164 THEORY OF STATISTICS. 
of the one set running vertically and the other horizontally, and 
the difference has no statistical significance, the word array 
has been suggested as a convenient term to denote either a row 
or a column. If the values of X in one array are associated 
with values of ¥ between the limits ¥,, —8 and Y, +38, ¥, may be 
termed the type of the array. (Pearson, ref. 6.) The special 
kind of contingency tables with which we are now concerned 
are called correlation tables, to distinguish them from tables 
based on unmeasured qualities and so forth. 
3. Nothing need be added to what was said in Chapter VI. as 
regards the choice of magnitude and position of class-intervals. 
When these have been fixed, the table is readily compiled by 
taking a large sheet ruled with rows and columns properly 
headed in the same way as the final table and entering a dot, 
stroke, or small cross in the corresponding compartment for each 
pair of recorded observations. If facility of checking be of 
great importance, each pair of recorded values may be entered 
on a separate card and these dealt into little packs on a board 
ruled in squares, or into a divided tray; each pack can then be 
run through to see that no card has been mis-sorted. The 
difficulty as to the intermediate observations—values of the 
variables corresponding to divisions between class-intervals—will 
be met in the same way as before if the value of one variable 
alone be intermediate, the unit of frequency being divided 
between two adjacent compartments. If both values of the pair 
be intermediates, the observation must be divided between four 
adjacent compartments, and thus quarters as well as halves may 
occur in the table, as, e.g., in Table III. In this case the statures 
of fathers and sons were measured to the nearest quarter- 
inch and subsequently grouped by l-inch intervals: a pair in 
which the recorded stature of the father is 60'5 in. and that of 
the son 62-5 in. is accordingly entered as 0°25 to each of the 
four compartments under the columns 59:5-605, 60:5-61'5, and 
the rows 61'5-62-5, 62:5-63'5. Workers will generally form 
their own methods for entering such fractional frequencies 
during the process of compiling, but one convenient method is 
to use a small x to denote a unit and a dot for a quarter; the 
four dots should be placed in the position of the four points 
of the x and joined when complete. It is best to choose the 
limits of class-intervals, where possible, in such a way as to avoid 
fractional frequencies. 
4. The distribution of frequency for two variables may be 
represented by a surface or solid in the same way as the frequency- 
distribution of a single variable may be represented by a plane 
figure. We may imagine the surface to be obtained by erecting