THEORY OF STATISTICS. 
EXERCISES. 
1. Deduce equation (11) from the equations for transformation of co-ordinates 
without assuming the normal distribution. (A proof will be found in ref. 10.) 
2. Hence show that if the pairs of observed values of ; and x, are repre- 
sented by points on a plane, and a straight line drawn through the mean, the 
sum of the squares of the distances of the points from.this line is a minimum 
if the line is the major principal axis. 
3. The coefficient of correlation with reference to the principal axes being 
zero, and with reference to other axes something, there must be some pair of 
axes at right angles for which the correlation is a maximum, ¢.e. is numerically 
greatest without regard to sign. Show that these axes make an angle of 45° 
with the principal axes, and that the maximum value of the correlation is— 
L3H 
~ 343 
4. (Sheppard, ref. 12.) A fourfold table is formed from a normal correla- 
tion table, taking the points of division between 4 and a, B and B, at the 
medians, so that (4)=(a)=(B)=(B)=N/2. Show that 
DJ 
T= COS (1 yi 
234