A text-book of assaying

by C. Beringer; J. J. Beringer

It will be seen that exactly half are within the probable error; but this, considering the small number of results, must be more or less of an accident; it is more to the point they are all well within the limits of error.

I have a large number of other results which with a single exception are all in accord with those given; and this exception only just overstepped the limits. It was like a case of nine trumps, which though in a sense possible, is very unlikely to happen in any one's experience.

But even now we are not quite in a position to answer the question with which we started. If you refer to it you will see that we are face to face with this problem: the limit of variation on the 1000 who died would be say 70,[127] ignoring decimals. But if we calculate on the number who did not die, viz.—699,000,[128] we shall get a variation 26 times as great as this. But it is evident the variation must be the same in each case. I submitted this kind of problem also to the test of experiment, the results of which gave me great faith in Poisson's formula.

Imagine two hundred pennies in a bag all heads up. Any shaking will spoil this arrangement and give a certain proportion of tails. And, further, the probable effect of shaking and turning will be to reduce the preponderance of heads or tails whichever may be in excess. This of course is the reason why we are so unlikely to get more than 120 of them in either position.

But if the two hundred pennies are increased to 20,000 by adding pennies which have tails on both sides, then the shaking or mixing would be less effective. We should still expect as an average result to get the 100 heads but in 20,000 instead of 200. The variation will be 28 or 29 on the 100 instead of 20. And this is a better limit in such cases. Taking 28 as the limit of error on 100 instances and proportionally increasing the others so that the mean error becomes 7.8 and the probable error 5.6, we may now calculate the answer without gross mistake.

The probable variation on the 1000 deaths by accident will be 18, the mean variation will be 24.6, and the limits of variation 88.5. One such table showing in five years a mean number of deaths of about 1120 per annum gives an annual deviation of about 50 up or down of this. It will be seen at once that an improvement of 30 or 40 in any one year would be without meaning, but that an improvement of from 100 to 200 would indicate some change for the better in the circumstances of the industry. Before applying these principles to the elucidation of some of the problems of sampling it will be well to give Poisson's formula (in a modified form) and to illustrate its working.

Let x equal the number of cases of one sort, y the cases of the other sort, and z the total. In the example, z will be the 700,000 engaged in the industry; x will be the 1000 killed by accidents, and y will be the 699,000 who did not so die. The limit of deviation or error calculated by Poisson's formula will be the square root of 8xy/z. Replacing x, y and z by the figures of the example we get the square root of (8×1000×699000)/700,000, which works out to the square root of 7988.57, or 89.3. Which means that we may reasonably expect the number of deaths not to vary from 1000 by more than 89, i.e., they will be between 1090 and 910. It will be seen that this number is in very satisfactory agreement with 88.5 given by the rougher calculation based on my own experiments.

To come to the question of sampling. Consider a powder of uniform fineness and fine enough to pass through an 80 sieve. For purposes of calculation this may be assumed to be made up of particles of about one-eighth of a millimetre across (say roughly 1/200 of an inch); cubed, this gives the content as about 1/500 (strictly 1/512) of a cubic m.m. Now one cubic m.m. of water weighs 1 milligram; therefore 500 such particles if they have the specific gravity of water weigh 1 milligram, and otherwise weigh 1 milligram multiplied by the sp. gr.: 500 particles of ruby silver (Pyrargyrite)[129] will weigh 5.8 milligrams and will contain nearly 3.5 milligrams of silver.

Now suppose a portion of 3.2667 grams (1/10 Assay Ton) of silver ore to contain 500 such particles of ruby silver and no other material carrying silver: such an ore would contain 35 ozs. of silver to the ton. But the limits of variation on 500 particles would be 28[130] multiplied by the square root of 5, or 62 particles. Thus the limit of sampling error would amount to just one-eighth of the silver present, or say to rather more than 4 ozs. to the ton; the mean sampling error would be rather more than a quarter of this, or say about 1.3 ozs. to the ton.