The table shows that in thirty trials of ten throws each the actual experience coincided with the probable in eleven cases, that in two instances heads appeared eight times out of ten, and in one case only once. These results in groups of ten may be combined into groups of twenty, thirty, fifty, one hundred, or in a single group of three hundred, and comparisons may then be made of the fluctuations in those respective groups. By this arrangement the original data assumes the form shown on page 126:

In the above table the data are arranged in fifteen groups. of twenty throws each, ten groups of thirty, six of fifty, three of one hundred, and a single group of the three hundred throws and the number of times the coin fell heads or tails is shown for each group. The important fact to be considered is the relation between the probable and the actual experience in each grouping of the data. For instance, in twenty throws the probability is that heads will appear ten times, but the figures show that in one case this result occurred thirteen times and once only six; in thirty throws heads appeared as many as eighteen times in two instances and as few as eleven the same number of times. The following brief table shows the maximum and the minimum number of times the coin turned heads up in any single trial of the specified number of throws:

FLUCTUATIONS IN NUMBER OF TIMES HEADS

If this data are now reduced to the form of percentage the results can be more readily compared, for the amount of the

fluctuations will then have a common basis. It is understood that the probability of the coin falling heads up is and this will be represented by fifty per cent. The variation of the actual percentage from fifty per cent. will therefore be the measure of the variation. The table presented herewith gives the results obtained:

This table furnishes the basis for an important generalization with reference to the accuracy of the theory of probability. It shows that where the coin was thrown ten times the results varied from a minimum of ten per cent. to a maximum of eighty per cent.; where twenty throws were made the variation was less, viz, from thirty to sixty-five per cent.; and that as the number of throws increased the variation became smaller and smaller and the percentage of times heads appeared approached fifty, the true probable percentage. That the three hundred throws resulted in exactly one hundred and fifty heads must be regarded as an accident; but it can be said with equal certainty that it would be impossible out of any three hundred purely chance throws to get as many as eighty per cent. or as few as ten per cent. to fall heads up. The generalization referred to above is as follows: Actual experience may show a variation from the true "probable experience but as the number of trials is increased this variation decreases; and if a very great number of trials were taken the actual and the probable experience would coincide. Concretely, if the coin were flipped ten million times and it were a pure chance which way it would fall, the

actual results would be so near five million times heads that the difference would be negligible. This generalization is called the law of average. This law is fundamental to all insurance. Premium rates are based on probable losses and will not accurately measure the risk unless the actual experience approximates the probable. That this approximation shall be realized it is at all times necessary to deal with a sufficiently large number of cases to guarantee that great fluctuations in results will be eliminated, i.e. to insure the operation of the law of average. In other words prediction of the future in life insurance based on what has happened in the past can be made for a large group of persons; it cannot be made for a single individual. When a mortality table shows that persons of a certain age die at the rate of seven per thousand per year that does not mean that out of a group of one thousand exactly seven will die within a year, but that out of a large group, maybe containing many thousands, the deaths will occur at the rate of seven per thousand.

With reference to the prediction of future mortality rates the law of average has a double application. Future mortality will be measured on the basis of past mortality data. These data of the past will supposedly be an approximate measure of the law of mortality heretofore referred to. But the statistics used for this purpose must be of sufficiently general application and must include a sufficiently large group of individuals to insure the operation of the law of average. Only in case this is so will the data in question be a fair measure of the true law of mortality. Granted then that the collected data are approximately correct, they become a measure of future mortality, only in case the group among whom the probable deaths are to be estimated is large enough to guarantee an average death rate or the operation of the law of average within the group.

THE MEASUREMENT OF MORTALITY - MORTALITY TABLES

The establishment of any plan of insuring against premature death requires some means of giving mathematical values

to the chances of death, and the considerations advanced in the first division of this chapter show that the laws of probability can be used for this purpose as soon as trustworthy data are secured showing the course of past mortality. Mortality tables, as such data are called, are records of past mortality put into such form as can be used in estimating the course of future deaths.

Sources of Mortality Tables. There are two sources from which the best-known mortality tables in existence to-day have been obtained: (1) population statistics obtained from census enumerations, and the returns of deaths from registration offices, and (2) the mortality statistics of insured lives. Well-known examples of the former are the English life tables of Drs. Farr, Ogle, and Tatham, successively in charge of the General Registry Office of England and Wales. Dr. Farr's life table, for instance, was based on the registered deaths in England and Wales during the years 1838–54, and on the two census enumerations of population for 1841 and 1851.

Objection to Tables Based on Population Data.- For the purposes of measuring the mortality of insured lives, however, it is questionable whether statistics of a general population can be used. Such data, to be sure, would represent the average mortality of a population group and to that extent would approximate the true law of mortality. But for purposes of insurance this may or may not be the mortality rate desired. An insurance company wants a measure of the mortality occurring among insured lives and it is probable that this may differ from that of a specific population group. Insured lives are subject to special influences affecting mortality and these factors must be taken into consideration. The statement has been made that if an insurance company could insure every person who passed a certain corner in a large city until it had a large enough group to guarantee the operation of the law of average, the company could dispense with its medical examination. This is probably true, but the trouble is, when the matter of insurance is left to the choice