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s t=0.8747 log +9.94185

P= 7839

(Compare these examples with that of Case III.) The methods of computation shown in the foregoing cases must not be considered as limited to mere games of chance. They will apply generally to all kinds of statistical inquiries. Their capability of being directed to useful practical questions connected with business matters is elucidated in the following interesting and concluding example,

EXAMPLE. -According to the Life Office returns, upon which the “New Experience" life tables are founded, out of 41,385 lives (male and female) age 41, there were 445 deaths, giving 40,940 survivors at the expiration of one year :

1. Required the chance, deduced therefrom, that for age 41 the true

4 4 5+30, probability of dying in one year is contained between the limits

41385 inclusive ?

2. If another set of 41,385 lives be take at the same age 41, what is the probability that the number of deaths in one year will fall within the limits 445+30, inclusive ?

3. Assuming the observed probability of dying in one year, viz. : 445

to be the true rate for that year of age; what then would 41 385 be the probability that out of another set of 41,385 lives aged 41, the deaths in one year would be contained within the limits 445+30?

1
9 3

>

1. See CASE V.

a=445; B=40,940; x=30.
2a 890 log 2.94939
B 40,940 log 4.61215

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t=1.0279 log 10.01195

P= 8540 The probability of No. 1 is nearly equal to that of No. 3, and would obviously be identical with the latter if the limits were only slightly extended to 445+3 01

Thus we find that the formula of Cases III and V suggest the following interesting theorem.

41385

a

THEOREM. If in atß observations an event has been observed to happen a times and to fail 6 times, the probability that the true chance of

a+(+1) the event lies between the limits

is equal to the probability that

a +ß in a + ß observations the number of occurrences of the event shall lie between the limits a+x, assuming in the latter case that the true chance of the event is known to be

a+ In the two cases involved in this theorem, the difference of 1 in the values of x is deducible from the consideration that the ordinate (y) of the curve of facility of error may be regarded as practically representing the probability that an error shall fall within the limits x+}; and that the middle or maximum ordinate (Y) may similarly represent, the probability of the error being comprised within the limits +1.

The principal questions appertaining to the collection, classification, and discussion of statistics are briefly comprehended in the foregoing cases. The practical resolution of these questions only requires a slight knowledge of logarithms of numbers, and is effected by simply following out the method of working exhibited in the various examples. In the several calculations it will be perceived that instead of using negative indices to logarithms, the complements to 10 are put down and used as positive indices, according to the convenient custom of experienced computers in the higher departments of mathematics; thus, in all additions, subtractions, &c., it is only requisite to reject and borrow 10 whenever necessary.

In almost every branch of statistical inquiry the quantities which enter are subject to deviations from the normal values, arising from a combination of innumerable minute circumstances, of which the variations due to each are separately very small. The total change from the mean value is of course produced by the superposition of these minute variations, some of which tend towards an excess and others towards a defect. When the number of observations is small, they are not unlikely to comprise a number of exceptional cases, in which the greater part of the variations repeatedly tend in the same direction, namely, all towards excess or all towards defect, thus producing divergent results in that particular direction; and these exceptional cases may comprise such a proportion of the total observations as to considerably affect the average result.

But when the number of observations is large, the occurrence of such exceptional variations, in one direction only, becomes highly improbable, and the exceptions necessarily form such an inappreciable proportion of the total facts, that the disturbing effects become practically neutralized. Thus it is that, although but little dependence can be placed in deductions or conclusions drawn from small numbers, there will certainly be found to exist general laws amongst masses of phenomena, of whatever description, physical or moral, which in the invariable and perpetual recurrence of events of the same kind, as the numbers augment, necessarily approximate towards fixed and determinate ratios. It is also evident that every conclusion will have greater weight, and the approximation to normal results must be greatly assisted and expedited, when the more prominent circumstances which tangibly affect the observations admit of being wholly eliminated, or, still better, of being separately classified and tabulated.

In that case the phenomena appertaining to each set of circumstances or conditions would come under separate and distinct investigation, instead of the more complicated phenomena arising from their diversified combinations. If, however, the various conditions, which principally affect the quantities, do not admit of separation, there will be no alternative but to enlarge the number of observations, if such should be possible, until a reasonable presumption is shown that they present a sufficient accumulation of experience for the purpose of eliciting the required results with a very small probability of error.

As a concluding remark I may add that in this paper I have endeavoured, briefly and practically, to explain some of the most useful and important results and applications of the mathematical theory; and that I shall be glad if it should in any degree assist in directing more attention to the neglected logical science of statistics.

On the Law of the Ages at which Life Insurances are effected.

By S. C. CHANDLER, JR., Actuary of the Continental Life Insurance Company, of New York.

[Reprinted from the Spectator,of New York and Chicago.] IF a classification be made of any considerable number of life policies according to age at issue, a cursory inspection will show that the numbers at the different ages are not fortuitous, but are subject to some unknown, though plainly marked, mathematical relation; and the amenity of the results to law will be more evident as the number of policies employed in the classification is larger, and the effect of merely accidental fluctuations is consequently more nearly eliminated.

So far as my information extends, no investigation of the character of this law has ever been undertaken. Yet the subject seems to me worthy of consideration aside from the interest that attaches to it simply as a statistical inquiry. The development of such a distributive law would supply actuaries with a very efficient tool in many computations where, a high degree of precision not being essential, the approximate determinations afforded by its application would possess quite as much practical value as the more accurate results of laborious classifications or seriatim calculations.

Some attention was given to the problem about two years since, but the material then at hand was too meagre for a definite solution. The publication subsequently, however, of the "Mortality Experience of Life Assurance companies,” giving in detail the observations collected by the Institute of Actuaries, supplied abundant data for the desired investigation. To present the results, so far attained, of a discussion of this material, is the object of this paper.

The total number of lives embraced in the statistics of the twenty companies contributing their experience was 160,426. Of this number 11,146 were diseased lives, and 2,433 were exposed to extra risk from climate and occupation. Excluding these two classes, insured under what may be considered exceptional conditions, there remain 146,847 as the number of healthy lives of both sexes insured at all ages. In the seventh column of the annexed table is shown the distribution according to current age at entry. It will be seen that, disregarding minor fluctuations, the numbers first increase, and afterwards diminish, at varying rates, the maximum occurring about age 30. The course of the numbers is graphically exhibited in the accompanying chart in which the years of age are laid off on the axis of abscissas, the numbers insured on the axis of ordinates. The tendency towards a continuous curve is obvious.

Now if the number insured at each age is a function of the age, the area of that portion of the curve terminated by any ordinate will likewise be a function of the

age. senting by M the number insured at and over the age x,

M=f(x). The form of f(x) being found, the number at any given year of age will be the finite difference,

That is, repre

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