Explain the various methods for measuring Dispersion. Also give their merits and demerits?

Following are the important methods of studying dispersion -

(1)                              Numerical Methods :

1. Methods of limits b) Methods of average deviation:
   1. Range i)          Quartile Deviation
   2. Inter-quartile range ii)         Mean Deviation

• Percentile Range iii)        Standard Deviation

(2)                     Graphic Method - Lorenz Curve :

1. Range : The difference between the value of the smallest item and the value of the largest item of the series is called

Range = Largest item – Smallest item Co-efficient of Range = L - S

L + S

Merits of Range :

• Simple and easy to be
• It takes minimum time to
• Not necessary to know all the values, only smallest and largest value is
• Helpful in quality control of

Demerits of Range :

• Not based on all the
• Subject to fluctuation/uncertain
• Cannot be computed in case of open-end
• As it is not based on all the values it is not considered as a good or appropriate

1. Inter-Quartile Range : Inter-quartile range represents the difference between the third-quartile and the first quartile. It is also known as the range of middle 50%

Inter-quartile range = Q3 – Q1

Merits :

• It is easy to
• Can be measured in open end
• It is least affected by the uncertainty of the extreme

Demerits :

• It does not represent all the
• It is an uncertain
• It is very much affected by sampling fluctuations.
• Percentile Range : It is the difference between the values of the 90th and 10 th It is based on the middle 80% items of the series.

Percentile Range = P90 – P10

Merits and Demerits : Its use is limited. Percentile range has almost the same merits and demerits as those of inter- quartile range.

iv)                                Quartile Deviation / Semi – Interquartile Range :

Quartile Deviation gives the average amount by which the two quartiles differ from the median.  Quartile deviation is an absolute measure of dispersion.

It is calculated by using the formula -

Q.D. = Q3  – Q1  , coefficient of Q.D. =  Q3  – Q1 2 Q3 + Q1

Merits :

• It is easy to calculate and
• It has a special utility in measuring variation in open end
• QD is not affected by the presence of extreme values.

Demerits :

• It is very much affected by sampling fluctuations.
• It does not give an idea of the formulation of the
• It is not capable of further algebraic

1. Mean Deviation () : Mean Deviation is also known as average deviation or first measure of It is the

average difference between the items in a distribution and the median mean or mode of that series.

Computation of Mean Deviation :

• Least affected by the extreme

• It can be computed from any average, mean, median or

Demerits :

• Signs are ignored therefore mathematically it is incorrect and not a significant

• Cannot be compared if mean deviations of different series are based on different
• Does not give accurate

vi)                                 Standard Deviation (σ) :

Standard Deviation was introduced by Karl Pearson in 1823. It is the most important and widely used measure of studying dispersion, as it is free from those defects from which the earlier methods suffer and satisfies most of the properties of a good measure of dispersion.

Standard Deviation is the square root of the average of the square deviations from the arithmetic mean of a distribution.

Co-efficient of Standard Deviation :

Standard deviation is an absolute measure. When comparison of variability in two or more series is required, relative measure of standard deviation is computed which is called coefficient of standard deviation.

Computation of Standard Deviation :

Coefficient of S.D. = σ / X Variance = σ2

Coefficient of variation : Coefficient of variation is used for the comparative study of stability or homogeneity in more than two or more series.

C.V =  σ  X  100 X

Merits :

• Based on all the
• Well-defined and definite measure of
• Least affected by sample
• Suitable for algebraic

Demerits :

• Standard Deviation     is    comparatively   difficult      to
• Much importance is given to the extreme values.

vii)                            Graphical Method - Lorenz Curve :

This curve was devised by Dr. Max O‘Lorenz. He used this technique to show inequality of wealth and income of a group of people. It is simple, attractive and  effective measure of dispersion, yet it is not scientific since it does not provide a figure to measure dispersion.

Merits :

• Easy to understand from the
• Comparison can be done in two or more
• Attractive &
• Concentration or density of frequency can be known from the

Demerits :

• It is not a numerical measure; therefore it is not definite as a mathematical
• Drawing of curve requires more time and

Q.2           What is the best method of measuring Dispersion. Write the formula for calculating combined S.D.

Ans.: Standard Deviation is the best method of measuring dispersion as deviations taken from mean and algebraic signs are not ignored and it is algebraically correct.

Formula for calculating combined Standard Deviation :

σ12              =          N1 (σ12 + D12) + N2 (σ22 + D22) + - - - - - - - - - -

N1 + N2

Where –

N1 & N2 à No. of items of two series σ1 & σ 2 à S.D. of the two series

D1 à X1 – X12 D2 à X2 – X12

X12 à Combined arithmetic mean

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