d 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, 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,