the conversion of results obtained by the
use of one series of test-weights into what would have been given
by another series, is a piece of simple arithmetic, the fact
ultimately obtained by any apparatus of this kind being the "just
distinguishable" fraction of real weight. In my own apparatus the
unit of weight is 2 per cent.; that is, the register-mark 1 means 2
per cent.; but I introduce weights in the earlier part of the scale
that deal with half units; that is, with differences of 1 per cent.
In another apparatus the unit of weight might be 3 per cent., then
three grades of mine would be equal to two of the other, and mine
would be converted to that scale by multiplying them by 2/3. Thus
the results obtained by different apparatus are strictly comparable.

A sufficient number of test-weights must be used, or trials made, to
eliminate the influence of chance. It might perhaps be thought that
by using a series of only five weights, and requiring them to be
sorted into their proper order by the sense of touch alone, the
chance of accidental success would be too small to be worth
consideration. It might be said that there are 5 x 4 x 3 x 2, or 120
different ways in which five weights can be arranged, and as only
one is right, it must be 120 to 1 against a lucky hit. But this is
many fold too high an estimate, because the 119 possible mistakes
are by no means equally probable. When a person is tested, an
approximate value for his grade of sensitivity is rapidly found, and
the inquiry becomes narrowed to finding out whether he can surely
pass a particular mistake. He is little likely to make a mistake of
double the amount in question, and it is almost certain that he will
not make a mistake of treble the amount. In other words, he would
never be likely to put one of the test-weights more than one step
out of its proper place. If he had three weights to arrange in their
consecutive order, 1, 2, 3, there are 3x2 = 6 ways of arranging them;
of these, he would be liable to the errors of 1, 3, 2, and of 2, 1, 3,
but he would hardly be liable to such gross errors as 2, 3, 1, or 3,
2, 1, or 3, 1, 2. Therefore of the six permutations in which three
weights may be arranged three have to be dismissed from consideration,
leaving three cases only to be dealt with, of which two are wrong
and one is right. For the same reason there are only four reasonable
chances of error in arranging four weights, and only six in
arranging five