IX.—CORRELATION. 165 
at the centre of every compartment of the correlation-table a 
vertical of length proportionate to the frequency in that com- 
partment, and joining up the tops of the verticals. If the 
compartments were made smaller and smaller while the class- 
frequencies remained finite, the irregular figure so obtained would 
approximate more and more closely towards a continuous curved 
surface—a frequency-surface — corresponding to the frequency- 
curves for single variables of Chapter VI. The volume of the 
frequency-solid over any area drawn on its base gives the 
frequency of pairs of values falling within that area, just as the 
area of the frequency-curve over any interval of the base-line gives 
the frequency of observations within that interval. Models of 
actual distributions may be constructed by drawing the frequency- 
distributions for all arrays of the one variable, to the same scale, 
on sheets of cardboard, and erecting the cards vertically on a 
base-board at equal distances apart, or by marking out a base- 
board in squares corresponding to the compartments of the 
correlation-table, and erecting on each square a rod of wood of 
height proportionate to the frequency. Such solid representations 
of frequency-distributions for two variables are sometimes termed 
stereograms. 
5. It is impossible, however, to group the majority of 
frequency-surfaces, in the same way as the frequency-curves, 
under a few simple types: the forms are too varied. The simplest 
ideal type is one in which every section of the surface is a sym- 
metrical curve—the first type of Chap. VL (fig. 5, p. 89). Like 
the symmetrical distribution for the single variable, this is a very 
rare form of distribution in economic statistics, but approximate 
illustrations may be drawn from anthropometry. Fig. 29 shows 
the ideal form of the surface, somewhat truncated, and fig. 
30 the distribution of Table III., which approximates to the same 
type,—the difference in steepness is, of course, merely a matter of 
scale. The maximum frequency occurs in the centre of the 
whole distribution, and the surface is symmetrical round the 
vertical through the maximum, equal frequencies occurring at 
equal distances from the mode on opposite sides. The next 
simplest type of surface corresponds to the second type of 
frequency-curve—the moderately asymmetrical. Most, if not all, 
of the distributions of arrays are asymmetrical, and like the dis- 
tribution of fig. 9, p. 92: the surface is consequently asymmetrical, 
and the maximum does not lie in the centre of the distribution. 
This form is fairly common, and illustrations might be drawn 
from a variety of Sources—economics, meteorology, anthropometry, 
ete. The data of Table IL. will serve as an example. The total 
distributions and the distributions of the majority of the arrays 
i: