John Blakeman 
333 
errors due to random sampling from a general population whose regression is linear, 
or whether the values are such as to indicate significant non-linearity. 
For this purpose we compare the calculated value of ^, or ^, or with its 
probable error, where, if the standard deviations of ^, w are written , , 2^- 
the probable errors, say E^, E^, E^, are defined by the equation £' = •674492. 
While Karl Pearson has given in his memoir a formula for S,, and the 
formula for 2^ is well-known, no formula has yet been given for calculating any 
such quantity as S^, 2^.. 2^- for the testing of linear regression. 
It is the object of the present investigation to determine such formulae ; the 
obvious thing to do would be to determine 2^, as f is the quantity occurring 
naturally in Karl Pearson's work ; but the formula for 2^ is so very much 
simpler that it has been made the basis of this investigation ; it is then shown 
how to determine 2^ and 2;^. from 2^- and other known quantities. 
Thus it is at the worker's choice to test linearity of regression by comparing 
any one of the quantities with its probable error. 
Following Karl Pearson's notation n.^^ will denote the number of individuals 
in the a.'p-array of y's, Hy^ the number of individuals in the 7/s-array of and nx^y^ 
the number of individuals in the sub-group common to these two arrays. N is 
the total number of individuals in the table ; x, y are the means of the two 
characters, while ax, o-y are their standard deviations; is the mean of the 
iCp-array of y's while o-„^ is the standard deviation of that array and /V,„, o-jj/, 77, r 
are quantities defined by the following equations, denoting a double summation: 
J^Puv = S' {nx^y^ {X, - xr (ys - yr}> 
=S{nx^(yx^-yr'}, 
Najif = S [hx^ {yx^ - y)'\ {i.e. o-j/ = \^], 
0-31 Pn 
v = — > = - — . 
O-y O-x^-y 
and further we shall define a quantity \uv by the equation 
NK, = 8'{'nx^(x,-xr{yx^-yr}. 
The fundamental results on which our work is based maybe stated as follows: — 
Let n, n'he the number of individuals falling within any two mutually exchisive 
groups, then if S„, 2,i', be the standard deviations of ?i, 71' and the correlation 
between deviations in n and n due to random sampling, we have 
^-' = ^(1-]^)' ^'^' 
71)1 
