VL—THE FREQUENCY-DISTRIBUTION. 87 
to the maximum that a histogram is, on the whole, the better re- 
presentation of the distribution of frequency, and in such a 
distribution as that of Table IV. the use of the histogram is 
almost imperative. 
12. If the class-interval be made smaller and smaller, and at 
the same time the number of observations be proportionately in- 
creased, so that the class-frequencies may remain finite, the 
polygon and the histogram will approach more and more closely 
to a smooth curve. Such an ideal limit to the frequency-polygon 
or histogram is termed a frequency-curve. In this ideal frequency- 
curve the area between any two ordinates whatever is strictly 
proportional to the number of observations falling between the 
corresponding values of the variable. Thus the number of 
observations falling between the values z, and z, of the variable 
in fig. 4 will be proportional to the area of the shaded strip in the 
figure; the number of observed values greater than z, will 
similarly be given by the area of the curve to the right of the 
ordinate through z,, and so on. When, in any actual case, the 
number of observations is considerable—say a thousand at least 
—the run of the class-frequencies is generally sufficiently 
smooth to give a good notion of the form of the ideal distri- 
bution ; with small numbers the frequencies may present all 
kinds of irregularities, which, most probably, have very little 
significance (¢f. Chap. XV. § 15, and § 18, Ex. iv.). The forms 
presented by smoothly running sets of numerous observations 
present an almost endless variety, but amongst these we notice 
a small number of comparatively simple types, from which many 
at least of the more complex distributions may be conceived as 
compounded. For elementary purposes it is sufficient to consider 
these fundamental simple types as four in number, the symmetri- 
cal distribution, the moderately asymmetrical distribution, the 
extremely asymmetrical or J-shaped distribution, and the U-shaped 
distribution. 
13. The symmetrical distribution, the class-frequencies decreas- 
ing to zero symmetrically on either side of a central maximum. 
Fig. 5 illustrates the ideal form of the distribution. 
Being a special case of the more general type described under 
the second heading, this form of distribution is comparatively rare 
under any circumstances, and very exceptional indeed in economic 
statistics. It occurs more frequently in the case of biometric, more 
especially anthropometric, measurements, from which the following 
illustrations are drawn, and is important in much theoretical work. 
Table VI. shows the frequency-distribution of statures for adult 
males in the British Isles, from data published by a British 
Association Committee in 1883, the figures being given separately