PPT on measurement of trend in sales of a company

1. Measurement of trend in sales
of a company
J KRISHNA SINDHU

2. Method of Least Squares
● This is the best method for obtaining the trend values.
● The line of best fit is a line from which the sum of the deviations of various points is zero.
● The sum of the squares of these deviations would be least when compared with other fitting methods.
The following two conditions are satisfied:
(i) The sum of the deviations of the actual values of Y and Ŷ (estimated value of Y) is Zero. that is
Σ(Y–Ŷ) = 0.
(ii) The sum of squares of the deviations of the actual values of Y and Ŷ (estimated value of Y) is least. that
is
Σ(Y–Ŷ)2 is least

3. Procedure to Calculate:
(i) The straight line trend is represented by the equation Y = a + bX …(1)
where Y is the actual value, X is time, a, b are constants
(ii) The constants ‘a’ and ‘b’ are estimated by solving the following two normal
Equations ΣY = n a + b ΣX ...(2)
ΣXY = a ΣX + b ΣX2 ...(3)
Where ‘n’ = number of years given in the data.
(iii) By taking the mid-point of the time as the origin, we get ΣX = 0

4. (iv) When ΣX = 0 , the two normal equations reduces to
The constant ‘a’ gives the mean of Y and ‘b’ gives the rate of change (slope).
(v) By substituting the values of ‘a’ and ‘b’ in the trend equation (1), we get the Line of Best Fit.
Problem: Given below are the data relating to the sales of a product in a company
Fit a straight line trend by the method of least squares and tabulate the trend values.
Years 2015 2016 2017 2018 2019
Sales 30 50 75 80 40

5. a=ΣY/N =275/5=55
b=ΣXY/X^2=50/10=5 Y(t)=a+bX : 55+5(-2)=45
55+5(-1)=50
55+5(0)=55
55+5(1)=60
55+5(2)=65
YEAR
(x)
SALES
(Y)
X=(x-A)
=(x-2017)
X^2 X.Y Y(t)=a+bX
2015 30 -2 4 -60 45
2016 50 -1 1 -50 50
2017 75 0 0 0 55
2018 80 1 1 80 60
2019 40 2 4 80 65
N=5 ΣY=275 ΣX^2=10 ΣXY=50

6. 2020- Y(t)=a+bX= 55+5(3)=75
YEAR
(x)
SALES
(Y)
X=(x-A)
=(x-2017)
X^2 X.Y Y(t)=a+bX
2015 30 -2 4 -60 45
2016 50 -1 1 -50 50
2017 75 0 0 0 55
2018 80 1 1 80 60
2019 40 2 4 80 65
2020 3 75
2021 4

7. CONCLUSION
Unlike the other methods, under this method, it is quite possible to forecast any past
or future values perfectly, since the method provides us with a functional relationship
between two variables in the form of a trend line equation

8. THANK YOU