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:
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 × 100 and divided by the length of it; e.g. if a skull is 8 in. long and 6 in. broad, (6×100)/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 if at o (the average or mean) the index is 78, and if the line pa divides the right side of the fig. into halves, then op is the probable error. If the measurement at p is 80, the probable error is 2. Similarly, on the left hand, the probable error is op', and the measurement at p' is 76. We may infer, then, that the skull of the man before us is more likely to have an index of 78 than any other; if any other, it is equally likely to lie between 80 and 76, or to lie outside them; but as the numbers rise above 80 to the right, or fall below 76 to the left, it rapidly becomes less and less likely that they describe this skull.
In such cases as heights of men or skull measurements, where great numbers of specimens exist, the average will be actually presented by many of them; but if we take a small group, such as the measurements of a college class, it may happen that the average height (say, 5 ft. 8 in.) is not the actual height of any one man. Even then there will generally be a closer cluster of the actual heights about that number than about any other. Still, with very few cases before us, it may be better to take the median than the average. The median is that event on either side of which there are equal numbers of deviations. One advantage of this procedure is that it may save time and trouble. To find approximately the average height of a class, arrange the men in order of height, take the middle one and measure him. A further advantage of this method is that it excludes the influence of extraordinary deviations. Suppose we have seven cephalic indices, from skeletons found in the same barrow, 75½, 76, 78, 78, 79, 80½, 86. The average is 79; but this number is swollen unduly by the last measurement; and the median, 78, is more fairly representative of the series; that is to say, with a greater number of skulls the average would probably have been nearer 78.