Period Abridged Life Tables  

An excel spreadsheet template for calculating life expectancy is available.

Life tables conceptually trace a cohort of newborn babies through their entire life under the  

Age Interval  (x to x+n)
The period of life between two exact ages;  e.g. “20 to 25” means the 5-year interval between the 20^th and 25^th birthdays.

Age Specific Mortality Rate  (_nM_x)
The age specific mortality rate , _nM_{x = n}D_x/_nP_x  where _nD_x is the number of deaths occurring to persons aged n to x+n, and _nP_x is the number of persons aged x to x+n alive at the mid-point of the period under consideration.  Mortality rates are usually presented as deaths per x persons per year, where x is a convenient population base, e.g. 10,000.  If a longer period is used to count the number of deaths (e.g. 5 years as in the life table template) then an adjustment must accordingly be made to calculate the annual age specific mortality rate by dividing the derived rate by the period of observation years.

Probability of Dying (_nq_x)
To estimate the exact probability of dying between age x and x+n (_nq_x) the deaths to persons aged x to x+n must be related to the true population at risk.  The mid-period population estimate must be adjusted by adding half the number of deaths occurring over the period to it.  This assumes that deaths are evenly spread throughout the time period and across the period of life under consideration.

In general for age groups;  _nq_x  =  2 x n(_nM_x) / 2 + n(_nM_x)

The assumption that deaths are spread evenly across the time period of life under consideration is particularly unrealistic in the case of infants, where the majority of deaths occur within the first few days of life.  To calculate the probability of dying for infants therefore, the infant mortality rate is used, as the number of livebirths is an exact estimate of the population at risk of dying.  Therefore for infant deaths we have;

               ₁q₀  =  ₁D₀/B  where B = number of livebirths over period of consideration.

Another exception occurs in the final open-ended age group (85+ in the lifetable templates).  As everybody within this age group must die, the probability of dying is equal to 1,  i.e.  _¥q_n = 1.000.

Persons Alive  (l_x)
The number of persons living at the beginning of the indicated age interval (x) out of the total number of births assumed as the radix of the life table.

The number of persons alive at the beginning of an age interval (l_x+n) is equal to the number alive at the beginning of the previous age interval (l_x), minus the numbers of persons dying within that previous age interval  over the time period considered (_nd_x);

                                           i.e.    l_x+n  =  l_x - _nd_x

Persons Dying  (_nd_x)
The number of persons dying within the indicated age interval (x to x+n) out of the total number of births assumed in the table.

The number of persons dying within a particular age interval (_nd_x) is equal to the number persons alive at the beginning of that age interval (l_x) multiplied by the probability of dying within that age interval (_nq_x),   i.e.    _nd_x = l_x x _nq_x

Person-Years Lived in Age Interval  (_nL_x)
The number of person-years that would be lived within the indicated age interval (x to x+n) by the assumed cohort of 100,000 births.  Again, assuming that deaths are spread equally across the age intervals and also across the period of consideration, then the number of person-years lived in an age interval is calculated as the mean of the populations at the beginning and end of each age interval, multiplied by the length of each age interval in years    i.e.     _nL_x  =  n/2 x (l_x + l_x+n)

A difficulty arises with the estimation of ₁L₀,that is the average number of infants alive who have not yet reached their first birthday.  Again the problem is that it cannot be assumed that infant deaths occur uniformly throughout the 0 to 1 age interval.  For this reason ‘separation factors’ are used to weight the average of l₀ and l₁ as follows;

                                               ₁L₀  =  0.3l₀ + 0.7l₁

An additional problem lies with _¥L₈₅, the average number of persons alive over age 85.  This is approximated from   _¥L₈₅  =  _¥d₈₅/_¥M_85.

Person Years Lived From Age x  (T_x)
The total number of person-years that would be lived after the beginning of the indicated age interval by the assumed cohort of 100,000 births.  This is calculated by simply cumulating the _nL_x column from the oldest to the youngest age.

Life Expectancy  (e_x)
The average remaining lifetime (in years) for a person who survives to the beginning of the indicated age interval.  Calculated by dividing the total number of person-years lived from age x (T_x) by the number of persons alive at age x (l_x)

                                    i.e.   e_x  =  T_x/l_x