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
