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, 75-1/2, 76, 78, 78, 79, 80-1/2, 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.

To make a single measurement of a phenomenon does not give one much confidence.  Another measurement is made; and then, if there is no opportunity for more, one takes the mean or average of the two.  But why?  For the result may certainly be worse than the first measurement.  Suppose that the events I am measuring are in fact fairly described by Fig.  II, although (at the outset) I know nothing about them; and that my first measurement gives p, and my second s; the average of them is worse than p.  Still, being yet ignorant of the distribution of these events, I do rightly in taking the average.  For, as it happens, 3/4 of the events lie to the left of p; so that if the first trial gives p, then the average of p and any subsequent trial that fell nearer than (say) s’ on the opposite side, would be better than p; and since deviations greater than s’ are rare, the chances are nearly 3 to 1 that the taking of an average will improve the observation.  Only