Miscellanea 
317 
The first of these two terms is that which is obtained in the ordinary way, so the contri- 
bution of each array should be corrected by the addition of the second term and rf l itself by 
the addition of 
Now if Professor Pearson's correction (ii) has been made we may take the point whose 
coordinates are (x p , y p ) to lie on the regression line, and if further we assume the regression 
line to be linear throughout the x p group and to be inclined at an angle of tan ~ 1 r p — to the 
horizontal we have 
Hence (iv) becomes 
y p =x p .r p ^ and y Xp =x p ,r p ^ 
gf V^K (ay -Ay)' 2 } 
Now S {rip (x p - Xpf) is the second moment of the x p group about its own mean and when the 
distribution is known can often be approximately evaluated. Similarly when the distribution is 
known r p can be estimated and the correction to i; 2 calculated group by group. 
But by making certain assumptions we can very much simplify the work, and a practical 
test, in which the assumptions are not justified, will show the sort of errors which are 
introduced. 
The first assumptions are that the regression is linear and the arrays homoscedastic. In 
this case of course r p is constant and equal to 77 ; we are practically determining a value of r by 
the rj method. 
The correction then becomes 
S[S{m p (x p -x p ) 2 }], 
or writing X 2 = S [S {m p (x p — x p )' 2 }] and H 2 for the raw value of rj 2 after using Pearson's correction, 
we get from (iii) x] 2 = H 2 + rj 2 \' 2 or 
H' 2 
r = {\^) (V1 >- 
To obtain a value for X 2 we still require to postulate something of the nature of the distri- 
bution and I propose to treat (i) of the case where the unit of grouping is constant and small 
enough for the frequency in each group to be considered to be distributed as a trapezium, 
and (ii) of the case where the frequency distribution is normal. 
(i) First to find the second moment of a trapezium about its mean. 
Let z s and % be the ordinates forming the 'walls' of the trapezium and let the group 
unit be h. 
Then y = z s + j "" ^ x is the equation to the 'roof referred to the 'floor' and left hand 
' wall ' as axes. The area is clearly ^ 8 "*"^ . 
The mean is at 
2 2 \(z<-t,)h? z t h*\ _h 2i s , + z s 
(it 
— r I yx dx = 
■ z s'l J 0 ' 
