Page 417 - DMTH404_STATISTICS

Unit 30: Method of Least Square

Solution. Notes

Calculation Table

Census Population ( t 1966) log Y X log Y X 2
Year t Y X = 5
1941 319 5 15038 75190 25
.
.
.
.
.
.
1951 361 3 15575 4 6725 9
.
.
1961 439 1 16425 16425 1
.
.
1971 54 8 1 17388 17388 1
.
.
.
1981 683 3 18344 55032 9
.
.
.
.
.
1991 84 4 5 19263 9 6315 25
Total 0 102033 30395 70
.
.
10.2033 3.0395
From the above table, we get A = = 1.70 and B = = 0.043
6 70
Further, a = antilog 1.70 = 50.12 and b = antilog 0.043 = 1.10
Thus, the fitted trend equation is Y = 50.12(1.10) ,
X
Origin : 1st July, 1966 and unit of X = 5 years.
The trend values can be computed by the equation Y = antilog [1.70 + 0.043X]. Further, the
prediction of population for 2001 is obtained by substituting X = 7, in the above equation.
 Y = antilog[1.70 + 0.043  7] = antilog[2.001] = 100.2 crores
30.1.4 Merits and Demerits of Least Squares Method
Merits
1. Given the mathematical form of the trend to be fitted, the least squares method is an
objective method.
2. Unlike the moving average method, it is possible to compute trend values for all the
periods and predict the value for a period lying outside the observed data.
3. The results of the method of least squares are most satisfactory because the fitted trend
satisfies the two important properties, i.e., (i) S(Y - Y ) = 0 and (ii) S(Y - Y ) is minimum.
2
o t o t
Here Y denotes the observed value and Y denotes the calculated trend value.
o t
The first property implies that the position of fitted trend equation is such that the sum of
deviations of observations above and below this is equal to zero. The second property
implies that the sum of squares of deviations of observations, about the trend equation,
are minimum.
Demerits
1. As compared with the moving average method, it is a cumbersome method.

2. It is not flexible like the moving average method. If some observations are added, then the
entire calculations are to be done once again.
3. It can predict or estimate values only in the immediate future or past.

4. The computation of trend values, on the basis of this method, doesn't take into account the
other components of a time series and hence not reliable.

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