Research Methodology - Dr. Krishan K. Pandey

Testing of Hypotheses-II 309
(ii) If N is larger than 7, we may use χ 2 value to be worked out as: χ 2 = k(N – 1). W with
d.f. = (N – 1) for judging W’s significance at a given level in the usual way of using χ 2
values.
(f) Significant value of W may be interpreted and understood as if the judges are applying
essentially the same standard in ranking the N objects under consideration, but this should
never mean that the orderings observed are correct for the simple reason that all judges
can agree in ordering objects because they all might employ ‘wrong’ criterion. Kendall,
therefore, suggests that the best estimate of the ‘true’ rankings of N objects is provided,
when W is significant, by the order of the various sums of ranks, R j . If one accepts the
criterion which the various judges have agreed upon, then the best estimate of the ‘true’
ranking is provided by the order of the sums of ranks. The best estimate is related to the
lowest value observed amongst R j .
This can be illustrated with the help of an example.
Illustration 9
Seven individuals have been assigned ranks by four judges at a certain music competition as shown
in the following matrix:
Individuals
A B C D E F G
Judge 1 1 3 2 5 7 4 6
Judge 2 2 4 1 3 7 5 6
Judge 3 3 4 1 2 7 6 5
Judge 4 1 2 5 4 6 3 7
Is there significant agreement in ranking assigned by different judges? Test at 5% level. Also point
out the best estimate of the true rankings.
Solution: As there are four sets of rankings, we can work out the coefficient of concordance (W) for
judging significant agreement in ranking by different judges. For this purpose we first develop the
given matrix as under:
Table 12.9
K = 4 Individuals ∴ N = 7
A B C D E F G
Judge 1 1 3 2 5 7 4 6
Judge 2 2 4 1 3 7 5 6
Judge 3 3 4 1 2 7 6 5
Judge 4 1 2 5 4 6 3 7
Sum of ranks (R ) 7 13 9 14 27 18 24 ∑ R =
j j
dRj − Rji
2
81 9 49 4 121 4 64 ∴ s = 332
112