Infant mortality (q₀)

For interim life tables covering the period year T to year T+2 inclusive, infant deaths at <4weeks, 1-2 months, 3-5 months, 6-8 months and 9-11 months are summed separately for males and females over the three years T, T+1 and T+2. The ‘at risk’ population is then derived for each group from the monthly birth figures, separately for males and females, as follows (where B_XxxT = Births in Month Xxx of calendar year T):

<4 weeks:
1-2 months:
3-5 months:
6-8 months:
9-11 months:

Each of the total groups of deaths is then divided by the appropriate at risk population calculated above and the results totalled to give q₀.

The m₀ shown in the life table is calculated from q₀ using the formula:

Calculation of q_x above age 0

First m_x is calculated for each age by dividing the sum of the deaths at age x in each of the three years by the sum of the mid year population aged x last birthday for each of the three years. The corresponding q_x is then derived using the formula:

The construction of the life table

Starting with a radix of 100000 simultaneous births (l₀), the life table population is calculated by multiplying l₀ by q₀ to give d₀, the number of deaths aged 0. The resulting d₀ is then subtracted from the l₀ to give l₁. Similarly l₂ is l₁ less l₁ times q₁ and so on.

Generally:

The calculation of expectation of life at each age

In order to calculate the expectation of life at exact age x the number of 'years alive' at each individual age (L_x) needs to be calculated. For ages above 1, where deaths can be assumed to occur linearly over a year of age, this can be taken as
.

Below age 1, this assumption is unrealistic. L₀ is calculated using the following formula:

, where a₀ is the average age of death of those dying within the first year of life. By making assumptions for the average age of death for each of the periods used for the infant death calculation (see below), a₀ can be calculated. The assumed average ages at death are as follows:

Age at death	Assumed average age at death (months)	Notes
<4 weeks	0.2	Based on analysis of England and Wales data for deaths under 1 month
1-2 months	1.5
3-5 months	4.0
6-8 months	7.0
9-11 months	10.0

Summing the L_x column from age x to the oldest age gives the total number of years lived (T_x) from age x. The expectation of life at exact age x is given by dividing the number of years lived by the number at that age i.e.