Below are the scores assigned by each of 5 judges to each of 40 competitors in a
synchronized swimming event at the 1986 National Olympic Festival in Houston, Texas.
33.1 32.0 31.2 31.2 31.4
26.2 29.2 28.4 27.3 25.3
31.2 30.1 30.1 31.2 29.2
27.0 27.9 27.3 24.7 28.1
28.4 25.3 25.6 26.7 26.2
28.1 28.1 28.1 32.0 28.4
27.0 28.1 28.1 28.1 27.0
25.1 27.3 26.2 27.5 27.3
31.2 29.2 31.2 32.0 30.1
30.1 30.1 28.1 28.6 30.1
29.0 28.1 29.2 29.0 27.0
27.0 27.0 27.3 26.4 25.3
31.2 33.1 31.2 30.3 29.2
32.3 31.2 32.3 31.2 31.2
29.5 28.4 30.3 30.3 28.4
29.2 29.2 29.2 30.9 28.1
32.3 31.2 29.2 29.5 31.2
27.3 30.1 29.2 29.2 29.2
26.4 27.3 27.3 28.1 26.4
27.3 26.7 26.4 26.4 26.4
27.3 28.1 28.4 27.5 26.4
29.5 28.1 27.3 28.4 26.4
28.4 29.5 28.4 28.6 27.5
31.2 29.5 29.2 31.2 27.3
30.1 31.2 28.1 31.2 29.2
31.2 31.2 31.2 31.2 30.3
26.2 28.1 26.2 25.9 26.2
27.3 27.3 27.0 28.1 28.1
29.2 26.4 27.3 27.3 27.3
29.5 27.3 29.2 28.4 28.1
28.1 27.3 29.2 28.1 29.2
31.2 31.2 31.2 31.2 28.4
28.1 27.3 27.3 28.4 28.4
24.0 28.1 26.4 25.1 25.3
27.0 29.0 27.3 26.4 28.1
27.5 27.5 24.5 25.6 25.3
27.3 29.5 26.2 27.5 28.1
31.2 30.1 27.3 30.1 29.2
27.0 27.5 27.3 27.0 27.3
31.2 29.5 30.1 28.4 28.4
One may wonder if there is significant evidence of some degree of agreement among the judges in a sport for which
the judging is highly subjective. Kendall's coefficient of concordance can be used to test the null hypothesis
of no agreement (randomness) against the general alternative (of some degree of agreement). To do this test with
StatXact, one needs 40 columns of 5 instead of 5 columns of 40. The data in the form below can be pasted into
StatXact's CaseData editor to create 40 columns of length 5.
33.1 26.2 31.2 27.0 28.4 28.1 27.0 25.1 31.2 30.1
32.0 29.2 30.1 27.9 25.3 28.1 28.1 27.3 29.2 30.1
31.2 28.4 30.1 27.3 25.6 28.1 28.1 26.2 31.2 28.1
31.2 27.3 31.2 24.7 26.7 32.0 28.1 27.5 32.0 28.6
31.4 25.3 29.2 28.1 26.2 28.4 27.0 27.3 30.1 30.1
29.0 27.0 31.2 32.3 29.5 29.2 32.3 27.3 26.4 27.3
28.1 27.0 33.1 31.2 28.4 29.2 31.2 30.1 27.3 26.7
29.2 27.3 31.2 32.3 30.3 29.2 29.2 29.2 27.3 26.4
29.0 26.4 30.3 31.2 30.3 30.9 29.5 29.2 28.1 26.4
27.0 25.3 29.2 31.2 28.4 28.1 31.2 29.2 26.4 26.4
27.3 29.5 28.4 31.2 30.1 31.2 26.2 27.3 29.2 29.5
28.1 28.1 29.5 29.5 31.2 31.2 28.1 27.3 26.4 27.3
28.4 27.3 28.4 29.2 28.1 31.2 26.2 27.0 27.3 29.2
27.5 28.4 28.6 31.2 31.2 31.2 25.9 28.1 27.3 28.4
26.4 26.4 27.5 27.3 29.2 30.3 26.2 28.1 27.3 28.1
28.1 31.2 28.1 24.0 27.0 27.5 27.3 31.2 27.0 31.2
27.3 31.2 27.3 28.1 29.0 27.5 29.5 30.1 27.5 29.5
29.2 31.2 27.3 26.4 27.3 24.5 26.2 27.3 27.3 30.1
28.1 31.2 28.4 25.1 26.4 25.6 27.5 30.1 27.0 28.4
29.2 28.4 28.4 25.3 28.1 25.3 28.1 29.2 27.3 28.4
Once you've got the 40 columns of length 5 in Var1 through Var40 of StatXact's CaseData editor, use
Nonparametrics > K Related Samples > Kendall's W...
Then put all 40 variables into the Populations (Treatments) box. I gave up on obtaining an exact
p-value and used the Monte Carlo option (using 1,000,000 trials and a fixed seed of 23456). The
point estimate of the p-value is 0, meaning that none of the one million random permutations resulted
in a test statistic value as extreme as the observed value of the test statistic (about 0.7626). A
99% confidence interval for the exact p-value is about [ 0, 4.6 x 10^{-6} ). (The asymptotic p-value
is about 7.8 x 10^{-10}.) (Note: One gets exactly the same results if StatXact is used to do Friedman's
test on the data (because the two tests are equivalent).)
So even though the coefficient of concordance (about 0.7626) is less than 1 (because there is not
perfect agreement in the way the 5 judges order the 40 competitors), there is very strong evidence
that there is some degree of agreement.