

NATURE 



437 



THURSDAY, SEPTEMBER 8, 1892. 



THE HIGHER THEORY OF STATISTICS. 

 Die Grundziige der Theorie der Statistik. Von Harald 

 Wesiergaard, Professor an der Universitat zu Kopen- 

 hagen. (Jena : Verlag von Gustav Fischer.) 



THIS is an important contribution to the Calculus of 

 Probabilities and the higher theory of Statistics. 

 The foundation of experience on which the whole edifice 

 of probabilities is based has been strengthened and ex- 

 tended by the new material which Prof. Westergaard has 

 deposited. Here, for instance, is one of his experiments : — 

 From a bag containing black and white balls in equal 

 numbers, he drew (or caused to be drawn) a ball 10,000 

 times, the ball being replaced in the bag and the bag 

 shaken up after each extraction. He records not only 

 the total numbers of each colour, but also the number of 

 white balls in each of 100 batches, each numbering 100 

 balls, also in 50 batches each of 200 balls, and so on. 

 The diminution of the relative deviation from the average 

 as the size of the batch is increased comes out clearly. 

 On an equally large scale Prof. Westergaard has observed 

 the proportion of prizes to blanks in batches of tickets 

 drawn at a lottery ; and the frequency with which 

 different numbers, drawn under conditions such that one 

 number was as likely as another, were observed to occur 

 actually. He has similarly tabulated the frequency with 

 which the different digits I, 2, 3, &c., terminate certain 

 officially recorded amounts, the " kontis " of a savings 

 bank, of which documents he has examined 10,000. These 

 experiences afford new confirmation to the first principles 

 of the calculus : namely, the fundamental fact of statistical 

 regularity which the definition of probability involves, 

 and the postulate that certain events are independent of 

 each other in such wise that, if the probability of each be 

 \, the probability of the double event is a quarter. 



Ascending from these simple experiences, Prof. 

 Westergaard reaches by a new and easy route the 

 formula for the measure, or modulus, of the extent to 

 which the observed number — eg. of white balls in a batch 

 of 100 or 1000 — is likely to differ from the most probable 

 number ; in the instance just given 100/ or \ooop, if p is 

 the probability of drawing a white ball. The sought ex- 

 pression, it is presumed, must be a symmetric function of 

 the probability of the event (drawing a white ball), which 

 we have called p, and the complementary probability 

 (drawing a non-white ball), viz., i - p. This hint enables 

 us to decipher from the reco rds of e xperience that the 

 modulus is proportioned to '^p(\ - P). The influence of 

 the size of the batch upon the extent of the deviation is 

 similarly elicited from observation. Thus with a mini- 

 mum of mathematical equipment, by easy steps and 

 through an unpretentious a posteriori gate, we are led 

 into the very stronghold of Probabilities — if not to the 

 law of error itself, at any rate to one of its most important 

 properties. 



Prof. Westergaard has not only popularized the law of 

 error, he has also proved it. He has added considerably 

 to its evidence, by observing in an immense number of 

 instances the exact correspondence between fact and 

 theory. We must content ourselves with citing one set of 

 NO. I 193, VOL. 46] 



instances. Ten thousand balls having been drawn at 

 random, as above mentioned, and the composition of 

 each batch of a 100 being examined, it was found that for 

 twenty- five out of a hundred such groups the number of 

 whiteballslaybetween49and5i (inclusive)— limits distant 

 ± I from 50, the most probable number. For forty out 

 of the hundred groups the number of white balls lay 

 between the limits 50 ± 2. And so on. The observations 

 are exhibited and compared with theory in the annexed 

 table :— 



The multiplication of correspondences like this, the 

 concatenation of evidence in favour of the law of error 

 which the author has put together in his fifth chapter, 

 is very cogent. 



Another sort of verification to which the law of error is 

 submitted is to compare it with the explicit binomial to 

 which the exponential law is an approximation. This 

 approximation is closer than may be supposed. For 

 example, if a hundred balls be taken at random, each 

 ball being as likely to be black as white, the probability 

 of obtaining exactly 50 balls, as evaluated by the 

 binomial theorem, is oSo, as approximately determined by 

 the exponential law of error is also •080. The proba- 

 bility of obtaining either 49, 50, or 51 is, according to the 

 exact calculation, "236, according to the approximative 

 formula, also "236. And so on. 



Among other contributions to the calculus which Prof. 

 Westergaard has either adduced from authors rarely 

 read, or himself deduced, may be noticed his elegant 

 treatment of the case where the probabilities of two 

 alternative events (say, drawing white or black balls) 

 are not equal (p. 70). Suppose that the probability, say 

 p, of one event is very small, then the formula for the 

 deviation of the number of white balls actually drawn 

 from the number most likely to be drawn, viz. np, admits 

 of simplification. The "mean error" { = modulus-T-\^-2.) 

 is in general ... (v«/(i-/); in the particular case it 

 becomes approximately 'Jnp. A further simplification 

 may be explained by an example. Suppose that we know 

 the number of deaths, say 900, per unit of time in a cer- 

 tain population. Then, without knowing the number of 

 the population, or without taking the trouble of referring 

 to it and calculating the death-rate, we may determine 

 approximately the fluctuation to which the number of 

 deaths is liable. For the measure of that fluctuation 

 the " mean error," is approximately 'JtTp ; n being the 

 number of population, a large number, and/ the death- 

 rate, a small fraction. Now np is 900, and accordingly 30 

 is the mean error, that is, assuming that the urn in which 

 the lots of Fate are shaken — "omnium Versatur urna 



U 



