The 43,315 males surviving to the end of the first year of life out of 51,195 born will each have lived a complete year in the first year, or among them 43,315 years. Similarly the 40,452 males will live among them 40,452 further complete years, and so on, until all the males started with become extinct at the age of 105. Evidently, therefore, the total number of complete years lived by the 51,195 males started with at birth will be
43,315 + 40,452 + 39,456 + 38,723 + ... + 10 + 6 + 4 + 3 + 2 + 1 = 2,206,174 years, this sum being obtained by adding together the numbers living at each age beyond (i.e. below on this table) the age in question right down to its last item. This number of years is lived by 51,195 males. Hence the number of complete years lived by, i.e. the expectation of life of, each male
= 2,206,174 ∕ 51,195 = 43·09 years.
This is the curtate expectation of life. It deals only with the complete years of life, not taking into account that portion of life-time lived by each person in the year of his death, which may be assumed to be on an average half a year. Hence the complete expectation of life according to the above table is 43·59 years.
In the following table the expectation of life (complete) for various towns and for England is given:—
Life Table.—Expectation of Life at Birth.
| Male. | Female. | ||
| English Life Table, | 1838-54 (Farr) | 39·91 | 41·85 |
| „ | 1871-80 (Ogle) | 41·35 | 44·62 |
| „ | 1881-90 (Tatham) | 43·66 | 47·18 |
| London, 1881-90 (Murphy) | 40·66 | 44·91 | |
| Brighton, 1881-90 (Newsholme) | 43·59 | 49·25 | |
| Manchester City, 1881-90 (Tatham) | 34·71 | 38·44 | |
| Glasgow, 1881-90 (Chambers) | 35·18 | 37·70 | |
Formulæ of varying degrees of accuracy have been devised for giving in the absence of a Life Table an approximation to the expectation of life.
Willich’s Formula is as follows:—If x = expectation of life, and a = present age, then x = 2 ∕ 3 (80-a). Thus, at the age of 50 years the expectation of life, according to this formula, is 20 years. By the English life-table for 1881-90 it was 18.82 for males, and 20·56 for females. Farr’s formula is based on the birth and death-rates. If b = birth-rate and d = death-rate per unit of population, then
Expectation of life = (2 ∕ 3 × 1 ∕ d) + (1 ∕ 3 × 1 ∕ b).
Thus b for England and Wales, 1889-98 = 30·3 ∕ 1,000 = ·0303.
and d for England and Wales, 1889-98 = 18·4 ∕ 1,000 = ·0184.
(2 ∕ 3 × 1 ∕ ·0303) + (1 ∕ 3 × 1 ∕ ·0184) = 47.2 years, as compared with the expectation of life for 1881-90 shown in the above table.
| In a life-table the number out of which one dies annually |  | are |
| the mean age at death | identical | |
| and the expectation of life | in value |