This document discusses various measures of dispersion, which describe how spread out or varied the values in a data set are. It describes absolute measures like range and relative measures like coefficient of range. It also discusses interquartile range, mean deviation, standard deviation, and the Lorenz curve. Quartile deviation and mean deviation are simple measures but are less accurate and reliable. Standard deviation is a more certain measure but gives more importance to extreme values. The Lorenz curve measures deviation from equal distribution for variables like income, wealth, wages and more.

1. Chapter – 11
Measures of Dispersion

3. ABSOLUTE AND RELATIVE MEASURES OF
DISPERSION
When dispersion of
the series is
expressed in terms
of the original unit of
the series, it is called
absolute measure of
dispersion.
The relative measure
of dispersion
expresses the
variability of data in
terms of some
relative value or
percentage.

4.  Range
R = Upper Limit of the Last Class Interval –
Lower Limit of the First Class Interval
Or
R = H – L
Where R=Range; H=Highest value in the series;
L=Lowest value in the series
 Coefficient of range
CR = H – L
H + L
Where CR=Coefficient of range; H=Highest value in
the series; L=Lowest value in the series

5. • Inter Quartile Range = Q3 – Q1
• Quartile Deviation = Q3 – Q1
2
 Also called Semi-inter Quartile Range
• Coefficient of Quartile Deviation
Coefficient of QD=
Q3 – Q1÷ Q3 + Q1 = Q3 – Q1
2 2 Q3 + Q1

6. QUARTILE DEVIATION
Merits
• Simple
• Less effect of
Extreme Values
Demerits
• Not based on all
Values
• Formation of
Series not known
• Instability

7. • Mean deviation is the arithmetic
average of deviations of all the values
consideration taken from a statistical
average (mean, median or mode) of
series.
• In taking deviation of values,
algebraic signs + and – are not taken
into consideration, that is negative
deviations are also treated as positive
deviations

8. – If deviations are taken from median
• MDm = ∑ | X – M | or ∑ | dm |
N N
– If deviations are taken from arithmetic
average
• MDx = ∑ | X – X | or ∑ | d x |
N N
Where, MD = Mean deviation ; X – M =
Deviation from the median ; X - X =
deviation from the arithmetic average ; N =
Number of items

9. • Coefficient of MD from Mean = MDx
X
= Mean Deviation
Arithmetic Mean
• Coefficient of MD from Median = MDm
M
= Mean Deviation
Median
• Coefficient of MD from Mode = MDz
Z
= Mean Deviation
Mode

10. MEAN DEVIATION
Merits
• Simple
• Based on all
Values
• Less Effect of
Extreme Values
Demerits
• Inaccuracy
• Not Capable of
Algebraic
Treatment
• Unreliable

11. • Standard Deviation is the Square root of the
Arithmetic Mean of the squares of deviations
of the items from their mean value. This is
generally denoted by (sigma) of the Greek
language.
• COEFFICIENT OF STANDARD DEVIATION = σ
X

12. Merits
• Based on all Values
• Certain Measure
• Little Effect of a Change in Sample
• Algebraic Treatment
Demerits
• Difficult
• More Importance to Extreme Value

13. • Lorenz Curve is a measure of deviation of actual
distribution from the line of equal distribution.
• Lorenz curve as a measure of dispersion is
presently applied to the following parameters,
viz.,
1. Distribution of income
2. Distribution of wealth
3. Distribution of wages
4. Distribution of profits
5. Distribution of production
6. Distribution of population

14. • By ‘Less than’ or‘More than’ ogives methoda frequency distribution series is
first converted into a less than ormorethan cumulative series as in the cas
e of ogives, data arepresented graphically to make a ‘less than’ or ‘more
than’ ogive, N/2 item of the series is determined and from this point (on
the y-axis of the graph)a perpendicular is drawn
to the right to cut the cumulative frequency curve.The medianvalue is the o
newhere cumulative frequency curvecuts corresponding to x-axis.
• Less than and more than ogive curve method present the data graphical
ly in the form of ‘less than’ and ‘more than’
ogives simultaneously. The two ogives aresuperimposed upon eachother t
o determine the median value. Mark the point where the ogive curvecut
each other, draw a perpendicular from that point on x-
axis, the corresponding value on the x-axis would be the median value.

15. • Prepare a histogram from the data.
• Find out the rectangle whose height is the highest.
This will be the modal class.
• Draw two lines -
one joining the top right point of the rectangle preceding the modal class
with top right point of the modal class.
• The other joining the top left point of the modal
classwith the top left point of the post modal class. From the point of i
ntersection of these two diagonal lines, drawa perpendicular on horizo
ntal axisi.e. xaxisthe point where this perpendicular linemeets x-
axis, gives us the value of mode.