groups except for differences in medians. The Mann-Whitney U test can then analyze specific sample pairs.

1. KRUSKAL-WALLIS TEST
In statistics, the Kruskal–Wallis one-way analysis of variance by ranks (named after William Kruskal
and W. Allen Wallis) is a non-parametric method for testing whether samples originate from the
same distribution. It is used for comparing more than two samples that are independent, or not
related. The parametric equivalence of the Kruskal-Wallis test is the one-way analysis of variance
(ANOVA). The factual null hypothesis is that the populations from which the samples originate
have the same median. When the Kruskal-Wallis test leads to significant results, then at least one
of the samples is different from the other samples. The test does not identify where the
differences occur or how many differences actually occur. It is an extension of the Mann–Whitney
U test to 3 or more groups. The Mann-Whitney would help analyze the specific sample pairs for
significant differences.
Since it is a non-parametric method, the Kruskal–Wallis test does not assume a normal
distribution, unlike the analogous one-way analysis of variance. However, the test does assume an
identically shaped and scaled distribution for each group, except for any difference in medians.
Kruskal–Wallis is also used when the examined groups are of unequal size (different number of
participants).[1]
Method:
1. Rank all data from all groups together; i.e., rank the data from 1 to N ignoring group
membership. Assign any tied values the average of the ranks they would have
received had they not been tied.
2. The test statistic is given by:
where:
o is the number of observations in group
o is the rank (among all observations) of observation from group
o is the total number of observations across all groups
o
o
o is the average of all the .

2. 3. Notice that the denominator of the expression for is exactly
and . Thus
Notice that the last formula only contains the squares of the average ranks.
4. A correction for ties can be made by dividing by , where
G is the number of groupings of different tied ranks, and ti is the number of tied
values within group i that are tied at a particular value. This correction usually
makes little difference in the value of K unless there are a large number of ties.
5. Finally, the p-value is approximated by . If some values are
small (i.e., less than 5) the probability distribution of K can be quite different from
this chi-squared distribution. If a table of the chi-squared probability distribution is
available, the critical value of chi-squared, , can be found by entering the
table at g − 1 degrees of freedom and looking under the desired significance or
alpha level. The null hypothesis of equal population medians would then be rejected
if . Appropriate multiple comparisons would then be performed on
the group medians.
6. If the statistic is not significant, then there is no evidence of differences between the
samples. However, if the test is significant then a difference exists between at least
two of the samples. Therefore, a researcher might use sample contrasts between
individual sample pairs, or post hoc tests, to determine which of the sample pairs are
significantly different. When performing multiple sample contrasts, the Type I error
rate tends to become inflated.

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The Kruskal-Wallis test evaluates whether the population medians on a dependent variable are
the same across all levels of a factor. To conduct the Kruskal-Wallis test, using the K independent
samples procedure, cases must have scores on an independent or grouping variable and on a
dependent variable. The independent or grouping variable divides individuals into two or more
groups, and the dependent variable assesses individuals on at least an ordinal scale. If the
independent variable has only two levels, no additional significance tests need to be conducted
beyond the Kruskal-Wallis test. However, if a factor has more than two levels and the overall test
is significant, follow-up tests are usually conducted. These follow-up tests most frequently involve
comparisons between pairs of group medians. For the Kruskal-Wallis, we could use the Mann-
Whitney U test to examine unique pairs.
ASSUMPTIONS UNDERLYING A MANN-WHITNEY U TEST
Because the analysis for the Kruskal-Wallis test is conducted on ranked scores, the population
distributions for the test variable (the scores that the ranks are based on) do not have to be of any
particular form (e.g., normal). However, these distributions should be continuous and have
identical form.
Assumption 1: The continuous distributions for the test variable are exactly the same (except their
medians) for the different populations.
Assumption 2: The cases represent random samples from the populations, and the scores on the
test variable are independent of each other.
Assumption 3: The chi-square statistic for the Kruskal-Wallis test is only approximate and becomes
more accurate with larger sample sizes.
The p value for the chi-square approximation test is fairly accurate if the number of cases is
greater than or equal to 30.
EFFECT SIZE STATISTICS FOR THE MANN-WHITNEY U TEST
SPSS does not report an effect size index for the Kruskal-Wallis test. However, simple indices
can be computed to communicate the size of the effect. For the Kruskal-Wallis test, the median
and the mean rank for each of the groups can be reported. Another possibility for the Kruskal-
Wallis test is to compute an index that is usually associated with a one-way ANOVA, such as eta
square (h2), except h2 in this case would be computed on the ranked data. To do so, transform the
scores to ranks, conduct an ANOVA, and compute an eta square on the ranked scores. Eta square
can also be computed directly from the reported chi-square value for the Kruskal-Wallis test with
the use of the following equation:

4. n2=X2/N-1Where N is the total number of cases
THE RESEARCH QUESTIONS
The research questions used in this example can be asked to reflect differences in medians
between groups or a relationship between two variables.
1. Differences between the medians: Do the medians for change in the number of days of cold
symptoms differ among those who take a placebo, those who take low doses of vitamin C, and
those who take high doses of vitamin C?
2. Relationship between two variables: Is there a relationship between the amount of vitamin C
taken and the change in the number of days that individuals show cold symptoms?