Download Measures of Dispersion

Measures of Dispersion
Introduction
Properties of good measure of variation
Methods of Variation
Meaning of Dispersion
Dispersion is the measure of the variation
of the items.
Dispersion is the extent to which values in
a distribution differ from the average of
the distribution.
Dispersion is the scatteredness of the data
series around it average.
Example
Series A
100
100
100
100
100
Series B
100
105
103
90
102
Series C
1
489
2
3
5
Significance of measuring
dispersion
Determine the
reliability of an
average
Serve as a
basis for the
control of the
variability
To compare
the variability
of two or more
series and
Facilitate the
use of other
statistical
measures.
Properties of a good measure of
variation
It should be rigidly defined.
It should be easy to understand and easy to calculate.
It should be based on all the observations of the data.
It should be easily subjected to further mathematical
treatment.
It should be least affected by the sampling fluctuation .
It should not be unduly affected by the extreme values.
Lorenz curve
Standard
Deviation
Mean
Deviation
Interquartile
Range &
Quartile
Deviation
Range
Methods of
measuring dispersion
Absolute & Relative measures
of variation
Absolute
Relative
• Measure the dispersion in the original unit of
the data.
• Variability in two or more distribution can be
compared provided they are given in the
same unit.
• Measure of dispersion is free from unit of
measurement of data.
• It is the ratio of a measure of absolute
dispersion to the average, from which
absolute deviations are measured.
• It is called co-efficient of dispersion.
Range
Range is defined as the difference between the
value of smallest item and value of largest item
in the distribution.
Range = Highest value of an observation (H) –
Smallest value of an observation (L)
Coefficient of range = (H-L)/ (H+L)
Quality control
Daily Life
Uses of
Range
Weather
Forecasting
Fluctuations in
the share prices
Advantages of Range
• Easy to understand and compute
• Gives a quick idea about variation
• It is rigidly defined
Disadvantages of Range
• It is not based on each and every
observation of the distribution
• It can not be computed in case of
open-ended series
• It can not be put to further
mathematical treatment
• Affected by extreme values
Interquartile Range
Q3-Q1
Quartile Deviation
(Q3-Q1)/2
Coefficient of Quartile Deviation
Q3-Q1
Q3-Q1
Mean Deviation
The average of difference of the values of items
from some average of the series (ignoring
negative sign), i.e. the arithmetic mean of the
absolute differences of the values from their
average .
The average used is either the arithmetic mean
or median
Calculation of mean deviation
Mean Deviation
 Individual series
 Discrete series
 Continuous series
Coefficient of mean deviation
(M. D.)
Advantages of Mean deviation
• It is based on each and every item of
distribution
• It is less affected by extreme values
Disadvantages of Mean deviation
• It is not capable of further algebraic
treatment
• It ignores signs and hence makes the
method non- algebraic
Numerical (Individual Series)
The wheat production (in Kg) of 20
acres is given as: 1120, 1240, 1320,
1040, 1080, 1200, 1440, 1360, 1680,
1730, 1785, 1342, 1960, 1880, 1755,
1720, 1600, 1470, 1750, and 1885.
Find the quartile deviation and
coefficient of quartile deviation.
Discrete Series
Marks
No. of
students
20
30
40
50
60
70
8
12
20
10
6
4
Practical
Age (in
Years)
No. of
students
4-6 6-8
30
40
8-10 10-12 12-14 14-16 16-18 18-20
50
60
40
30
20
10
Standard Deviation
Standard Deviation is the square
root of the mean of the squared
deviation from the arithmetic mean.
It is denoted by Greek letter σ (read
as sigma).
Standard Deviation
 Combined Standard Deviation
 S.D. of n natural no.
 sum of the squares of the deviations of
items from their AM is minimum.
 Variance & Coefficient of variation
 Merits
 Demerits
Calculation of standard
deviation in individual series
Direct
Method
Calculation of standard
deviation in individual series
Short cut Method
d=X-A
𝜎=
∑𝑑 2
∑𝑑
−
𝑁
𝑁
2
Practical
Income of members are given below:
4000, 4200, 4400, 4600, 4800
Calculate Standard Deviation
Calculation of standard deviation in group
data (discrete and continuous series)
Direct
Method
Calculation of standard deviation in group
data (discrete series)
Short cut Method
𝜎=
∑𝑓𝑑 2
𝑁
∑𝑓𝑑
−
𝑁
2
Practical
X
3.5
4.5
5.5
6.5
7.5
8.5
9.5
F
3
7
22
60
85
32
8
Calculation of standard deviation
in group data (continuous series)
Short cut Method
𝜎=
∑𝑓𝑑 2
𝑁
h = class interval
∑𝑓𝑑
−
𝑁
2
×ℎ
Mathematical Properties of
Standard Deviation
1.
Combined standard deviation
 µ1 = X1
 µ2 = X2
 µ = X12
Mathematical Properties of
Standard Deviation
2. Standard deviation of N natural
Numbers
𝜎=
1
𝑁2 − 1
12
3. Sum of the squares of the
deviations of items from their AM is
minimum.
4.Percentage of area measured in terms of
standard deviation & mean in normal curve
 Variance is square of standard
deviation
 Coefficient of Variation
=
𝜎
𝑀𝑒𝑎𝑛
× 100
Advantages of standard
deviation




Most widely used measure of dispersion.
Capable of mathematical treatment.
Does not ignore the algebraic signs.
Provides a unit of measurement for the
normal distribution.
Disadvantage of standard
deviation
 It gives more weight to extreme
values and less to those which are
near the mean
Relation between mean deviation & standard
deviation under normal distribution
Mean deviation =
4
𝜎
5
Lorenz Curve
Profits earned
Rs. in ’000
No. of companies
Area A
No. of companies
Area B
6
6
2
25
11
38
60
13
52
84
14
28
105
15
38