The Mismeasure of Man by Stephen Jay Gould

THE REAL ERROR OF CYRIL BURT

Rogers Hornsby hit .424 (1924—I was a bratty little kid like that);

others know Billy Martin only as a figure in Lite beer commercials.

3. The differences represent disparities in native ability that
   cannot be erased even by intense training. (The situation would be
   even more complex if the sample included both boys and girls of
   conventional upbringing. The correlation might then be attributed
   primarily to a fourth cause—sexual differences; and we would
   have to worry, in addition, about the cause of the sexual difference:
   training, inborn constitution, or some combination of nature and
   nurture).
   In summary, most correlations are noncausal; when correla-
   tions are causal, the fact and strength of the correlation rarely spec-
   ifies the nature of the cause.

Correlation in more than two dimensions

These two-dimensional examples are easy to grasp (however
difficult they are to interpret). But what of correlations among
more than two measures? A body is composed of many parts, not
just arms and legs, and we may want to know how several measures
interact during growth. Suppose, for simplicity, that we add just
one more measure, head length, to make a three-dimensional sys-
tem. We may now depict the correlation structure among the three
measures in two ways:

1. We may gather all correlation coefficients between pairs of
   measures into a single table, or matrix of correlation coefficients
   (Fig. 6.2). The line from upper left to lower right records the nec-
   essarily perfect correlation of each variable with itself. It is called
   the principal diagonal, and all correlations along it are 1.0. The
   matrix is symmetrical around the principal diagonal, since the cor-
   relation of measure 1 with measure 2 is the same as the correlation
   of 2 with 1. Thus, the three values either above or below the prin-
   cipal diagonal are the correlations we seek: arm with leg, arm with
   head, and leg with head.


2. We may plot the points for all individuals onto a three-
   dimensional graph (Fig. 6.3). Since the correlations are all positive,
   the points are oriented as an ellipsoid (or football). (In two dimen-
   sions, they formed an ellipse.) A line running along the major axis
   of the football expresses the strong positive correlations between
   all measures.