Background Notes

Life Tables presented here are period life expectancies.  Period expectation of life at a given age for 2010-12 is the average number of years a person would live if he or she experienced age-specific mortality rates for that time period throughout his or her life.  It is therefore not the number of years someone of that age could actually expect to live because death rates are likely to change in the future.

The basic assumption is that a given cohort of births, (100,000), start in a given year.  The mortality rates for each age are used to calculate how many of the cohort will reach each year of age until eventually all members of the cohort have died.  This enables the total number of years lived by the cohort to be calculated.  When this total is divided by the number of persons in the cohort, (100,000), the result is the average number of years lived in the cohort, or the mean expectation of life at birth.  The total  number of years lived by the cohort from any given age can also be calculated and, when divided by the number of survivors in the cohort entering upon that year of age, the figure obtained is the expectation of life in years for those persons.

Life Tables were constructed for males and females which are representative of the mortality experience in Ireland in 2011 by using the 2010, 2011 and 2012 estimates and census of population (usually resident) and deaths registered in the three years.  The life table should reflect the normal mortality conditions at about the time of the Census. 

Glossary of technical terms  

x  the exact age of the person, that is, on his or her birthday.
l_x the number of persons surviving to exact age x out of the original 100,000 aged 0.
d_x the number of deaths in the year of age x to x+1 out of l_x persons who enter  that year.
p_x the probability of surviving a year, or the ratio of the number completing the year of age x to x+1 to the number entering on the year.
q_x the rate of mortality, the probability of dying in a year, or the ratio of the number of deaths in the year of age x to x+1 to the number entering on the year.
L_x the population to be expected according to the Life Table aged between x and x+1 years, assuming deaths occur evenly over the year.
T_x the expected number of person years to be lived by the survivors at age x.
e⁰_x life expectancy at age x for each person surviving, or the total future life time in years which will on average be passed  through by persons aged exactly x.

Examples

Figures from the Male Irish Life Table No. 16 are used in the examples below.  Please note that totals may not add up due to rounding.

The first column of the life table, l_x equals the number of persons surviving in the life table at each exact age x, in other words the January population. l₀ represents the life table population of new born children or those aged exactly zero.  If we let l₀ equal 100,000 then for example, l₅ is the number of persons surviving on their fifth birthday, which in this case equals 99,560.

The second column of the life table, d_x equals the expected number of deaths of persons aged age x in the life table.

equation 1                                   

d_x= l_x−  l_x+1  

Equation 1 tells us that the number of deaths equals the number of persons surviving at age x less the number of persons surviving at age x+1.

e.g. for males aged 5                    

d₅= l₅ − l₆

= 99,560 – 99,550

= 10

The third column of the life table, p_x equals the probability of surviving from exact age x to x+1.  This is simply the ratio of those completing the year of age x to x+1 to the number entering the year.  For example, p₅ is the probability of surviving ones fifth year, which in this case equals 0.99990.
    
equation 2                                    

p_x= l_(x+1) / l_x

Rewriting equation 2 where age x = 10, we see the number of persons surviving to their eleventh birthday equals the number of persons at their tenth birthday multiplied by the probability of their surviving to their eleventh, the remainder having of course died.  Migration is ignored in a life table as the population if closed.

l₁₁ = l₁₀.p₁₀ 

= 99,509 x 0.99992

= 99,502

The fourth column of the life table, q_x equals the probability of dying between one birthday and the next.  This may also be called the risk of dying in a life table year, in other words the risk of dying at a particular age. The probability of dying and the probability of survival equal unity.  In other words one can only be alive or dead.

equation 3

p_x+ q_x = 1    

From equations 1, 2 and 3:

equation 4

q_x= d_x / l_x   

So the probability of dying is the ratio of the number of deaths at exact age x divided by the number of persons surviving at that exact age.  Hence we say the life table is based on ‘current mortality rates only and that no assumptions are made about future changes’.

The fifth column of the life table, L_x equals the number of years survived by the life table cohort between the ages x and x+1, in other words the July population.  Assuming a uniform distribution of deaths over a year of age and using equation 1 we find:

equation 5

L_x= l_x− (d_x / 2)

= l_x − (( l_x− l_x+1 ) / 2)

= (l_x + l_x+1 ) / 2             (x>0)

e.g. for age 1 this means

L₁= l₁− (d₁ / 2) = 99,621 – 35/2 = 99,604 

or 

L₁= (l₁+1₂ ) / 2= (99,621 + 99,586)/2 = 99,604

This cannot be used at age 0 as infant deaths are not evenly distributed (i.e. they are non-linear over a year).  For example, in 2011 34% of all infant deaths occurred on their first day of life.

The sixth column of the life table, T_x equals the total number of years which will be survived at age x, l_x.  So if L_x is person years, then T_x is cumulated person years , i.e.

equation 6

 T_{x = x}∑¹⁰⁵L_x 

e.g.  T₁₀₂ = L₁₀₂+L₁₀₃+L₁₀₄+L₁₀₅

The final column of the life table, e⁰_x is the life expectancy in years

equation 7

 e⁰_x=T_x / l_x  

e⁰₀ represents life expectancy at birth and it is broadly used to express the level of mortality.  Life expectancy is the average number of additional years a person would live if current mortality trends were to continue.  The expectation of life at birth represents the mean length of life of individuals who are subjected since birth to current mortality trends.  Life expectancy is usually compiled on the basis of a life table showing the probability of dying at each age for a given population according to the age specific death rates prevailing in a given period.

Further information

From equation 3 we see the link between the probability of surviving with that of dying, therefore we can make assumptions on the probability of surviving from the probability of dying.  This is what is referred to in population projections as the mortality assumption.

S_x= L_x ⁄ L_x-1

The survivorship ratio at age x, S_x, equals the ratio of those surviving between ages x and x+1 and those surviving between the ages of x-1 and x, e.g. the ratio of those aged 5 – 9 surviving to age 10 -14 is calculated as follows:

S_10-14 =₁₀∑¹⁴ L_x / ₅∑⁹ L_x

Similarly, the probability of a man aged 20 dying before his 50th birthday is calculated as follows:

q_x=1−p_x

=1− (l_x+1 ) / (l_x ) 

= ( l_x− l_x+1 ) / l_x

 therefore

q_20-50 = ( l₂₀− l₅₀ ) / l₂₀

= (99,237 − 95,409) / 99,237

= 0.039 = 3.9%

Hide Background Notes