PRESENTATION ON

GOMPERTZ GROWTH CURVE &

RATIO METHOD

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POPULATION PROJECTION

Assumption that the past trends will continue to operate in

the future. The reliability and usefulness of projections

depend on the assumptions and their closeness to reality.

When an element of judgment is added to the projections, it

becomes a forecast. Forecasts enjoy the advantage of

being based upon the assumption or a set of assumptions

which are likely to be realized in the near future and can

yield a relatively more realistic

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GOMPERTZ GROWTH CURVE

The Gompertz curve was originally derived to estimate human

mortality by Benjamin Gompertz.

It is a sigmoid function.

Describes a growth pattern that is initially quite slow, increases

for a period of time, and then tapers off as the population

approaches a growth limit.

It is very similar to neighborhood and/or city growth patterns

over the long term

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GOMPERTZ FORMULA:

The standard Gompertz curve:

Where, y is the variable representing performance, L is the upper limit, e is the base of the natural logarithms [Euler's Number: e = 2.71828...] t is time, and b and k are the coefficients obtained by fitting the curve

to the data.

ktbeLey

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Properties of Gompertz Curve:

The initial value is zero at time, t = - and a value of L at time,t = +. The curve is not symmetrical. The inflection point occurs at t = (ln b) /k, when y = L/e.

By taking the logarithm of the Gompertz curve twice, we obtain:

When Y is regressed on t, the constant term is ln b and the slope term is k.

ktby

LY

lnlnln

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Examples of Gompertz curve are:

Mobile phone uptake, where costs were initially high (so

uptake was slow), followed by a period of rapid growth,

followed by a slowing of uptake as saturation was reached.

Population in a confined space, as birth rates first increase

and then slows as resource limits are reached

Modeling of growth of tumors.

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RATIO METHOD

The Ratio Method projects population growth for a sub-area

using population projection for a larger or parent population.

A regional projection can be used to project the population

size of districts, and a projection for the country can be used

to project the population size of a region.

However, use of good judgment in population projections is

important since one should consider the specific features of

the region, its history, development, and socio-economic

factors.

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FOR EXAMPLE:

Supposing that a certain province (state) contained 25 per

cent of the population of the entire country in 1920 and 28 per

cent in 1950.

One may infer that it will contain about 29 per cent in 1960,

30 per cent in 1970, etc., unless other wise there is a reason

to assume.

This method, however, cannot always be used without some

modification if some patently observed results are to be

avoided.

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ASSUMPTION:

Local population change is highly dependent on what happens

to the population in the surrounding regions or states.

The ratio method can be used to project population growth for

the local area if two conditions are met: similar population

patterns exist for both the local and the parent population; and

it is expected to continue in the future.

The ratio method also assumes that a projection exists for the

larger or parent population.

As indicated before, projections for large areas such as a

country, province, or region tend to be more reliable than those

produced for smaller locals.

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Ratio Method Equation:

This method is used for projecting the total

population size for 5 to10+ year periods.

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How this method is used:

By using historic census data to graph the sub-area population and the parent/state population to see if similar growth is taking place.

Collecting projections for the parent population, examining differences in the projections and determines why the differences exist, for what methods were used and what were the assumptions.

EXAMPLE:

Projects the total population of a region using the projection for

a country. A projection of a state or province could also be

used as the basis to project the size of a district. Once district

projections are available, it is possible to project the size of a

small town or village, provided that growth patterns are similar.

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Process of using this Ratio Method:

First, check whether similar growth patterns exist

among the sub-area and the parent population.

This Graph represents Population Trend of a Region.

Population Trend of the Volta Region of Ghana, 1960 to 2000.

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This graph represents Population Trend of a Country.

Population Trend of Ghana

In this case, the patterns of growth are similar but not identical.

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A reliable projection for the larger population is obtained in

this equation by knowing the projected size of the country

(Ghana) for year 2010 is 22,650,000.

Application of the Ratio Method Equation

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ADVANTAGES: Over extrapolation techniques:

Computation is fast; it only takes a few minutes once the data and a

reliable projection are available.

Revisions are simple, and the ratio method can be used to make long-

range projections.

DISADVANTAGES: Over extrapolation techniques:

The ratio method does not support the study of changes in births, deaths,

and migration.

In addition, it requires a reliable projection for the larger area.

Finally, it is highly dependent on the assumed relationship between the

sub-area and the larger or parent population.

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