# Guide to calculating national life tables

### Introduction

National life tables, which are produced annually for the UK and its constituent countries, provide period expectation of life. This is the average number of additional years a person can be expected to live for if he or she experiences the age-specific mortality rates of the given area and time period for the rest of his or her life.

Each life table is based on the population estimates and deaths by date of registration data for a period of 3 consecutive years. This helps to reduce the effect of annual fluctuations in the number of deaths caused by seasonal events such as winter ’flu. The national life tables are based on the mid-year population estimates and corresponding data on births, infant deaths and deaths by individual age from those years (the calculation of infant mortality also requires monthly births data for the year before the 3-year period – Appendix A).

Summary quality reports for the data used in the calculation of national life tables are available here: births (257.9 Kb Pdf) , deaths (222.3 Kb Pdf) and population estimates (236.8 Kb Pdf) .

### Life tables

Life tables are usually constructed separately for males and females because of their very different mortality patterns. A life table describes the course of mortality throughout the life cycle. A life table contains:

mx

The central rate of mortality, defined as the average annual number of deaths at age x last birthday in the 3-year period to which the national life table relates, divided by the average population at that age over the same period.

qx

The mortality rate between age x and (x +1); that is, the probability that a person aged x exactly will die before reaching age (x +1).

lx

The number of survivors to exact age x of 100,000 live births of the same sex who are assumed to be subject throughout their lives to the mortality rates experienced in the 3-year period to which the national life table relates.

dx

The number dying between exact age x and (x +1) described similarly to lx, that is, dx=lx–lx+1.

ex

The average period expectation of life at exactly age x, that is, the average number of years that those aged x exactly will live thereafter based on the mortality rates experienced in the 3-year period to which the national life table relates.

### Methodology

Starting with a radix of 100,000 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 d1 (where d1 = l1 x q1) and so on.

Generally:

dx= qx– lx lx+1= lx – dx

### 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 (Lx) 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:

L0 = a0l0+(1–a0)l1

where a0 is the average age of death of those dying within the first year of life (see Appendix A).

Summing the Lx column from age x to the oldest age gives the total number of years lived (Tx) from age x. The period expectation of life at exact age x is given by dividing the number of years lived by the number at that age, that is: