[Illustration:  FIG. 11.]

Here o is the average event, and oy represents the number of average events.  Along ox, in either direction, deviations are measured.  At p the amount of error or deviation is op; and the number of such deviations is represented by the line or ordinate pa.  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.