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mathematics, and which has been brought to so splendid a development by Laplace and Gauss. Only thus will statistics grow to be a science, in the full meaning of the word ; and only thus will really useful and trustworthy results be obtained. It may even be predicted that, in a future century, Mathematical Statistics will solve problems of the bare enunciation of which we have as yet no idea.

What has been here said of statistics in general, is not equally true of all the divisions of the science; for it is only the statistics of population which now manifestly admits of mathematical treatment, whereas the other divisions must for the present be excluded from it. In the statistics of population some steps have indeed been already made, which may be considered as an attempt to lay the foundation of mathematical statistics, but these steps are so far from satisfactory, and have made so little use of the means of investigation which analysis in its present advanced condition could have supplied, that it is necessary to disregard them and commence the investigation anew from the very beginning.

The first and most important idea with which the mathematical treatment of the statistics of a population has to deal, is that of mortality; for this enters into every question that can be proposed with regard to a population. For the purpose of answering enquiries as to the mortality that will prevail among a given population or any other given body of persons, it has been usual to form what is known as a mortality table, which represents the rate at which that body of persons will die off, on the supposition that the rate of mortality prevailing at one instant of time will remain unchanged thereafter. A large number of such mortality tables are to be found collected in statistical text books, but they are all, more or less, of doubtful value. Various attempts have also been made to find a formula that shall represent the number living at any age in the table; but these attempts were certain to fail, so long as the fundamental facts were deficient in the accuracy which, as stated above, was characteristic of Tycho's observations. Even the function, first proposed by Gompertz and since discussed by so many persons, y=ab*, which leads in its application to logarithmic integrals and gamma functions, must remain a very doubtful hypothesis until a sufficiently trustworthy collection of facts can be obtained to test its

accuracy. It is, indeed, no exaggeration to assert that all existing mortality tables, without exception, are far from being so trustworthy as could be desired. This is owing, partly to the imperfection of


the facts on which they are based, partly to the imperfection of the methods hitherto employed in their construction. As regards the former, the censuses in Germany are at present taken in so unsatisfactory a manner, that calculations based on them can only be considered as rough approximations to the truth. This is, at all events, true as regards the censuses within the limits of the Zollverein, whereas in Belgium and France, where to be sure they go to greater expense, matters appear to be in a better condition. Having elsewhere stated my views as to what science requires in a census, I will not repeat them here,* but will only add that, unfortunately, the public are but little disposed to assist in the matter. It is, for instance, well known that women universally state their ages too low; nay, at the last census in Hanover, two ladies positively refused to state their ages at all, so that the enumeraters could do nothing but guess at them. Under such circumstances, we must, perhaps, give up all idea of ever obtaining a completely trustworthy census. More accurate data can be obtained from the experience of Life Insurance Companies, Annuity Companies, Widows' Funds, and similar societies, but hitherto very few of these have been at the pains to extract and arrange,

with view to strict mathematical treatment, the statistical facts lying buried in their records; and we therefore possess much fewer mortality tables derived from such sources than might have been expected. But to tables of this kind the second of the above mentioned objections applies; for they are defective, in consequence of the defective methods employed in their construction. Even the two which at the present time are considered the best, and are extensively used, namely, Brune's Table, deduced from the experience of the General Widows' Fund, in Berlin, and that deduced from the experience of the 17 English Offices, are open to this objection; for the methods employed in their construction, so far as they are known, are far from being theoretically correct.

It must be admitted that the labour bestowed by statisticians on the formation of mortality tables is so far justified, that a mortality table presents in a very clear and intelligible form the facts to which it relates, and exhibits extremely well the instantaneous changes in a population. But, after all, it is only a popular form, insufficient for scientific purposes, and it has accordingly been forcibly pointed out by other writers † that the formation of a

* See my Essay, Zur Bevolkerungs-Statistik, in the Zeitschrift des Königlich Preussischen statistischen Bureau, 3rd year, Part I.

† See, especially, Fischer's “Grundzüge des auf die menschliche Sterblichkeit gegründeten Versicherungswesens,” Oppenheim, 1860.

mortality table cannot be the first step of scientific enquiry. The fundamental idea from which our treatment of the statistics of a population must start, is rather that of the probability of dying, or more exactly, the probability that a person who belongs to a definite group will die within a year. Statistics has hitherto lacked this idea, because, as already remarked, it has not had a correct idea of the calculus of probabilities in general ; and with the introduction of the above idea, the science becomes essentially a new one. We can, if we please, substitute for the above probability, that of living a year, which is the complement to unity of it. If either the one or the other of these probabilities is known for every year of age, it requires but little labour to deduce a mortality table.

The analytical expedients required for the solution of the problem have been so fully developed by Laplace, in his Théorie Analytique des Probabilités, that it is rather surprising they have not been earlier applied to it. It must be confessed, however, that it is but lately that the want has become fully felt, in consequence of various recent writers having attempted the solution of the questions under consideration. These have certainly, one after the other, made steps towards it, but have not yet reached the desired goal. We shall here endeavour, with the help of the above mentioned expedients, to make another step forward. If it has been observed that out of a group

of L

persons, there are L' alive at the end of a year, and there have been no entries into this group, nor withdrawals from it during the year (in which cases a special investigation would be required), it has been the custom hitherto to say that the probability, p, of living a year is given for each person of this group by the equation

L p=

L. But it is not forgotten to add that if, under the like circumstances, other values of L and L' were given, the value of p may generally be different. Hence it logically follows that the calculation gives the value of p only inexactly; and thus originates the strange idea prevailing among the unlearned that the calculus of probabilities gives inexact results. Now, in reality, the above form of expression is wholly incorrect. The value of p given by the above equation makes no claim whatever to be the true value of the unknown probability, but it is the most probable value resulting from the observations that have been made. Now to the most probable value there always belongs a probable error, and a mean

error, either of which may be used to measure the degree of trustworthiness of that most probable value. This probable and this mean error have, however, never yet been determined.

Again, if the probability p of living a year is given, and we wish, under the same circumstances as before, to know the number, l', which will be alive at the end of a year out of a new group of l persons, it has hitherto been the custom to say that it is given by the equation,

l'=lp. At the same time it is admitted, that actual observation may show a number different from l', so that in this case too the calculation may be charged with being inexact. But here, also, the usual form of expression, as given above, is inaccurate; for the value of l' given by the equation is not the true value, but the most probable value of the number alive at the end of a year. Hence this value also has a probable error, and a mean error, either of which measures its degree of trustworthiness; and these probable and mean errors have likewise never yet been determined.

These two cases furnish examples of a few of the gaps which statistics, as hitherto understood, has in vain sought to fill up. To fill them up, and to build further on the ground thus gained, is the object of the following investigations, which will be found to take an entirely new direction. The degree of precision in our treatment of the subject, and the important applications it admits of, will be seen so directly from the investigations themselves, that it appears unnecessary to say more of them here.



Chapter 1.
The probability of living a year is supposed to be given.

$1. Let p be the probability that a person of the age x will be alive at the end of a year.* This is most commonly given by means of a mortality table; the ratios of the numbers living at successive ages being equal to the different values of this probability. Then, according to well known principles, the probability that out of 1 persons of the age x, l' will be alive at the end of a year, is

* It may be remarked, in passing, that the following investigations admit of being applied to many other subjects besides mortality. Thus, for instance, we might have said, “Let p be the probability that an unmarried person of the age x will still be unmarried at the end of a year;” or “Let p be the probability that a building of class x will not be burnt down at the end of a year,” and so on; the nature of the particular subject indicating what modifications must be introduced in applying our investigations to such cases as these. We will not pursue this subject further on the present occasion.

12 P=



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Here l' can have any integral value from 0 to l inclusive.

The value of l' which makes P a maximum can be found by comparing the above value of P with the values it receives when I'-1 and I' +1 are substituted for 1'. It is given by the relations,

1-1'+1 1
1, 1-2


l' +1;

I'+1 I i.e.


p whence, if I is a large number, and lo denotes the most probable value of l', we have lo=lp.

(2) This is the number given in a mortality table as living at the age *+1, if I is the number living at the age 2.


$ 2. Putting l'=lo+k=lp+k, then by (1) the probability that the number alive at the end of a year will exceed by k the most probable number, is

12. P=

plp+k. (1-p)Wp=k .. (3) \p+k17(1-P)

ok Here k may have any integral value from - lp to Il-p).

If we assume that l, 1p+k, /(1-)-k are large numbers, and apply Stirling's formula

n=v2.nn+3,6, approximately, equation (3) becomes 1

+ P= 12 (1p+k)p+k+I{{(1–p) – k341042+7 p?p+k• (1–p)ąpk 1




k U1-P)-k (1+


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