Page 164 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Unit 7: Measures of Dispersion

Notes
Solution: We are given raw moments; 1, 16 and 40 , which should be converted
1 2 3
into central moments.
2

Now 2 2 1 = 16 1 = 15
3 2 3 = 40 3 16 + 2 = 86
3 3 2 1 1
86
1 1 3 = 1.48
(15) 2

7.8.6 Empirical Relation among Various Measures of Dispersions

Although much depends upon the nature of a frequency distribution, it has been observed that
for a symmetrical or moderately skewed distribution, the following approximate results hold
true.
5
QD 0.8453 (approximately ) × MD
6
2
QD 0.6745 (approximately ) × SD
3
4
MD 0.7979 (approximately ) × SD
5
or we can say that 6 SD 9 QD 7.5 MD
Also Range 6 SD

!
Caution Standard deviation is independent of change of origin but not of change of scale.
This implies that if a constant is added (or subtracted) to all the observations, the value of
standard deviation remains unaffected. On the other hand, if all the observations are
multiplied or divided by a constant, the standard deviation also gets multiplied (or divided)
by this constant.

Notes Choice of a Suitable Measure of Dispersion
The choice of a suitable measure depends upon: (i) The nature of available data, (ii) the
objective of measuring dispersion and (iii) the characteristics of the measure of dispersion.
The nature of available data may restrict the choice of a measure of dispersion. For example,
if the distribution has class intervals with open ends, one can only calculate quartile
deviation, percentile deviation, etc. On the other hand, if the objective is to know the
extent of variations in the values of a variable in a given time or situation, the calculation
of range may be more appropriate, e.g., maximum and minimum rainfall in a season,
maximum and minimum temperature on a particular day, etc. Similarly, Lorenz Curves
are generally used to compare the extent of inequalities of income or wealth in two or
more situations. Further, if one is interested in obtaining the magnitude of variation in
observations on the average from a central value, mean deviation and standard deviation
are used. In statistical analysis, the use of standard deviation is preferred to mean deviation
because of its several merits over the later. In the words of M.M. Blair, “These two measures
Contd...

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