Page 103 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes w d
2. X A i i (where di = Xi - A) (Using short-cut method),
w
w
i
wu X i A
3. X A i i h (where u i ) (Using step-deviation method)
w h
w
i
Remarks: If X denotes simple mean and X denotes the weighted mean of the same data, then
w
1. X = X , when equal weights are assigned to all the items.
w
2. X > X , when items of small magnitude are assigned greater weights and items of large
w
magnitude are assigned lesser weights.
3. X < X , when items of small magnitude are assigned lesser weights and items of large
w
magnitude are assigned greater weights.
6.2.3 Properties of Arithmetic Mean

Arithmetic mean of a given data possess the following properties:
1. The sum of deviations of the observations from their arithmetic mean is always zero.

According to this property, the arithmetic mean serves as a point of balance or a centre of
gravity of the distribution; since sum of positive deviations (i.e., deviations of observations
which are greater than X ) is equal to the sum of negative deviations (i.e., deviations of
observations which are less than X ).

2. The sum of squares of deviations of observations is minimum when taken from their
arithmetic mean. Because of this, the mean is sometimes termed as 'least square' measure
of central tendency.
3. Arithmetic mean is capable of being treated algebrically.

This property of arithmetic mean highlights the relationship between X , i f X and N.
i
According to this property, if any two of the three values are known, the third can be
easily computed.

4. If X and N are the mean and number of observations of a series and X and N are the
1 1 2 2
corresponding magnitudes of another series, then the mean X of the combined series of
N X N X
N + N observations is given by X 1 1 2 2
1 2
N 1 N 2
5. If a constant B is added (subtracted) from every observation, the mean of these observations
also gets added (subtracted) by it.

6. If every observation is multiplied (divided) by a constant b, the mean of these observations
also gets multiplied (divided) by it.

7. If some observations of a series are replaced by some other observations, then the mean of
original observations will change by the average change in magnitude of the changed
observations.

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