October 12

Ratios and Rates


*Ratio : x/y ; magnitude of x relative to y, used in demography for descriptive statistics

Example 1: Sex Ratio at Birth

Sex Ratio at Birth:
x= # of male births in a given year ;
y=100s of female births in same year

Consider Wales in 1985: x= 336,835 ; y= 3195.82 because # of females born = 319,582 but because we want the denominator to be 100s of females, we divide the total # of females born by 100.

x/y= 336,835/3195.82= 105.4 male births per 100 female births 105.4/100 is very typical for various reasons. One of the more commonly cited is the fact males have higher mortality rates from pre-conception (sperm) throughout life than their female counterparts and nature attempts to even out this phenomenon by ensuring that more males are conceived.

Consider China:
Year Male births/100 of female births
1970 ~= 106
1982 108.5
1987 110.9
1990 111.3
2000 116.9

The fact that the ratio is changing so rapidly and is so high above the norm. It implies something about China's practices. Particularly, sex selective abortions in which parents abort the pregnancy if the child is a female.

Example 2: Child per Woman ratio

Child per woman ratio (It may be affected by factors like migration but it is still considered an effective indicator of fertility levels):

x= # of living children 0-4 years old ; y= 1000s of 15-49 year old women
x/y= # of children under 5 per 1000 women of reproductive age

Consider India:
State Children 0-4 per 1000 women ages 15-49
Goa 1159-->family size=3-4
Tamil Nadu 1246-->family size=3-4
: :
: :
: :
Bihar 2622-->family size=4-5
Meghalaya 2646-->family size=4-5

These fertility rates tell us something abotut the socioeconomic status of the area. Poorer places generally have higher rates.

  • Most importantly (be sure to remember) Bihar is like the Mississippi of India!! (Joke)

Example 3: Dependency Ratio

Dependency Ratio: x=# of people 0-14 years + # of people 65 and older
y=100s of people 15-64

Consider Florida (2000)

x= 3,034,565+2,807,597=5,842,162
y=101,402.16 because there were 10,140,216 people ages 15-64 in Florida in 2000 and it was divided by 100

x/y~=58 dependents per 100 working age people in Florida
x/y~=48 " " Georgia

This is because Florida has more retirees than our neighbors to the north.


A rate is a special kind of ratio, with event counts in the numerator and person-years of risk (also called "exposure") in the denominator

  • Rate: a # of events of per person-year (or per person time period)

Exposure: time (person-years) spent at risk of an event
eg.a) I lived all of 2007 on Earth-> One person-year of exposure to a meteor strike
b) I was unmarried on 1 Jan 82
I was married on 27 Nov 82
I remained married for the duration of 1982 (and beyond)
Thus, I had 331/365=.907 person-years of exposure to risk of marriage.

Person-years: # of persons x number of years
eg.a) 1 person exposed for 2 years = 2 person-years
b) 2 people each exposed for 3 months = .5 person-years

  • Example

Paper Mem. # of Months Obs. Person-Years Obs. # of times Obs. Flossing
A 12 1 10
B 9 .75 50
C 3 .25 50
D 15 1.25 0
E 12 1 10
4.25 120

Rate= 120 flosses/4.25 person-years=28.24--> Rates can be bigger than one

Sometimes "rates" aren't really (events/exposure time) ratios
eg. Sometime "rates" have people in the denominator for whom the event is not possible
Crude birth rate = (# of birth in population in calendar year/mid-year population)
a) Mid-year population is a guess about total person-years lived by the population
b) Lots of these person-years are not spent at risk of birth (eg. baby-years, brownie-
years, gramps-years,…, man-years)

eg. Consider Mexico in 1996
Mexico's midyear population= 95,772,000
1996 births= 2,513,000
CBR=(2,513,000/9572000)=0.02624—>2% of Mexicans gave birth in 1996
->26.24 births per 1000 person-years

Crude death rate= (#of deaths in a year/mid-year population)
1996 deaths=439,000
CDR=(439,000/95,772,000)=0.00458—>.5% of Mexicans died in 1996
->4.58 deaths per 1000 person-years
1000 Mexicans on 1 Jan 96
-> 26.24 births expected over next year
-> 4.58 deaths " "
->1021.66 Mexicans on 31 Dec 96

(21.66/1000)=2.166% annual growth

Is this growth feasible? At a 2% annual growth rate, it would take Mexico's
population 35 years to double. That rate would be impossible to sustain over long periods of time.