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# Population Forecasting Methods | Formulas | Example Problems | Practice Problem

Population forecasting is a method to predict/forecast the future population of an area. Usually, the population at the design period of water supply systems is predicted to find the water demand at that time, as the systems are required to fulfill their purposes till the end of the design period. Methods to predict the population are discussed further.

## Arithmetical Increase or Arithmetical Mean Method

### Where Arithmetical Increase Method is used

The arithmetical Increase Method is mainly adopted for old and developed towns, where the rate of population growth is nearly constant. Therefore, it is assumed that the rate of growth of the population is constant. It is similar to simple interest calculations. The population predicted by this method is the lowest of all.

### Arithmetical Increase Method Derivation

dP/dt = K (say), where, dP/dt represents rate of growth of population.

Integrating the above equation over P1 to P2 over a time period of t1 to t2,

∫dP = K∫dt

[P2 - P1] = k * [t2 - t1]

P2 = P1 + K * Δt

P2 = P1 + x̄ * n

P2 = P1 + n

### Arithmetical Increase Method Formula

Pn = Po + nx̄,

where,

Po - last known population

Pn - population (predicted) after 'n' number of decades,

n - number of decades between Po and Pn and,

x̄ - the rate of population growth.

### Arithmetical Increase Method Example Problem

The following data (common data) will be used in the example problems for all other methods to be discussed.

 Year Population 1930 25000 1940 28000 1950 34000 1960 42000 1970 47000

Question: With the help of the common data find the population for the year 2020 using the arithmetic increase method.

Solution:

Step 1: Find the increase in population each decade.

 Year Population Increase 1930 25000 - 1940 28000 3000 1950 34000 6000 1960 42000 8000 1970 47000 5000

Step 2: Find the average rate of increase of population (x̄)

x̄ = (3000+6000+8000+5000)/4

x̄ = 22000/4

x̄ = 5500

Step 3: Find the number of decades (n) between the last known year and the required year

n = 5 (5 decades elapsed between 1970 and 2020)

Step 4: Apply the formula Pn = Po + nx̄,

P = P + (5 * 5500)

P = 47000 + 27500

P = 74,500. Therefore, population at 2020 will be 74,500.

## Geometrical Increase Method

### Where Geometrical Increase Method is used

This method is adopted for young and developing towns, where the rate of growth of population is proportional to the population at present (i.e., dP/dt ∝ P). Therefore, it is assumed that the percentage increase in population is constant. It is similar to compound interest calculations. The population predicted by this method is the highest of all.

### Geometrical Increase Method Derivation

Let's say, for the 0th-year population is P

For 1st year/decade, according to this method, the population would become,

P + (r/100)P, where r is the growth rate.

For 2nd year/decade, according to this method, population would become,

[P + (r/100)P] + (r/100)[P + (r/100)P]

= P[1+(r/100)]^2

Generalizing the above equation, we get,

Pn = Po[1 + (r/100)]^n

### Geometrical Increase Method Formula

Pn = Po[1 + (r/100)]^n,

where,

Po - last known population,

Pn - population (predicted) after 'n' number of decades,

n - number of decades between Po and Pn and,

r - growth rate = (increase in population/initial population) * 100 (%).

r could be found as arithmetic mean (i.e., (r1 + r2 + r3 ... rn)/n) or as a geometric mean (i.e., nth root of (r1 * r2 * r3 ... rn)), for the given data. According to Indian standards r should be calculates using geometric mean method.

### Geometrical Increase Method Example Problem

Question: With the help of the common data find the population for the year 2020 using the Geometrical increase method.

Solution:

Step 1: Find the increase in population each decade.

Step 2: Find the growth rate.

 Year Population Increase in population Growth rate 1930 25000 - - 1940 28000 3000 (3000/25000) * 100 = 12% 1950 34000 6000 (6000/28000) * 100 = 21.4% 1960 42000 8000 (8000/34000) * 100 = 23.5% 1970 47000 5000 (5000/42000) * 100 = 11.9%

Step 3: Find the average growth rate (r) using geometrical mean.

r = ∜(12 * 21.4 * 23.5 * 11.9)

r = 16.37 %

Step 4: Find the number of decades (n) between the last known year and the required year

n = 5 (5 decades elapsed between 1970 and 2020)

Step 5: Apply the formula Pn = Po[1 + (r/100)]^n

P = P[1 + (16.37/100)]^5

P = 47000[1.1637]^5

P = 1,00,300. Therefore, population at 2020 will be 1,00,300.

## Incremental Increase Method

### Where Incremental Increase Method is used

This method is adopted for average-sized towns under normal conditions, where the rate of population growth is not constant i.e., either increasing or decreasing. It is a combination of the arithmetic increase method and geometrical increase method. Population predicted by this method lies between the arithmetical increase method and the geometrical increase method.

### Incremental Increase Method Formula

Pn = (Po + nx̄) + ((n(n+1))/2)* ȳ,

where,

Po - last known population,

Pn - population (predicted) after 'n' number of decades,

n - number of decades between Po and Pn,

x̄ - mean or average of increase in population and,

ȳ - algebraic mean of incremental increase (an increase of increase) of population.

### Incremental Increase Method Example Problem

Question: With the help of the common data find the population for the year 2020 using the Incremental Increase method.

Solution:

Step 1: Find the increase in population in each decade.

Step 2: Find the incremental increase i.e., increase of increase.

 Year Population Increase in population Incremental Increase 1930 25000