[soniya]

Least square method of population projection

This method is applicable when time-series data is available. It is a simple method commonly used to make future projections on the basis of the past
trend. It is common to fit a straight line to the past observations.

y= a+bx

a= ƩY / N
b= Ʃxy /x2
y= last five year decades population x= -2,-1,0,1,2
N= 5 (for five year population data)

[Manish Jain]

Example of Least square method of population projection

Given Data

Year            population (in million)
1997           51.99
1998           52.49
1999           55.12
2000           54.07
2001           57.56

ΣY = Na + b. Σ X

ΣXY = a. Σ X + b. ΣX2

where:

ΣX = the sum of all observations of X
ΣY = the corresponding sum of all the Y observations
ΣXY = the sum of all the products of X and Y
ΣX2 = the sum of all the squares of X
N = total number of observations

Population  Projections Based on Method of Least Squares 

Year          population   S. No. of           Square of
                (in million)       Col. 1                   ‘X’
                      (Y)                   (X)                                                      X.Y

1997                51.99             1                       1                              51.99
1998                52.46             2                       4                              104.92
1999                55.12             3                      9                               165.36
2000               54.07              4                      16                             216.23
2001                57.56             5                       25                            287.80
Total     (ΣY) 271.20           (ΣX) 15           (ΣX2) 55                  ΣXY) 826.35

By using the following two equations, the value of Y = a + b . X can be calculated:


ΣY = Na + bΣ X =  271.20 = 5a + 15b ……………………………1

ΣXY = a Σ X + b ΣX2 =  826.33 = 15a + 55b ……………………………….2


Multiply equation (1) by 3 and subtract it from Equation (2) to get the value of ‘b’. By putting the value of ‘b’ in Equation (2), the value of ‘a’ can also be worked out: 
Thus, by interpolation we get the following equation: 

Y = 50.43 + 1.27 X (with 1997 as X=1) 

In the above example, by putting the values of ‘X’ we can get values for various years to draw the best-fitting lines and then project it in the future. Accordingly, the population projections for 2002 can be worked out as under: 

population in 2002 (Y) = 50.43 + 1.27 x 6 = 58.05, X = 6 in year 2002-03