# Expectation of Life

EXPECTATION OF LIFE. The term expectation of life denotes the average number of years which the persons of a specified age, taken one with another, will live according to a given mortality table. (See LIFE TABLE.) If of persons aged exactly x, survive to age x+i, 10)+2 to age x+2, etc., then the average number of completed years of future life-time which the persons will experience is (lx+, -}-lx+,+ etc.) This expression is represented by the symbol and is technically known as the curtate expectation of life. No allowance is included in the formula for that portion of life-time lived by each individual in the year of his death, which on the average will amount to approximately one-half of a year. The complete ex pectation of life, is therefore approximately equal to ex-F. Also may be derived directly by means of the formula and therefore denotes the population between the ages x and x+r, and denotes the total population at age x and over.

There can be few technical terms so generally misunderstood and misused as the expectation of life. Several additional phrases have theref ore been suggested as more appropriately de scribing the function, e.g., the mean after life-time, the average after life-time, the mean duration of life and the average dura tion of life.

Amongst non-technical people the expectation of life has ac quired a prestige greatly in excess of its actual merits. It is sel dom used by actuaries, yet there appears to be a wide-spread im pression that it forms the basis of all actuarial calculations in volving the probabilities of survivorship. One of the most con spicuous instances of this popular fallacy is the assumption that the value of a life annuity is equivalent to that of an annuity certain for a term of years corresponding to the expectation of life. The latter value is demonstrably in excess of the true value (see ANNUITY).

Another common misconception is that the expectation of life is an actuarial estimate of the number of years an individual may reasonably expect to live. The expectation of life is, however, an average value which is derived from a particular series of rates of mortality and, even if these rates accurately represent the future mortality experience at all ages, will correspond to the future life time of only a very small proportion of the individuals concerned, the superior vitality of those who survive for a longer term being counterbalanced by the heavier mortality amongst the others.

Objections have recently been urged against the use of the ex pectation of life as an index of future mortality experience, owing to the fact that the progressive improvement in vitality revealed by successive investigations has suggested that it has become un reliable as a measure of the average future life-time of a body of lives. In such circumstances the figure that would have been the true expectation of life can only be determined after all the lives, whose experience is being reviewed, have died.

The expectation of life provides a means by which the mortal ity experience of different investigations may be compared. In recent years its use for this purpose has been the subject of much adverse criticism, and certainly as an instrument for detailed comparison it is not so satisfactory as several other criteria, e.g., the probability of survivorship for an indicated period, say five or ten years. The expectation of life at a particular age does, how ever, provide a comprehensive figure which summarizes the ex perience at higher ages, and its value at birth is probably the best available statistical measure of the health conditions of a com munity. It is, therefore, of great value to medical officers and other statistical workers.

From the following table, which shows the expectation of life at specimen ages according to successive life tables, the improvement in the vitality of the population will be apparent.