*In the 1980 issue of *TEACHING STATISTICS*, D. Griffiths supports
the use of the Spearman rank correlation coefficient on the grounds that
"it is the one which is commonly
used." The claim may well be true. But it is a poor
excuse for ignoring practical and pedagogical advantages of the Kendall
coefficient.*

All too often statistics users compute a quantity like a correlation
coefficient without asking what the quantity means. In the Griffiths discussion,
what operational interpretation can we associate with the value** r**
for the Spearman correlation coefficient between the density of
public houses and the density of places of worship? We are given a formal
explanation for the rather awkward-looking algebraic structure of the Spearman
coefficient, but we are not told anything about how to interpret the result
of our computations.

The facts are that it is no easy matter to assign an operational interpretation to the Spearman coefficient. The Kendall coefficient, on the other hand, has an intuitively simple interpretation. What is more, its algebraic structure is much simpler than that of the Spearman coefficient. It can even be computed from the actual observations without first converting them to ranks.

A correlation coefficient is intended to measure "strength of relationship". But different correlation coefficients measure strength of relationship in different ways. A product moment coefficient, a Spearman coefficient, and a Kendall coefficient, all equal to1/3 mean three rather different things. Only the Kendall coefficient has a simple interpretation.

When statisticians talk of strength of relationship,
they usually have in mind the strength of the tendency of two variables,
X and Y, to move in the same (opposite) direction. The Kendall coefficient
measures this tendency in a very direct and easily understood way. Let
(X_{i}, Y_{i})
and (X_{j},Y_{j}) be a pair of (bivariate) observations.
If X_{j} -X_{i} and Y_{j} - Y_{i
}have the same sign, we shall say that the pair is *concordant*,
if they have opposite signs, we shall say that the pair is *discordant
*In the (x,y)-plane the points _{+}**
^{+} **form a concordant
pair, while the points

The maximum value +1 is achieved if all *n(n -*l)/2
pairs are concordant. Correspondingly the minimum
value -1 is achieved if all pairs are discordant. The quantity t is known
as the Kendall rank correlation coefficient tau.

We shall call t the
Kendall correlation coefficient in the (X*,
*Y)-population. It is an intuitively simple measure of strength
of relationship between X*
*and Y. For example, in a population with t
= 1/3** **two sets of observations (X_{i},
*Y _{i}) *and (X

On practical and pedagogical grounds, the Kendall coefficient has substantial advantages over the Spearman coefficient. This writer is not aware of any theoretical reasons for preferring the Spearman coefficient. Quite the contrary, Kendall S has much greater universality, so much so that a good deal of what is called non-parametric statistics can be built around S. A recent book by C. Leach does exactly that.

*Addendum*: A recent paper in *Teaching Statistics
*by D. Wilkie discusses a pictorial representation of the Kendall coefficient
in terms of a quantity *c *equal to the number of crossings among
the *n *lines which connect X - and Y-ranks. A little consideration
shows that discordant pairs of observations produce crossings while concordant
pairs do not. Thus Wilkie’s *c *equals our D*. *If, as Wilkie
seems to assume, there are no tied ranks, *C + D=* n(n*
-* 1)/2, and t
can be written as

[n(n-1)/2 - 2D]/[n(n-1)/2] = 1 - 4D/n(n-1)

which is the expression given by Wilkie. (Note: Wilkie’s t is our t.)

*University of Connecticut*

**References**

Griffiths, D. (1980). *A Pragmatic Approach to
Spearman’s Rank Correlation Coefficient. *Teaching Statistics 2, pp.
10?13.

Kruskal, W. *(1958). Ordinal Measures of Association.
*Journal of the American Statistical Association *53, *pp. 814?861.

Leach, C. (1979). *Introduction to Statistics:
A Nonparametric Approach for the Social Sciences. *Wiley.

Wilkie, D. (1980). *Pictorial Representation of
Kendall’s, Rank Correlation Coefficient*. Teaching
Statistics 2, pp. 76-78.

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