SADC Course in Statistics Basic summaries for demographic studies (Session 03)

SADC Course in Statistics

Basic summaries for demographic studies

(Session 03)

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Learning Objectives

At the end of this session, you will be able to

• interpret and use demographers’ shorthand notation

• correctly distinguish & use absolute & relative numbers, and ratios & rates

• explain the definitions, and thinking behind, different death rates

• recognise the detailed accuracy needed in demographic studies


Demographic “shorthand”

Standard symbols are very common in demography to represent common ideas/ data summaries too long-winded to write out in full each time ~ they are not 100% predictable so take care in interpretation.

Example ~ P usually represents a total population size; P0 and Pt may be size of

population at times 0 and t; but elsewhere Px could represent the population aged x

[between xth and (x+1)th birthdays]


Absolute & relative numbers

An actual count is referred to as “absolute”; a relative number could be a count expressed per 100, or per 1000 i.e. compared to (relative to) another number.

Example ~ 59 children left a school last term (absolute number). 590 children were registered at the start of term, so 59/590 = 0.1 or = 10% left (relative number).

Fractions can confuse laymen so we express them as “one in ten” or “ten per hundred”.


When to use absolute and relative

When you need to express the actual size of something, use absolute numbers, but for comparing things of different sizes usually “correct for” differences by using relative numbers.

Example ~ 12 cases of stochastic fever occurred in Mauritius in 2003 as opposed to 20 in Mozambique in same year. Per million population these figures equate to roughly 10.7 and 1.06, accounting for different population sizes.


Ratio

Of course a ratio is just one number divided by another.

Example ~ in the school’s entry year, grade 1, there were 50 girls and 47 boys: the Male/Female ratio is 47/50 [ or 0.94]In the school’s final year, grade 12, there were 10 girls and 17 boys: the Male/Female ratio is 17/10 [or 1.70]

Demographers call these “age-specific” or “grade-specific” sex ratios. M/F is standard definition of sex ratio [not F/M]


Proportion

A proportion is a number representing part, divided by a number representing the whole of the same thing. All proportions are ratios, not all ratios are proportions.

Example ~ the proportion of girls in grade 1(as above) is 50/(50 + 47) = 50/97 [or 0.5155]. The proportion of boys in grade 12 is 17/27 [or 0.6296]. The proportion of boys, in each case, is 1 – proportion of girls.


Rates – the time dimension

Ratios and proportions are point-in-time still snapshots. Where events occur through time - and where they are concerned with the speed (flow) - demographers use rates.

Example ~ death rate. In one calendar year 50,000 population members died. The mid-year population was 10,000,000. The “death rate per thousand (all ages)” was:-

[50,000 / 10,000,000] x 1000 =

5 deaths per 1000 per year


Ratio or rate?

Demography requires very precise language.The term “ratio” is only used for a static,

point-in-time measure with no associated concept of “speed”.

Even though it involves dividing one number by another, the term “rate” is specifically used for a quantity which involves a sense of “speed” and can be expressed as being “per unit of time”.

Misusing these terms is a naïve amateur error!


Some basic death rates: 1

• Crude death rate is number of deaths in the defined population in a one-year period (D) divided by the mid-year size of the population (P): m = [D/P]

• Age-specific death rate is no. of deaths of those aged x, in year (Dx)/ size of mid-year

population aged x (Px) : mx = [Dx/Px]

• Figures sometimes x1000 and expressed “per 1000 per year” since rates are fractions much less than 1.


Some basic death rates: 2

Note there is an age-specific death rate for every age.

• Sometimes an age-specific rate over a range of ages from exact age (birthday) x up to but not including exact age (x + n) :-

No. of deaths of those aged x up to but not including (x + n), in year (nDx) divided by

size of mid-year population aged x up to but not including (x + n) , (nPx)

nmx = [nDx/nPx]


Data for death rates: 1

Usually the number of deaths (D, Dx or nDx)

comes from death registration, if that is available and reasonably reliable. It may have to be estimated by other means e.g. scaled up from a survey, if not.

Usually the mid-year population estimate

(P, Px or nPx respectively) comes from a

census population (reported by age) projected forward from the last census date.


Data for death rates: 2

The 2 quite distinct sources/data-collection processes can each have their own sorts of inaccuracy, e.g. coverage errors, so typically such rates are affected by the combined set of errors ~ take care!

Professional demographers look carefully at patterns e.g. of age-specific death rates in several countries in several years and have mathematical methods to “smooth” values that are out of line e.g. if m23 is much bigger than m21 , m22 and m24 , m25


Mortality rates: 1

A key theoretical, rather than practical, idea is that of one “birth cohort”, envisaged as a set of people all born at the same time. From a cohort, certain numbers will die at each age. The number who die between exact ages x and (x+1) divided by the total number who lived to age x, is denoted qx. It

is the estimated probability of dying aged x.

In demographic jargon this is called the age-specific mortality rate.


Mortality rates: 2

• The probability of surviving from exact age x to exact age (x+1) is denoted px.

Of course px + qx = 1.

• Death rate figures (e.g. mx) derived from real

population figures broadly represent chances of dying, but are not probabilities.

• A standard approximation links mx and qx. If

mid-yr popn aged x is Px, & deaths in year are

Dx then start-of-year popn that suffered those

deaths was about [Px + ½ Dx]


Mortality rate algebra

So mx = [Dx/Px], while approximately

qx = Dx / [Px + ½ Dx]

= (Dx/Px) / [1 + ½ (Dx/Px)]

on dividing top and bottom by Px.

qx = mx / [1 + ½ mx] = 2mx / [2 + mx]

So data-derived death rate (mx) feeds into the

last formula to give estimated probability of dying (or mortality rate), qx.


Mortality rate arithmetic

In a population of 30-year-old males, the mid-year popn size was estimated at 112,250. In the course of the year 1500 deaths occurred to this population, so

mx = [Dx/Px] = 1500/112,250 = 0.01336

qx = 2mx / [2 + mx] = 0.01327

so where mx is quite small, there is not

much difference between the two rates.


Other demographic rates: 1

There are hundreds of other rates in regular use. Age-sex specific rates are derived for one sex only as on previous slide. Rates are “crude” if they are not specific: usually a whole range of ages lumped together.

Comparing separate populations that have different age-structures uses either full sets of age-specific rates or compromise summaries, “standardised rates”. See later module I4.


Other demographic rates: 2

This 1 session is about demographic thinking. It is not a demographic methods course!

Other very important indices relating to births include birth rate, (age-specific) fertility rates, total fertility rate and there are many others for different phenomena e.g. nuptiality rates, in- and out-migration rates. Mortality is conceptually easier in occurring to one individual. Other rates more directly involve >1 person.


Practical work follows to ensure learning objectives

are achieved…

Documents

SADC Course in Statistics Basic summaries for demographic studies (Session 03)