Page:Transactions NZ Institute Volume 29.djvu/79

It will be observed that the average deaths and average population in Table A are given in groups of five years. The next step in the construction of the final tables is to ascertain the population and deaths at each year of age. The method now generally adopted is that known as Milne's Graphic Method. After a very careful consideration of this method it was decided not to adopt it, but to use instead a mathematical process of distribution based on the method employed by G. W. Berridge ("Journal of the Institute of Actuaries," xiii., 220, and xiv., 244; "Text-book of the Institute of Actuaries," Part ii., p. 465). The results of the distribution are given in Table B. As a test of the smoothness of the distribution, the results were drawn to scale on large diagrams, of which Plates I. and II. are reduced copies.

The population and deaths from 5 to 75 were treated in this way, the figures relating to the first five years of life requiring special treatment.

From Table B the ratio of deaths to population at each age ($m_{x}$) is at once obtained, and these ratios are given in Table C.

The probability of living a year at each age ($p_{x}$) is derived immediately from $m_{x}$ by means of the relation $p_{x}={\frac {2-m_{x}}{2+m_{x}}}$. The columns headed $p_{x}$ in Table E, from 5 to 75, were calculated by means of this formula.

The ages 0 to 5 now require consideration. Table D gives the annual births and deaths of children under five years of age for each of the years 1880–92. From these figures, by means of a modification of the method used by Dr. Farr ("Journal of the Institute of Actuaries," ix., p. 134), the probabilities of living a year at each age were determined. The results, after a slight adjustment to make them join smoothly on to the rest of the table, are given in column $p_{x}$, ages 0–5, in Table E.

The probability of dying in the year at each age ($q_{x}$) is obtained from $p_{x}$ by subtracting $p_{x}$ from unity: thus, ${q_{x}}={1-p_{x}}$.

The next column in the order of formation is the $l_{x}$ column. Starting with an assumed 10,000 births ($l_{0}$), the number surviving the year ($l_{1}$) is obtained from the relation $l_{1}=l_{0}\times p_{0}$. Similarly the number who reach the age of two alive, out of 10,000 born alive, is $l_{2}=l_{1}\times p_{1}$ or generally for any year x, $l_{x}=l_{x-1}\times p_{x-1}$.

The difference between the number born, $l_{0}$, and the number surviving the first year, $l_{1}$, gives the number who die in the first year, $d_{0}$, or $d_{0}=l_{0}-l_{1}$. Similarly for the number who die in the second year, $d_{1}$, $d_{1}=l_{1}-l_{2}$, and generally for the number dying in the xth year $d_{x-1}=l_{x-1}-l_{x}$. In this manner the column $d_{x}$ was formed.