Statistical Methods for Psychology

Table 10.4 Judge’s rankings of pleasantness of colored patches

Colored Patches

Judges A B C D E F G H

1 12345678

2 21543876

3 13275684

4 21354786

5 31246578

6 21365487

g 119183028364539

310 Chapter 10 Alternative Correlational Techniques

On the other hand, if the judges showed no agreement, each column would have had

some high ranks and some low ranks assigned to it, and the column totals would have been

roughly equal. Thus, the variability of the column totals, given disagreement (or random

behavior) among judges, would be low.

Kendall used the variability of the column totals in deriving his statistic. He defined W

as the ratio of the variability among columns to the maximum possible variability.

Since we are dealing with ranks, we know what the maximum variance of the totals

will be. With a bit of algebra, we can define

where represents the column totals, N 5 the number of items to be ranked, and k 5 the

number of judges doing the ranking. For the data in Table 10.4,

As you can see from the definition of W, it is not a standard correlation coefficient. It

does have an interpretation in terms of a familiar statistic. However, it can be viewed as a

function of the average Spearman correlation computed on the rankings of all possible

pairs of judges. Specifically,

For our data,

Thus, if we took all possible pairs of rankings and computed for each, the average

would be .768.

rs rs

rs=

kW 21

k 21

=

6(.807) 21

5

=.768

rs=

kW 21

k 21

=.807

=

12(7052)

62 (8)(63)

2

3(9)

7

=

84624

18144

2

27

7

W=

12 gTj^2

k^2 N(N^22 1)

2

3(N 1 1)

N 21

(^) aT^2 j = 112192118213021282136214521392 = 7052
Tj

W=

12 gTj^2

k^2 N(N^22 1)

2

3(N 1 1)

N 21

W=

Variance of column totals

Maximum possible variance of column totals