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Correlation and Application of Statistics to Problems of Heredity 51

races*, which have been often discussed from early times up to the present day, both by artists and by anthropologists. The fact that the average ratio between the stature and the cubit is as 100 to 37t or thereabouts does not give the slightest information about the nearness with which they vary together. It would be an altogether erroneous inference to suppose their average proportion to be maintained so that where the cubit was, say, one-twentieth longer than the average cubit, the stature might be expected to be one-twentieth greater than the average stature, and conversely. Such a supposition is easily shown to be contradicted both by fact

and theory." (loc. cit. pp. 135-:-6.)

Let us now describe Galton's procedure. In the first place Galton does not use means, he uses throughout medians, both for his marginal totals and his arrays. Further he does not use standard deviations, he makes use of the quartile measurements. Thus if Q17 M and Q3 be the measurements at first, second and third quartile divisions, he takes M as his median and 2 (Q3- Q1) as his measure of variation. Thus his results, unlike our modern treatment, depend essentially on assuming that all his data follow.a normal (or" curve of errors") distribution 1. If M, be the median of any character c and bM, the median of an array of this character for a given value b of a second character c', then Galton plots

bMC-MC to b-Mil 2(Q.1-Q1C 2 (Q3-Ql)C'

In other words he reduces the deviation of an array median from the population median to its unit of variation obtained from the quartiles, and plots this to the deviation of the second character from its median reduced likewise to its own unit of variation. Then he plots

2(Q3-We 2 (Q3-Ql)C

where a is a value of the first character and ,M,, the median of the corresponding array of the second character, and thus gets a second series of points. He takes six or seven values of a and of b, plots two sets of six or seven points and notes that the first and second series of points are nearly on one and the same straight line§. He draws this straight line as closely as he can to the points and through the median, and reads off its slope. This slope is Galton's measure of co-relation. If we take the mean deviation of c' for a given value of c, Galton calls c the "Subject" and c' the "Relative," but perhaps it would be best to call the latter the "Co-relative." Galton's data consisted of about 350 males of 21 years and upwards, of whom the majority were young students, measured in his Laboratory in 1888. He deals with

* [The variation in the ratio of stature to cubit does, however, provide a means of determining the correlation. K.P.]

t [Rather 100 to 27 or thereabouts on Galton's numbers, i.e. 67.20" for stature and 18.05" for cubit. K.P.]

$ In the table given on p. 52 for the correlation of Stature and Left Cubit it is very difficult to see any approximation to normality in the distribution of stature.

In order to get the same straight line, if c be the subject and c' the co-relative, and the "subject" axis horizontal, then it is needful when c' is subject and c co-relative to plot c' along the same axis as was used in the first case for c. In other words the character axes must be interchanged.

7-2

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