Statistical Methods for Psychology

are data on the average weekly spending on alcohol and tobacco in 11 regions of Great

Britain. (We saw these data in Exercise 9.27.) The data follow, and I have organized the

rows to correspond to increasing expenditures on Alcohol. Though it is not apparent from

looking at either the Alcohol or Tobacco variable alone, in a bivariate plot it is clear that

Northern Ireland is a major outlier. Similarly the distribution of Alcohol expenditures is

decidedly nonnormal, whereas the ranked data on alcohol, like all ranks, are rectangularly

distributed.

Section 10.3 Correlation Coefficients for Ranked Data 305

Region Alcohol Tobacco RankA RankT Inversions

Northern Ireland 4.02 4.56 1 11 10

East Anglia 4.52 2.92 2 2 1

Southwest 4.79 2.71 3 1 0

East Midlands 4.89 3.34 4 4 1

Wales 5.27 3.53 5 6 2

West Midlands 5.63 3.47 6 5 1

Southeast 5.89 3.20 7 3 0

Scotland 6.08 4.51 8 10 3

Yorkshire 6.13 3.76 9 7 0

Northeast 6.19 3.77 10 8 0

North 6.47 4.03 11 9 0

Notice that when the entries are listed in the order of rankings given by Alcohol, there

are reversals (or inversions) of the ranks given by Tobacco (rank 11 of tobacco comes be-

fore all lower ranks, while rank 10 of tobacco comes before 3 lower ranks). I can count the

number of inversions just by going down the Tobacco column and counting the number of

times a ranking further down the table is lower than one further up the table. For instance,

looking at tobacco expenditures, row 1 has 10 inversions because all 10 values below it are

higher. Row 2 has only one inversion because only the rank of “1” is lower than a rank

of 2, and so on.

If there were a perfect ordinal relationship between these two sets of ranks, we would

not expect to find any inversions. The region that spent the most money on alcohol would

spend the most on tobacco, the region with the next highest expenditures on alcohol would

be second highest on tobacco, and so on. Inversions of this form are the basis for Kendall’s

statistic.

Calculatingt

There are n(n 2 1) 2 5 11(10) 2 5 55 pairs of rankings. Eighteen of those rankings are

inversions (often referred to as “discordant”). This is found as the sum of the right-most

column), and 37 of those pairs are not inversions (“concordant”) and this is simply the total

number of pairs (55) minus the number of discordant pairs (18).

We will let Cstand for the number of concordant pairs and Dfor the number of discor-

dant pairs. The difference between Cand Dis represented by S.

D 518 5 Inversions

C 537

S 5 C 2 D 519

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