John Blakeman 
347 
We conclude from this table : 
I. As regards the linearity of the regression in the statistical examples 
chosen, illustrations C and D are certainly skew. The skewness of these examples 
is obvious either from a glance at the diagrams accompanying Karl Pearson's 
memoir or from his discussion of the cubic and quartic regression curves for these 
cases. Illustrations A and B give > ^ ' ■^f^'luss between 2 and 3. The 
limiting value of these ratios which should denote significance is a matter which 
can only be decided as a result of large statistical experience. When the I'atio 
gets as low as 2 we are getting on the border line and for cases A and B we can 
only say that the skewness is probably significant. This is the kind of result 
we should expect from the corresponding diagrams of Karl Pearson's memoir. 
He finds, however, that he obtains so much better fits by curves of higher order 
that he concludes that the regression of A is parabolic and of B cubic. 
II. If we refer to equations (xxxii), (xxxiii), (xxxiv), (xxxv) we see exhibited 
the numerical contribution of each term of formula (xxii) for S^-. As regards 
the numerical values of the quantities which vanish for normal correlation we see 
that they by no means become zero in the examples chosen. 
Also in these equations I have grouped together the three p terms and the 
three X terms, but the numerical values of the separate terms are still considerable. 
When the whole expression for S^-" is summed however we find that the last three 
terms of these numerical equations so nearly cancel that the difference between 
and E'^ is at the most "004, a quantity which is of no significance in the 
value of a probable error. This is exactly the kind of result Karl Pearson 
found in comparing the values of E^ as found from the complex formula and the 
simple formula. There is thus exactly the same justification for the use of 
formula (xxiii) as for the formula S,,'- = ^ ^ . 
Comparing the values of Ec^, E^ with E\, E\ we see that the agreement is 
closer still, the difference being at the most of order '002 ; and if we except 
illustration G the agreement in the values of E^ is remarkably close, being of 
the order "0003 at the most. 
Thus our numerical work leads us to the conclusion that we have the same 
justification for the use of the simple formulae (xxiii), (xxvii), and (xxix) for 
E^, E^, E^ as for the use of the accepted simple formulae for E,., Ey,. 
These simple formulae have been obtained by making approximations of a 
statistical nature, i.e. approximations suggested entirely by statistical experience 
apart from arithmetic. I proceed to examine the formulae thus obtained to see if 
any simplification can be made by arithmetical approximation. 
