John Blakeman 349 
I append the values given by (xxxix), for the fom- statistical illustrations: 
A 
B 
C 
D 
r 
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•2039 
2-6883 
-0302 
2-5711 
-4433 
13-7078 
•2016 
13-3086 
Even when - is not very small 
ViY 
E'i •67449-2 ^•Vl-2(7?^-?-^) + 7?^-r^' 
and it is obvious that if there be any doubt as to linearity of regression, when 
is not small, then r and t? must be both small and = -- 
good arithmetical apj^roximation for equation (xxvii), 
E\ •67449 ■ 
will still be a 
General Conclusion. 
Thus to test whether the regression of any frequency distribution is linear or 
not we have three possible methods, viz. : — to compare the value of either ■sb-, 
or ^, with its probable error. As regards the best quantity to choose for this we 
may first say that tir has no advantages whatever. In favour of the use of ^ 
we may say that the terms of are just rj and r, which are the quantities in 
terms of which we naturally judge our distribution ; we must be careful however 
in using formula (xxix) for 1^^ to remember that r is supposed positive ; i.e. if r is 
negative for any case we must change its sign before substituting in the formula 
and we then get the probable error of the difference between 77 and the absolute 
magnitude of r. 
^ however is the term occurring naturally in the work and is the one quantity 
of the three with a direct physical meaning; i.e. Ogives the mean square deviation 
of the distribution from the regression line. This will probably be sufficient to 
assure that tests for linearity will be conducted in terms of ^. 
Finally, 
(i) A simple test for linearity of regression which will be sufficient in very 
many cases is 
•67449' 
1V?< £• 
