Statistical Methods for Psychology

Ranking Data

Students occasionally experience difficulty in rankinga set of measurement data, and this

section is intended to present the method briefly. Assume we have the following set of data,

which have been arranged in increasing order:

5, 8, 9, 12, 12, 15, 16, 16, 16, 17

The lowest value (5) is given the rank of 1. The next two values (8 and 9) are then as-

signed ranks 2 and 3. We then have two tied values (12) that must be ranked. If they were

untied, they would be given ranks 4 and 5, so we split the difference and rank them both

4.5. The sixth number (15) is now given rank 6. Three values (16) are tied for ranks 7, 8,

and 9; the mean of these ranks is 8. Thus, all are given ranks of 8. The last value is 17,

which has rank 10. The data and their corresponding ranks are given below.

X: 58912 12 1516161617

Ranks: 1 2 3 4.5 4.5 688810

Spearman’s Correlation Coefficient for Ranked Data ( )

Whether data naturally occur in the form of ranks (as, for example, when we are looking at

the rankings of 20 cities on two different occasions) or whether ranks have been substituted

for raw scores, an appropriate correlation is Spearman’s correlation coefficient for

ranked data (rs).(This statistic is sometimes referred to as Spearman’s rho.)

Calculating

The easiest way to calculate is to apply Pearson’s original formula to the ranked data.

Alternative formulae do exist, but they have been designed to give exactly the same answer

as Pearson’s formula as long as there are no ties in the data. When there are ties, the alter-

native formula lead to a wrong answer unless a correction factor is applied. Since that cor-

rection factor brings you back to where you would have been had you used Pearson’s

formula to begin with, why bother with alternative formulae?

The Significance of

Recall that in Chapter 9 we imposed normality and homogeneity assumptions in order to

provide a test on the significance of r(or to set confidence limits). With ranks, the data

clearly cannot be normally distributed. There is no generally accepted method for calculat-

ing the standard error of for small samples. As a result, computing confidence limits on

is not practical. Numerous textbooks contain tables of critical values of , but for N 28

these tables are themselves based on approximations. Keep in mind in this connection that a

typical judge has difficulty ranking a large number of items, and therefore in practice Nis

usually small when we are using.

Kendall’s Tau Coefficient (t)

A serious competitor to Spearman’s is Kendall’st.Whereas Spearman treated the ranks

as scores and calculated the correlation between the two sets of ranks, Kendall based his

statistic on the number of inversionsin the rankings.

We will take as our example a dataset from the Data and Story Library (DASL)

Web site, found at http://lib.stat.cmu.edu/DASL/Stories/AlcoholandTobacco.html. These

rs

rs

rs Ú

rs rs

rs

rs

rs

rs

304 Chapter 10 Alternative Correlational Techniques

ranking

Spearman’s
correlation coeffi-
cient for ranked
data (rs)
Spearman’s rho

Kendall’st