Karl Pearson and Egon S. Pearson 
149 
This is done in Diagram I. But what we actually desire is to compare the obser- 
vations and the regression lines as given by the present polychoric method with 
those obtained by product-moment methods. 
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Stature of Father in Inches. 
Diagram I. 
Our actual data from which the table on p. 135 was obtained are given m 
Table XV. The following are the values of the constants in inches : 
Mean Stature of Father : x = 67"-878. 
Mean Stature of Son : y = 68"-84.5. 
Standard Deviation of Father : — 2"'6576. 
Standard Deviation of Son : o-,, = 2"-6885. 
Correlation of Father and Son : r = -5189 ± -0160. 
In Diagram II the regression line (slope, •524.5) with means of the arrays as 
dark circles is given. Against this we have put as hollow circles the values of 
hg- and kg. multiplied by their respective s.D.'s to indicate the result as worked 
out in the present paper. The closeness of the polychoric coefficient "5204 and 
the product-moment coefficient does not permit of two regression lines being 
drawn. It will be seen that the fit to the observations by use of broad categories 
and the polychoric method is I'eally quite as satisfactory as the fit by the product- 
moment method. But the amount of arithmetical work is incomparably greater 
by the former, even if it be less than Ritchie-Scott's process with 49 cells would be. 
Accordingly we now proceeded to investigate the extent to which approxi- 
mations shortening the arithmetic would introduce serious error. The first question 
to be answered is: To what extent in finding the means k<,. of the arrays is it 
needful to use the actual value of the correlation coefficient as found for each 
column ? In order to test this we proceeded to find the for each columnar 
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