Page 151 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes Alternative Method

From the last column of the above table, we have
Sum of squares = 15196
15196
Mean of squares = = 199.95
76
Thus, 2 = Mean of squares – Square of the mean = 199.96 – (14) = 3.96
2
Short-cut Method

Before discussing this method, we shall examine an important property of the variance (or
standard deviation), given below:
The variance of a distribution is independent of the change of origin but not of change of scale.

Change of Origin

If from each of the observations, X , X ...... X , a fixed number, say A, is subtracted, the resulting
1 2 n
values are X – A, X – A ...... X – A.
1 2 n
We denote X – A by d , where i = 1, 2 ...... n the values d , d ...... d are said to be measured from
i i 1 2 n
A. In order to understand this, we consider the following figure.

Values : 0 1 2 3 4 5 6 7 8
(= – 3) Values : –3 –2 –1 0 1 2 3 4 5

In the above, the origin of X values is the point at which X = 0. When we make the transformation
i i
d = X – 3, the origin of d values shift at the value 3 because d = 0 when X = 3.
i i i i i
The first part of the property says that the variance of X values is equal to the variance of the d i
i
values, i.e., 2 2 .
X d
Change of Scale

To make change of scale every observation is divided (or multiplied) by a suitable constant. For
example, if X denotes inches, then Y = X /12 will denote feet or if X denotes rupees, then Y = 100
i i i i i
X i
X = will denote paise, etc.
i 0 01
.
We can also have simultaneous change of origin and scale, by making the transformation
X i A
u
i , where A refers to change of origin and h refers to change of scale.
h
According to second part of the property 2 2 or 2 2 .
X Y X u

The Relation between 2 and 2
X u

2 1 2
Consider X f X i X .... (1)
i
N

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