From the above assumption it follows that the symmetrical curve describes only a ‘homogeneous class’ of measurements; that is, a class no portion of which is much influenced by conditions peculiar to itself.  If the class is not homogeneous, because some portion of it is subject to peculiar conditions, the curve will show a hump on one side or the other.  Suppose we are tabulating the ages at which Englishmen die who have reached the age of 20, we may find that the greatest number die at 39 (19 years being the average expectation of life at 20) and that as far as that age the curve upwards is regular, and that beyond the age of 39 it begins to descend regularly, but that on approaching 45 it bulges out some way before resuming its regular descent—­thus: 

[Illustration:  FIG. 12.]

Such a hump in the curve might be due to the presence of a considerable body of teetotalers, whose longevity was increased by the peculiar condition of abstaining from alcohol, and whose average age was 45, 6 years more than the average for common men.

Again, if the group we are measuring be subject to selection (such as British soldiers, for which profession all volunteers below a certain height—­say, 5 ft. 5 in.—­are rejected), the curve will fall steeply on one side, thus: 

[Illustration:  FIG. 13.]

If, above a certain height, volunteers are also rejected, the curve will fall abruptly on both sides.  The average is supposed to be 5 ft. 8 in.

The distribution of events is described by ‘some such curve’ as that given in Fig. 11; but different groups of events may present figures or surfaces in which the slopes of the curves are very different, namely, more or less steep; and if the curve is very steep, the figure runs into a peak; whereas, if the curve is gradual, the figure is comparatively flat.  In the latter case, where the figure is flat, fewer events will closely cluster about the average, and the deviations will be greater.

Suppose that we know nothing of a given event except that it belongs to a certain class or series, what can we venture to infer of it from our knowledge of the series?  Let the event be the cephalic index of an Englishman.  The cephalic index is the breadth of a skull x 100 and divided by the length of it; e.g. if a skull is 8 in. long and 6 in. broad, (6x100)/8=75.  We know that the average English skull has an index of 78.  The skull of the given individual, therefore, is more likely to have that index than any other.  Still, many skulls deviate from the average, and we should like to know what is the probable error in this case.  The probable error is the measurement that divides the deviations from the average in either direction into halves, so that there are as many events having a greater deviation as there are events having a less deviation.  If, in Fig. 11 above, we have arranged the measurements of the cephalic index of English adult males, and