It may be doubted whether many not specially trained in the study of statistics could answer such a question as the following:--Seven hundred thousand men being employed, there are, in a given year, one thousand deaths from accident. a.s.suming the conditions to remain unaltered, within what limits could one foretell the number of deaths by accident in any other year?
On the other hand, there is a widespread belief in the efficacy of what is called the law of averages. Even the ordinary newspaper reader is accustomed to look on the national death-rate or birth-rate as a thing capable of being stated with accuracy to one or two places of decimals, and he knows that the annual number of suicides is practically constant.
If a man played whist often and kept a record of the number of trumps n each hand, he would find fortune treated him quite fairly; in a year"s play the average number would deviate very little from the theoretical average, _i.e._, one-quarter of thirteen. And a knowledge of this truth is useful, and that not merely in keeping e.j.a.c.u.l.a.t.i.o.ns in due restraint.
But every good player knows more than this: he has a sense of what variations in the number of trumps may reasonably be expected. For example, he will be prepared to risk something on neither of his opponents having more than five trumps, and will accept it as a practical certainty that no one has more than eight. Much of what is known as good judgment is based on a proper estimate of deviations from the average. The question has an important bearing on sampling, as may be seen from the fact that shuffling and dealing at cards are but modifications of the well-known mixing and quartering of the sampler.
Because of this bearing on sampling and for other reasons, I became many years ago much interested in the question, and gave to its solution perhaps more labour than it was worth. In books on Medical Statistics the answer to the question is stated in a mathematical formula, called Poisson"s formula, which, in a modified form, I shall give further on.
But this did not satisfy me, because I wanted to learn what a reasonably safe _limit of error_ actually meant, and this could be best learnt by experiment; so with the help of some friends I went in for a thorough course of penny-tossing.
Tossing a penny twenty times, an average result would be ten heads and ten tails. To find the deviations from this, we tossed two hundred twenties, _i.e._, four thousand times. Of the two hundred, thirty-three gave the exact average, viz.:--10 heads; sixty-four gave an error of one, viz.:--9 or 11 heads; forty-nine, an error of two; twenty-six, an error of three; twenty, an error of four; eight gave an error of five, and this limit was not exceeded. From these we may say that six is a reasonably safe limit of error. Ninety-seven cases, say one-half, gave an error not exceeding one; and the mean error is 1.8.
In other words, in twenty tosses you will not get more than 16 nor less than 4 heads; you are as likely as not to get 9, 10, or 11 heads; and lastly, if you lost in twenty throws all heads or tails over 10 your average loss would be 1.8 penny, or say roughly 2d. on the twenty throws.
It was necessary to compare these with another series containing a larger average, say that of 100 heads in 200 throws. I confess the labour of tossing pennies two hundred at a time was little to our taste.
So from a bag of pennies borrowed from the bank, we weighed out samples containing two hundred, and for an evening we were busy counting heads and tails in these. The heads in sixty samples ranged from 80 to 114.
One hundred heads occurred seven times. The extent and frequency of the errors is shown in the table.
------+-------+------+-------+------+------- Error.|No. of |Error.|No. of |Error.|No. of | Times.| | Times.| | Times.
------+-------+------+-------+------+------- 1 | 8 | 6 | 3 | 11 | 1 2 | 5 | 7 | 3 | 14 | 3 3 | 6 | 8 | 3 | 15 | 1 4 | 3 | 9 | 7 | 18 | 2 5 | 6 | 10 | 1 | 20 | 1 --------------------------------------------
We may call the limit of error 21. Twenty-nine results out of sixty, say one-half, had an error not exceeding 4; and the mean error is 5.6. In comparing these with the series 10 in 20 we must, working by rule, divide not by 10 but by 3.16, the square root of 10; for if we multiply an average by any number[126] the error is also multiplied but only by the square root of the number. The error varies as the square root of the number. Now
21/3.16 = 6.6 = limit of error for 10 in 20.
5.6/3.16 = 1.8 = mean error " " "
4/3.16 = 1.2 = probable error " " "
It will be seen that these calculated results agree fairly well with those actually obtained. The rule by which these calculations are made is important and will bear further ill.u.s.tration. To calculate the number of heads in 3200 throws, we have to find the limit of error on a true average of 1600 in 3200. This being 16 times the average of 100 in 200, the corresponding errors must be multiplied by 4. This gives
214 = 84 = limit of error.
5.64 = 22.4 = mean error.
44 = 16 = probable error.
The results I have actually obtained with these large numbers are hardly enough to base much on, but have a value by way of confirmation.
Expecting 1600 heads, the actual numbers were 1560, 1596, 1643, 1557, 1591, 1605, 1615, 1545.
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 circ.u.mstances 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 ill.u.s.trate 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 8_xy_/_z_. Replacing _x_, _y_ and _z_ by the figures of the example we get the square root of (81000699000)/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 pa.s.s through an 80 sieve. For purposes of calculation this may be a.s.sumed 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 a.s.say 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.
On the other hand, if one took for the a.s.say a charge six times greater (say about 20 grams), the number of particles would be 3000 and the limits of variation would be 28 multiplied by the square root of 30, or 153 particles, which is very closely 1/20 of the silver present, or say 1.75 ozs. to the ton, whilst the mean error would amount to about .5 ozs. to the ton.
To work these examples by Poisson"s formula let us a.s.sume the gangue to have a mean sp. gr. of 3. Then 500 particles would weigh 3 milligrams; and 3.2609[131] grams would contain 543,500 particles. There would be then 500 of ruby silver and 543,500 of gangue, together 544,000, and the formula gives the square root of (8500543500)/544000, which works out to 63 particles as against 62 by the other method.
A practical conclusion from this is of course that either the ore must be powdered more finely or a larger portion than 3 grams must be taken for the a.s.say. Moreover, it is evident that on such an ore no small sample must be taken containing less than several million particles.
Consider now a copper ore of the same uniform fineness containing particles of copper pyrites (sp. gr. 4) of which 1000 particles will weigh 8 milligrams, mixed with gangue of which 1000 particles weigh 6 milligrams.
If one gram of such ore contain .5 gram of copper pyrites (= .1725 gram copper) and .5 gram of gangue, these will contain 62,500 and say 83,500 particles respectively. Altogether 146,000 particles. With Poisson"s formula this gives the limit of sampling error as the square root of (86250083500)/146000 or 521 particles. But a variation of 521 on 62,500 is a variation of .83 per cent. The percentage of copper in the ore is 17.25 per cent., and .83 per cent. of this is .14 per cent. The limits of sampling error, therefore, are 17.11 per cent. and 17.39 per cent. Again, it must be remembered that the mean sampling error would be a little over one-quarter of this, or say from 17.2 per cent. to 17.3 per cent. The practical conclusion is that a powder of this degree of fineness is not fine enough. In the last place let us consider a similar iron ore containing 90 per cent. of haemat.i.te (sp. gr. 5) and 10 per cent. of gangue (sp. gr. 3), 1 gram of such ore will contain 90,000 particles of haemat.i.te weighing .9 gram and containing .63 gram of iron with say 16,500 particles of gangue weighing .1 gram. Altogether 106,500 particles.
Poisson"s formula then gives the limits of variation as the square root of (89000016500)/106500 or 334 particles. But 334 on 90,000 is 0.23 on 63.0, which is the percentage of iron present. The limits of sampling error then are 62.77 per cent. and 63.23 per cent. and the mean variation is from 62.94 per cent. to 63.06 per cent.
These examples are worthy of careful consideration, and it must be remembered that the calculations are made on the a.s.sumption that the ore is made up of uniform particles of mineral of such fineness as would pa.s.s easily through an 80 sieve, but which does not pretend to represent with great exactness the fineness of the powdered ore customary in practice. They show that having pa.s.sed through such a sieve is no proof of sufficient powdering, not that all ores powdered and so sifted are unfit for a.s.saying. This last would be an absurd and illogical conclusion.
If an ore be powdered to a fairly fine sand and then be pa.s.sed through a series of sieves, say a 40, 60, and 80, in such a state that little or none remains on the first, but the others retain a large proportion; then of that which comes through the 80 sieve, perhaps two-thirds by weight may be even coa.r.s.er than the powder I have used in the example.
Of the rest most may be of about half this diameter; the weight of the really fine powder may be quite inconsiderable. On the other hand, if the grinding be continued until, on sifting, little or nothing that is powderable remains on the sieves; then in the sifted product the proportions will be very different. This last, of course, is the only right way of powdering. Also it is evident that so much depends on the manner of powdering that nothing precise can be stated as to the average coa.r.s.eness of the powder. Suppose, however, by good powdering a product is obtained which may be represented by a uniform powder with particles 1/20th of a millimetre in diameter (say roughly 1/500 inch). Compared with the previous powder, the diameter has been divided by 2.5; their number, therefore, in any given weight has been increased by the cube of 2.5, which is 15.6. But the value of a sample varies as the square root of the number of particles. Hence the reduction in size and consequent increase in number has made the sample nearly four times better than before; and it will be seen that this brings the sampling error within tolerable limits.
There are one or two words of warning which should be given. In the first place, using a 90 sieve instead of an 80 must not be too much relied on; the powder I took in the example would pa.s.s through it. It is a question of good powdering rather than of fine sifting. In the second place, a set of, say half-a-dozen, a.s.says concordant within 1 oz. where the theory gives 4 ozs. as the limit of error does not upset the theory: the theory itself states this as likely. It is the error you _may_ get in one or two a.s.says out of a hundred, not the error you are _likely_ to get in any one a.s.say, which is considered under the heading "limit of error."
Accepting the result just arrived at that a portion of 1 gram may be safely taken for an a.s.say if the particles are 1-20th of a millimetre in diameter, the further question remains as to what weight of the original sample must be reduced to this degree of fineness. This may be answered on the principle that the same degree of excellence should be aimed at in each of a series of samplings. This principle is ill.u.s.trated in the table on page 2.
A fine sand, such as would pa.s.s a 40 sieve but be retained on a 60 sieve, would be fairly represented by particles one-quarter of a millimetre in diameter. This being five times coa.r.s.er, to contain the same number of particles must be 125 times (the cube of 5) as heavy; therefore 125 grams of it can be taken with the same degree of safety as 1 gram of the finer powder. Of such a sand about this weight should be taken and reduced to the finer powder. If the ore were in coa.r.s.e sand, say in particles 1 millimetre in diameter, this would be four times as coa.r.s.e as that last considered, and we should have to take 64 times as much of it: 64 times 125 grams is 8 kilos, or say roughly from 15 to 20 lbs. This should be crushed to the finer size and mixed; then from 100 to 150 grams should be taken and ground to the finest powder.
There is, however, a reason why, on the coa.r.s.er stuff, a smaller proportion may safely be used. This becomes more evident if we consider a still coa.r.s.er sample. A heap of ore in stones about 2 inches across would be 50 times coa.r.s.er than the sand, and an equivalent sample would need to be 125,000 times heavier; this would amount to about 1000 tons.
Experienced samplers would say that under such conditions so large a sample was hardly necessary.
This is because I have a.s.sumed in the calculations that the grains of copper pyrites, for example, were all copper pyrites and the particles of gangue were free from copper. This would be true or nearly so for the very fine powder, but far from true in the case of the ore heap. In the heap probably few of the stones would be pure ore and still fewer would be free from copper. The stones would differ among themselves in their copper contents only within certain comparatively narrow limits. And it is evident that, if replacing one stone by another, instead of resulting in the gain or loss of all the copper one or other contained, merely affected the result to one-tenth of this amount, then a sample of 1-100th of the weight (say 10 tons) would be equally safe.
It should be remembered, however, that while the man who samples on a large scale can safely and properly reduce the size of his samples on this account, yet the principle is one which counts less and less as the stuff becomes more finely divided, and ought to be ignored in the working down of the smaller samples which come to the a.s.sayer.
FOOTNOTES:
[126] The 10 in 20 multiplied by 10 = 100 in 200.
[127] Multiply the errors for 100 by the square root of 10.
[128] Multiply the errors for 100 by the square root of 6990.
[129] Sp. Gr. 5.8. Silver 60 per cent.
[130] Taking 28 as the limit of variation on 100.
[131] The weight of the ore less the weight of ruby silver in it.