Life Expectancy is a measure of how long a person may live based on their year of birth, their current age and other demographic factors including sex. A widely used measure of life expectancy is life expectancy at birth.
Life expectancy is an important measure for superannuation planning, health care planning and other important issues in society. Life tables based on births and mortality rates in different countries show life expectancies at various ages and for different population groups such as men and women.
To learn more about how Life Expectancy is calculated go to the University of California, San Diego (UCSD) website using the button below.
If the link is outdated visit UCSD at 'socialsciences.ucsd.edu' and search for more information
END LIFE EXPECTANCY EXPLANATION
It is estimated that a Total Fertility Rate (TFR) of 2.1 (TFR = 2.1) will maintain the population level of a developed country but, this may need to be considerably higher for countries with high mortality rates.
Calculating the Total Fertility Rate
Using made up data for the year 2013.
| Age | Births | Population | ASFR (2) |
|---|---|---|---|
| 15-19 | 20 | 22500 | 0.89 |
| 20-24 | 2600 | 20100 | 129.35 |
| 25-29 | 2400 | 18960 | 126.58 |
| 30-34 | 2200 | 17800 | 123.60 |
| 35-39 | 900 | 16800 | 53.57 |
| 40-44 | 400 | 16500 | 24.24 |
| 45-49 | 10 | 14000 | 0.71 |
| TOTAL | 8530 | 126660 | 458.95 |
Calculate the Total Fertility Rate, given the Age Specific Fertility Rates in a given year, for the following population (work to two decimal places):
| Age | Births | Population | ASFR (2) |
|---|---|---|---|
| 15-19 | 167 | 34670 | 4.82 |
| 20-24 | 3200 | 31567 | 101.37 |
| 25-29 | 4300 | 43200 | 99.54 |
| 30-34 | 3615 | 41389 | 87.34 |
| 35-39 | 1450 | 32877 | 44.10 |
| 40-44 | 816 | 23900 | 34.14 |
| 45-49 | 70 | 15400 | |
| TOTAL | 13618 | 223003 |
The ASFR value is 4.55, the total is 375.86
TFR =END FERTILITY CALCULATIONS
As with fertility, age specific mortality rates can also be calculated. Each country uses data on mortality (deaths) to calculate life expectancy.
Mortality Rate Calculations
To calculate a simple (Crude) Mortality Rate or CMR for short, we divide the number of deaths in a population during a specified time period (year), by the size of the population in that period.
The Crude Mortality Rate for a country of 4,100,000 with 29,000 deaths in a given year is:
What is the Crude Mortality Rate (to two decimal places) for a country of 61,345,000 with 429,000 deaths in a given year?
END MORTALITY CALCULATIONS
Life Tables can be constructed using mortality rates usually age specific mortality rates. They are usually constructed separately for men and women because of the different mortality rates.
Other characteristics such as ethnicity can be used to distinguish different risks. A hypothetical cohort of 100 000 new born male or female babies are followed for their entire life. The cohort is used to derive the probability of surviving any particular year of age, remaining life expectancy for people at different ages and other calculations described in the following examples.
EXAMPLES
New Zealand
The Life Tables used for the following examples, and the exercises that follow, have been produced by Statistics New Zealand.
They have been developed from periodic government censuses. Because ethnicity is self-perceived, people can identify with Māori ethnicity even though they are not descended from a Māori ancestor. Conversely, people may choose to not identify with Māori ethnicity even though they are descended from a Māori ancestor. The Māori tables refer to persons of 'half or more' New Zealand Māori origin or of 'solely' New Zealand Māori origin.
Life Table Structure and Definitions
A shortened and modified version of the full New Zealand Māori Male Life Table 2005-2007 takes the form shown below. Other full tables for Māori females and for non-Māori males and females can be abbreviated similarly.
| Age (X) | NUMBER ALIVE | NUMBER ALIVENEXT YEAR OF AGE | YEARS OF LIFEREMAINING |
|---|---|---|---|
| 0 | 100 000 | 99 401 | 70.36 |
| 1 | 99 246 | 99 210 | 69.90 |
| 2 | 99 173 | 99 146 | 68.95 |
| 3 | 99 119 | 99 096 | 67.98 |
| : | |||
| 99 | 273 | 219 | 1.88 |
| 100 | 165 | 131 | 1.74 |
EXAMPLE 1
The numbers who die in the next year after x are the numbers alive at x minus the number alive at x+1.
For the first year this is 100,000 – 99,246 = 754
For the second year this is 99,246 – 99,173 = 73
EXAMPLE 2
The probability of living another year for an individual who reaches exact age x is number alive at exact age x+1 divided by number alive at exact age x.
For a randomly selected Māori male aged precisely two years and zero days the probability of living to age 3 is 99,119/99,173 = 0.99946.
EXAMPLE 3
The probability of dying in the next year for an individual who reaches the exact age x is one minus the probability of living.
For a randomly selected Māori male aged precisely two years and zero days the probability of dying next year is 1 – 0.99946 = 0.00054
The calculations from examples 1 to 3 are reported in the full Life Table in their own columns. The next examples refer to the Māori Male Table, again using the 'Life Tables' link above.
EXAMPLE 4
Find the fraction of Māori boys now aged 18 who are expected to live a further three years.
Initially, this fraction would appear to be the number alive at exact age 21 divided by the number alive at exact age 18. But in practice, ages are spread over the year, not concentrated exactly on the birth date. The number alive at exact age 22 divided by the number alive at exact age 19 is equally inaccurate. The estimate is given by the compromise fraction which averages these two, namely, NUMBER ALIVE YEAR 21 / NUMBER ALIVE YEAR 18.
Under the 2005-2007 mortality conditions this is...
EXAMPLE 5
Thirty 70 year old and twenty 80 year old Māori men are paid annuities of $20,000 per year by an insurance company for the rest of their lives. Assuming exact ages, how much would the insurance company pay?
From the column it can be seen that a 70 year old Māori male can expect to live 10.98 years while an 80 year old Māori male can expect to live 6.23 years. Therefore:
EXAMPLE 6
Find the probability that a newborn Māori boy dies between the exact ages of 10 and 20.
Out of the 100,000 hypothetical newborn Māori boys, there are...
who die between the exact ages of 10 and 20. Hence probability of dying is...
EXAMPLE 7
A Māori boy is nominally aged 19. Calculate the chances that he dies before reaching the age of 50, and that he dies in his fifties.
The number of the original cohort of 100,000 reaching nominal age 19 is ½(98,228 + 98386) = 98,307 while the number of the original cohort surviving to their 50th birthday is 89,719.
Therefore, the number who die before 50 is 98,307 – 89,719 = 8588, and the probability of this boy dying before the age of 50 is:
Also, the probability that one of these boys dies in his fifties becomes:
END LIFE TABLE EXAMPLES
Using the examples above to guide you, work through the following Life Table exercises.
All of the exercises below use the Life Tables produced by the New Zealand Department of Statistics, that were used in our examples above. These are 'Appendix 1: New Zealand period life tables, 2005–07'.
EXERCISES
New Zealand
Use the Māori Male Life Table (2005-2007) to answer the questions below.
EXERCISE 1
What is the probability that a non-Māori man exactly 20 years old will die before he reaches the age of 60 years? Give the answer to five decimal places.
EXERCISE 2
a. What are the chances that a newly-born Māori boy survives to his 60th birthday?
b. A Māori male is (nominally) aged 18. What are the chances that he dies before reaching the age of 60? (Hint: Calculate the number nominally 18 and subtract the number with exact age 60).
So the chances of dying before reaching age 60 is 0.19271.
c. If two hundred 60 year old and one hundred 65 year old Māori men are paid pensions of $20 000 a year for the rest of their lives, how much money would you expect the superannuation fund to pay out?
Total years = 200(17.02) + 100(13.82) = 4786 assuming exact ages. At $20,000 per year, amount paid = $95,720,000.
d. Repeat questions 'c' for Māori women and then for non-Māori men. (Use the Māori women and non-Māori men New Zealand Life Tables.)
Māori Women
Non-Māori Men
EXERCISE 3
The numbers of New Zealand non-Māori male infants in various age groups in a city are as follows:
| Age | 5-(6) | 6-(7) | 7-(8) | 8-(9) |
| Number | 2083 | 2527 | 2404 | 2625 |
Neglecting migration, predict the population of 13, 14, 15 and 16 year old non-Māori boys at high school eight years later.
13-(14)
=14-(15)
=15-(16)
=16-(17)
=The answer is:
=END LIFE TABLE EXERCISES