143,629 851 3,563 3,371 +192 +054 20 142,778 142,778 0 5,725 5,703 +22+004 ⚫000 728 1,309-581-798. 53 1,865 1,779 +86 +046 189 2,790 2,763+27 +010 260 332 392 440 5,130 4,981 +149 +029 478 506 525 5,645+114 +020 537 541 There being no direct means of arriving at a knowledge of the form of the required function, we must have recourse to tentative processes. The transcendental expression which has been adopted. as best representing the observations is given below. The form is unsatisfactory, and I am strongly induced to believe that it is susceptible of modification and improvement. It will be observed that of the total number of insurers 4,069, or two and three-quarters per cent, entered below the age of twenty years. Probably a large proportion of these held children's endowments. What influence this fact had in the result there is no means but all attempts to find an expression which would in which a is the total number insured at all ages; x the e the age above 20 years; and b, c, k, and p, are constants de upon the observations themselves. This equation, although an empirical one, gives results which present a close ag with the recorded numbers at each age. Taking the logarithms of both members of (1) and put Differentiating with respect to ẞ, y, p, and k, we obtain where he is the modulus of the common system of logarith the logarithm of the Naperian base. Rough determinations of the constants b, c, p, and having been obtained by a combination of graphical and nu processes, the values of the differential coefficients and the al term of (3) were calculated for twelve different values of x. solution of the resulting equations gave three independent minations of each of the corrections dß, dy, dk, and dø, the of which being applied to the assumed values, furnished a s approximation to the constants. These again, being subst in (2) and (3) for all values of x, final corrections were de from the equations of condition by the method of least squa In the solution a certain indeterminateness, arising from nature of the equations of condition, becomes apparent; the of the corrections depending greatly upon the number of equ employed in the formation of the normal equations. This fac the systematic deviations shown by the residuals, confirm me conviction that the equation (1) does not represent the true which we are in search, though affording a close approximati it. The adopted values of the constants given below, were obt by first determining dẞ and dy in terms of dk and do from normal equations in dẞ and dy, using 55 equations of con values dk and do derived, by an exercise of judgment, from a consideration of the results of several groupings of the equations of condition. Applying the corrections in this way deduced, we find b=1.0052, c=1.00159, k=1.86, p=2°30′; and these adopted values, and that of a=142,778 substituted in (1) give 1.86 M=142778 × 1·0052- x 1·0015916 sin r(2.5), For a comparison of the formula with actual experience the values of M have been computed by means of (4) and inserted in the second column of the table. The first differences, being the numbers insured at the several ages according to the formula, are given in the sixth column. The discordances between the computed and the actual numbers, with the proportion which these bear to the former, are exhibited in columns adjacent. Though the signs of the differences between the computed and the actual values of M show that systematic errors have not been entirely eliminated, the deviations are confined within narrow limits. The average discordance up to age 70, without regard to sign, is less than one per cent (0·008); while in but one instance (age 59) is the discordance so large as three per cent of the computed value of M. Turning to the number insured at each age, it will be seen that the formula presents a very satisfactory accordance with experience. The signs in the column of differences seem immethodical, there being thirty-two changes of sign between 20 and 87, the age at which the number insured becomes zero by the formula; while the more prominent discrepancies are traceable, on reasonable grounds, to a sufficient cause. To facilitate comparison of the calculated with the observed numbers, the values in the sixth column have also been plotted in the chert already referred to and connected by the dotted line The Considerable instruction may be derived from an exam of the residual errors. In following down column eight th attracted by large negative residuals at ages 20, 30, 40, 60. The corresponding effect in the chart is a prominence of the ages mentioned. A similar effect, though less not is exhibited at 35, 45 and 65. There would thus seem tendency to excess of the actual numbers over those indica theory at ages which are multiples of five years; a tendenc marked at those ages which are likewise multiples of ten ye Now, does this peculiarity arise from a real excess of app for insurance at the ages indicated, or from a tendency to mis ment such as would lead persons slightly older or younger say, for instance, 30 years, to represent themselves from igno carelessness or intention, as exactly 30 years of age? Mr. N Willey has suggested that there may be certain epochs in approaching which men are reminded of the ebb of their exis and take more readily and seriously into consideration the s of provision for their families against their death. This hypo is not, perhaps, altogether a fanciful one, and the pheno observed may possibly be due in some part to such a cause. more probable explanation, however, it seems to me, is the s above proposed. A similar interpretation of analogous in larities in the statistics of our volunteer soldiery in the late has been given by that distinguished authority, Dr. B. A. Go It becomes interesting to ascertain how nearly the for deduced from the experience of the English offices-of whic began operations one hundred and fifty years ago, and the you was nineteen years old at the time the data were collected-r sents the relative proportions of applicants for insurance at diff ages in American companies. Some material for a comparis readily available in the classifications, made for valuation purp of policies issued by the company with which I am conne Since it would be manifestly improper, in such comparison include endowment insurances, I have taken simply the life pol issued from January 1, 1869, to July 1, 1871. The distribu * Ages of the United States volunteer soldiery. Statistical Bureau, United Sanitary Commission. New York, 1866. |