1 THEORY OF STATISTICS. 
Example i., and in that case, of course, the standard deviations 
will also require reduction to the mean. 
As the arithmetical process of calculating the correlation co- 
efficient from a grouped table is of great importance, we give two 
illustrations, the first economic, the second biological. 
Example ii., Table VIII.—The two variables are (1) X, the 
percentage of males over 65 years of age in receipt of Poor-law 
relief in 235 unions of a mainly rural character in England and 
Wales ; (2) Y, the ratio of the numbers of persons given relief  out- 
doors” (in their own homes) to one “indoors” (in the workhouse). 
The figures refer to a one-day count (Ist August 1890, No. 36, 
1890), and the table is one of a series that were drawn up with 
the view to discussing the influence of administrative methods on 
pauperism. (Economic Journal, vol. vi., 1896, p. 613.) 
The arbitrary origin for X was taken at the centre of the fourth 
column, or at 17'5 per cent. ; for ¥ at the centre of the fourth 
row, or 3-5. The following are the values found for the constants 
of the single distributions :— 
£= - 01532 intervals= — 0"77 per cent., whence J, = 
16-73 per cent. 
o,=1'29 intervals = 6-45 per cent. 
7j= + 0°36 intervals or units, whence J, = 3-86. 
0, =2'98 units. 
To calculate 3(é7), the value of & is first written in every 
compartment of the table against the corresponding frequency, 
treating the class-interval as the unit: these are the figures in 
heavy type in Table VIII. In making these entries the sign of 
the product may be neglected, but it must be remembered that 
this sign will be positive in the upper left-hand and lower right- 
hand quadrants, negative in the two others. The frequencies are 
then collected as shown in columns 2 and 3 of Table VIIIa., 
being grouped according to the value and sign of é&y. Thus for 
én=1, the total frequency in the positive quadrants is 13+ 85 
= 215, in the negative 14+6=20: for &=2, 10+45+1+45 
=20 in the positive quadrants, 5+2+1+35=11'56 in the 
negative, and so on. When columns 2 and 3 are completed, they 
should first of all be checked to see that no frequency has been 
dropped, which may be readily done by adding together the totals 
of these two columns together with the frequency in row 4 and 
column 4 of Table VIII. (the row and column for which én=0), 
being careful not to count twice the frequency in the compartment 
common to the two; this grand total must clearly be equal to the 
total number of observations &, or 235 in the present case. The 
algebraic sum of the frequencies in each line of columns 2 and 3 is 
+892