a_. At _s_ the
deviation is _os_; and the number of such deviations is expressed by
_sb_. As the deviations grow greater, the number of them grows less. On
the other side of _o_, toward _-x_, at distances, _op'_, _os'_ (equal to
_op_, _os_) the lines _p'a'_, _s'b'_ represent the numbers of those
errors (equal to _pa_, _sb_).

If _o_ is the average height of the adult men of a nation, (say) 5 ft. 6
in., _s'_ and _s_ may stand for 5 ft. and 6 ft.; men of 4 ft. 6 in. lie
further toward _-x_, and men of 6 ft. 6 in. further toward _x_. There
are limits to the stature of human beings (or to any kind of animal or
plant) in both directions, because of the physical conditions of
generation and birth. With such events the curve _b'yb_ meets the
abscissa at some point in each direction; though where this occurs can
only be known by continually measuring dwarfs and giants. But in
throwing dice or tossing coins, whilst the average occurrence of ace is
once in six throws, and the average occurrence of 'tail' is once in two
tosses, there is no necessary limit to the sequences of ace or of 'tail'
that may occur in an infinite number of trials. To provide for such
cases the curve is drawn as if it never touched the abscissa.

That some such figure as that given above describes a frequent
characteristic of an average with the deviations from it, may be shown
in two ways: (1) By arranging the statistical results of any homogeneous
class of measurements; when it is often found that they do, in fact,
approximately conform to the figure; that very many events are near the
average; that errors are symmetrically distributed on either side, and
that the greater errors are the rarer. (2) By mathematical demonstration
based upon the supposition that each of the events in question is
influenced, more or less, by a number of unknown conditions common to
them all, and that these conditions are independent of one another. For
then, in rare cases, all the conditions will operate favourably in one
way, and the men will be tall; or in the opposite way, and the men will
be short; in more numerous cases, many of the conditions will operate in
one direction, and will be partially cancelled by a few opposing them;
whilst in still more cases opposed conditions will approximately balance
one another and produce the average event or something near it. The
results will then conform to the above figure.

From the above assumption it follows that the symmetrica