*POPULAR SCIENCE MONTHLY.*

normally to a fair degree of approximation, and consequently we may infer that our results are trustworthy indications of real facts.

A talent is a sum whose exact value few of us care to know, although we all appreciate the inner sense of the beautiful parable. I will, therefore, venture to adapt the phraseology of the allegory to my present purpose by substituting for 'talent' the words 'normal-talent.' The value of this normal-talent in respect to each and any specified quality or faculty is such that one-quarter of the people receive for their respective shares more than one normal-talent *over* and *above* the average of all the shares. Our normal-talent is therefore identical with what is technically known as the 'probable error.' Therefrom the whole of the following table starts into life, evolved from that of the *probability integral* It expresses the distribution of any normal

Table I.—*Normal Distribution (to the nearest per ten thousand and to the nearest per hundred).*

-4° | -3° | -2° | -1° | M+1° | +2° | +3° | +4° | Total | ||

ν and below |
u | t | s | r | R | S | T | U | V and above. | |

35 | 180 | 672 | 1613 | 2500 | 2500 | 1613 | 672 | 182 | 35 | 10,000 |

2 | 7 | 16 | 25 | 25 | 16 | 7 | 2 | 100 |

quality, or any group of normal qualities, among 10,000 persons in terms of the normal-talent. The M in the upper line occupies the position of Mediocrity, or that of the average of what all have received: the 1°, 3°, etc., and the 1°, 2°, etc., refer to normal talents. These numerals stand as graduations at the heads of the vertical lines by which the table is divided. The entries between the divisons are the numbers per 10,000 of those who receive sums between the amounts specified by those divisions. Thus, by the hypothesis, 2,500 receive more than M but less than M1°, 1,613 receive more than M1° but less than M2°, and so on. The terminals have only an inner limit, thus 35 receive more than 4°, some to perhaps a very large but indefinite amount. The divisions might have been carried much farther, but the numbers in the classes between them would become less and less trustworthy. The left half of the series exactly reflects the right half. As it will be useful henceforth to distinguish these classes, T have used the *capital* or large letters, E, S, T, U, V, for those above mediocrity and corresponding italic or small letters, *r, s, t, u, v,* for those below mediocrity, *r* being the counterpart of R, *s* of S, and so on.

In the lowest line the same values are given, but more roughly, to the nearest whole percentage.

It will assist in comprehending the values of different grades of