Transcript D_Statistics _chapter_5

©. S. Bhattacharjee
Ch 5_1
©. S. Bhattacharjee
Ch 5_2
What is meant by variability?
Variability refers to the extent to which the
observations vary from one another from some
average. A measure of variation is designed to state
the extent to which the individual measures differ on
an average from the mean.
Continued…..
©. S. Bhattacharjee
Ch 5_3
What are the purposes of measuring
variation ?
Measures of variation are needed for four basic
purposes:
To determine the reliability of an average;
To serve as a basis for the control of the variability;
To compare two or more series with regard to their
variability;
To facilitate the use of other statistical measures
©. S. Bhattacharjee
Ch 5_4
What are the properties of a good
measure of variation ?
A good measure of variation should possess the
following properties:
It should be simple to understand.
It should be easy to compute.
It should be rigidly defined.
It should be based on each and every observation of
the distribution.
It should be amenable to further algebraic treatment.
It should have sampling stability.
It should not be unduly affected by extreme
observations.
©. S. Bhattacharjee
Ch 5_5
What are the methods of studying variation ?
The following are the important methods of studying
variation:
The Range
The Interquartile Range or Quartile Deviation.
The Average Deviation
The Standard Deviation
The Lorenz Curve.
Of these, the first four are mathematical and the last
is a graphical one.
©. S. Bhattacharjee
Ch 5_6
What is meant by range ?
The range is defined as the distance between the
highest and lowest scores in a distribution.
It may also be defined as the difference between the
value of the smallest observation and the value of the
largest observation included in the distribution.
©. S. Bhattacharjee
Ch 5_7
What are the usages of range ?
Despite serious limitations range is useful in the
following cases:
Quality control: Range helps check quality of a
product. The object of quality control is to keep a
check on the quality of the product without 100%
inspection.
Fluctuation in the share prices: Range is useful in
studying the variations in the prices of stocks and
shares and other commodities etc.
Weather forecasts: The meteorological department
does make use of the range in determining the
difference between the minimum temperature and
maximum temperature.
©. S. Bhattacharjee
Ch 5_8
What are the merits of range ?
Merits:
Among all the methods of studying variation, range
is the simplest to understand and the easiest to
compute.
It takes minimum time to calculate the value of
range. Hence, if one is interested in getting a quick
rather than a very accurate picture of variability, one
may compute range.
©. S. Bhattacharjee
Ch 5_9
What are the limitations of range ?
Limitations:
Range is not based on each and every observation
of the distribution.
It is subject to fluctuations of considerable
magnitude from sample.
Range cannot be computed in case of open-end
distributions.
Range cannot tell anything about the character of
the distribution within two extreme observations.
©. S. Bhattacharjee
Ch 5_10
Example: Observe the following three series
Series A: 6,
Series B: 6
46
6
46
6
46
6
46
46
46
46
46
46
46
46
Series C: 6
10
15
25
30
32
40
46
In all the three series range is the same (i.e., 46-6=40),
but it does not mean that the distributions are alike. The
range takes no account on the form of the distribution
within the range. Range is, therefore, most unreliable as
a guide to the variation of the values within a
distribution.
©. S. Bhattacharjee
Ch 5_11
What is meant by inter-quartile range or
deviation?
Inter-quartile range represents the difference between
the third quartile and the first quartile. In measuring
inter-quartile range the variation of extreme
observations is discarded.
Continued…..
©. S. Bhattacharjee
Ch 5_12
What is inter-quartile range or deviation
measured ?
One quartile of the observations at the lower end and
another quartile of the observations at the upper end of
the distribution are excluded in computing the interquartile range. In other words, inter-quartile range
represents the difference between the third quartile
and the first quartile. Symbolically,
Inerquartile range = Q3 – Q1
Very often the interquartile range is reduced to the
form of the semi-interquartile range or quartile
deviation by dividing it by 2.
©. S. Bhattacharjee
Ch 5_13
The formula for computing inter-quartile deviation is
stated as under:
Q.D. 
Q  Q1
3
2
Q.D. = Quartile deviation
Quartile deviation gives the average amount by which the
two Quartiles differ from the median. In asymmetrical
distribution, the two quartiles (Q1 and Q3 ) are equidistant
from the median, i.e., Median ± Q.D. covers exactly 50 per
cent of the observations.
©. S. Bhattacharjee
Ch 5_14
When quartile deviation is very small it describes high
uniformity or small variation of the central 50%
observations, and a high quartile deviation means that
the variation among the central observations is large.
Quartile deviation is an absolute measure of variation.
The relative measure corresponding to this measure,
called the coefficient of quartile deviation, is
calculated as follows:
Co-efficient of Quartile deviation

Q  Q1
3
Q3

Q1
Coefficient of quartile deviation can be used to
compare the degree of variation in different
distributions.
©. S. Bhattacharjee
Ch 5_15
How is quartile deviation computed?
The process of computing quartile deviation is very
simple. It is computed based on the values of the upper
and lower quartiles. The following illustration would
clarify the procedure.
Example:
You are given the frequency distribution of 292 workers
of a factory according to their average weekly income.
Calculate quartile deviation and its coefficient from the
following data:
Continued…………
©. S. Bhattacharjee
Ch 5_16
Example:
Weekly Income
No. of workers
Below 1350
8
1350-1370
16
1370-1390
39
1390-1410
58
1410-1430
60
1430-1450
40
1450-1470
22
1470-1490
15
1490-1510
15
1510-1530
9
1530 & above
10
Continued…………
©. S. Bhattacharjee
Ch 5_17
Example:
Calculation of Quartile deviation
Weekly Income
No. of workers
c.f.
Below 1350
8
8
1350-1370
16
24
1370-1390
39
63
1390-1410
58
121
1410-1430
60
181
1430-1450
40
221
1450-1470
22
243
1470-1490
15
258
1490-1510
15
273
1510-1530
9
282
1530 & above
10
292
N = 292
©. S. Bhattacharjee
Continued…………
Ch 5_18
Example:
N
292
 146 th observatio n
Median = Size of th observatio n 
2
2
Median lies in the class 1410 - 1430
N
 p.c. f
Medain  L  2
i
f
146  121
 1410
 20
60
 1410 8  333  1418.333
N
292
Q1  Size of th observation 
 73rd observation
4
4
Q1 lies in the class 1390 1410.
©. S. Bhattacharjee
Continued…………
Ch 5_19
Example:
N
 p.c. f .
Q1  L  4
i
f
73  63
 1390
 20  1390 3  448  1393 448
58
3N
Q3  Size of
th observation
4
3  292

 219 th observation Q3 lies in the class 1430 1449
4
3N
 p.c. f .
219  181
Q3  L  4
 i  1430
 20  1430 19  1449
f
40
Q  Q1 1449 1393 448 55  552
55552
Coeffiecnt of Q.D.  3



 0  020.
Q3  Q1 1449 1393 448 2842 448 2842 448
©. S. Bhattacharjee
Ch 5_20
What are the merits of quartile deviation?
Merits:
In certain respects it is superior to range as a
measure of variation
It has a special utility in measuring variation in case
of open-end distributions or one in which the data
may be ranked but measured quantitatively.
It is also useful in erratic or highly skewed
distributions, where the other measures of variation
would be warped by extreme value.
The quartile deviation is not affected by the presence
of extreme values.
©. S. Bhattacharjee
Ch 5_21
What are the limitations of quartile
deviation?
Limitations:
Quartile deviation ignores 50% items, i.e., the first 25%
and the last 25%. As the value of quartile deviation does
not depend upon every observation it cannot be regarded
as a good method of measuring variation.
It is not capable of mathematical manipulation.
Its value is very much affected by sampling fluctuations.
It is in fact not a measure of variation as it really does not
show the scatter around an average but rather a distance
on a scale, i.e., quartile deviation is not itself measured
from an average, but it is a positional average.
©. S. Bhattacharjee
Ch 5_22
What is average deviation?
Average deviation refers to the average of the absolute
deviations of the scores around the mean.
It is obtained by calculating the absolute deviations of
each observation from median ( or mean), and then
averaging these deviations by taking their arithmetic
mean.
How is it calculated?
Continued…….
©. S. Bhattacharjee
Ch 5_23
Ungrouped data
The formula for average deviation may be written as:
A.D.( Med.)
X


 Med
N
If the distribution is symmetrical the average (mean or
median) ± average deviation is the range that will
include 57.5 per cent of the observation in the series. If
it is moderately skewed, then we may expect
approximately 57.5 per cent of the observations to fall
within this range. Hence if average deviation is small,
the distribution is highly compact or uniform, since
more than half of the cases are concentrated within a
small range around the mean.
©. S. Bhattacharjee
Ch 5_24
Ungrouped data
The relative measure corresponding to the average
deviation, called the coefficient of average deviation, is
obtained, by dividing average deviation by the
particular average used in computing average
deviation. Thus, if average deviation has been
computed from median, the coefficient of average
deviation shall be obtained by dividing average
deviation by the median.
A.D.
Coefficien t of A.D.Med. 
Median
If mean has been used while calculating the value of
average deviation, in such a case coefficient of average
deviation is obtained by dividing average deviation by
the mean.
©. S. Bhattacharjee
Ch 5_25
Example:
Calculate the average deviation and coefficient of average
deviation of the two income groups of five and seven
workers working in two different branches of a firm:
Branch 1
Income (Tk)
4,000
4,200
Branch II
Income (Tk)
3,000
4,000
4,400
4,600
4,200
4,400
4,600
4,800
4,800
5,800
©. S. Bhattacharjee
Continued…..
Ch 5_26
Calculation of Average deviation
Branch 1
│X- Med
Income (Tk)
Branch II
Med.=4,400
Income (Tk)
│X- Med
Med.= 4,400
4,000
400
3,000
1,400
4,200
200
4,000
400
4,400
0
4,200
200
4,600
200
4,400
0
4,800
400
4,600
200
4,800
400
5,800
1,400
N= 5
│X- Med =1,200
N=7
│X- Med = 4000
Continued…..
©. S. Bhattacharjee
Ch 5_27
Brach I:
X  Med. 1200

A.D. 

 240
N
5
A.D.
240
Coeff . of A.D. 

 0  054
Median 4,400
X  Med. 4,000

A.D. 

 571 43
Brach II:
N
7
57143
coeff . of A.D. 
 0  13
4,400
©. S. Bhattacharjee
Ch 5_28
Grouped data
In case of grouped data, the formula for calculating
average deviation is :
A.D.( Med.)
f


X  Med
N
Continued………..
©. S. Bhattacharjee
Ch 5_29
Example:
Calculation of Average Deviation from mean from the
following data:
Sales
(in thousand Tk)
No. of days
10 – 20
3
20 – 30
6
30 – 40
11
40 – 50
3
50 – 60
2
Continued……..
©. S. Bhattacharjee
Ch 5_30
Calculation of Average deviation
Sales
(in thousand
Tk)
m.p
X
f
X
 35 
10
fd
XX
f XX
(=d)
10 – 20
15
3
–2
–6
18
54
20 – 30
25
6
–1
–6
8
48
30 – 40
35
11
0
0
2
22
40 – 50
45
3
+1
+3
12
36
50 – 60
55
2
+2
+4
22
44
N = 25
fd = –5
f X  X
= 204
Continued……
©. S. Bhattacharjee
Ch 5_31
f

A.D. 
X  A
XX
N
 fd
5
 i  35 
 10  35  2  33
25
N
204
A.D. 
 8  16
25
Thus the average sales are Tk. 33 thousand per day
and the average deviation of sales is Tk. 8.16
thousand.
©. S. Bhattacharjee
Ch 5_32
What are the areas suitable for use of
average deviation?
It is especially effective in reports presented to the
general public or to groups not familiar with statistical
methods.
This measure is useful for small samples with no
elaborate analysis required.
Research has found in its work on forecasting business
cycles, that the average deviation is the most practical
measure of variation to use for this purpose.
©. S. Bhattacharjee
Ch 5_33
What are the merits of average deviation?
Merits:
The outstanding advantage of the average deviation is
its relative simplicity. It is simple to understand and
easy to compute.
Any one familiar with the concept of the average can
readily appreciate the meaning of the average
deviation.
It is based on each and every observation of the data.
Consequently change in the value of any observation
would change the value of average deviation.
©. S. Bhattacharjee
Ch 5_34
What are the merits of average deviation?
Merits:
Average deviation is less affected by the values of
extremes observation.
Since deviations are taken from a central value,
comparison about formation of different distributions
can easily be made.
©. S. Bhattacharjee
Ch 5_35
What are the limitations of average deviation?
Limitations:
The greatest drawback of this method is that
algebraic signs are ignored while taking the
deviations of the items. If the signs of the deviations
are not ignored, the net sum of the deviations will be
zero if the reference point is the mean, or
approximately zero if the reference point is median.
The method may not give us very accurate results.
The reason is that average deviation gives us best
results when deviations are taken from median. But
median is not a satisfactory measure when the degree
of variability in a series is very high.
©. S. Bhattacharjee
Continued…….
Ch 5_36
What are the limitations of average deviation?
Limitations:
Compute average deviation from mean is also not
desirable because the sum of the deviations from
mean ( ignoring signs) is greater than the sum of the
deviations from median (ignoring signs).
If average deviation is computed from mode that also
does not solve the problem because the value of
mode cannot always be determined.
It is not capable of further algebraic treatment.
It is rarely used in sociological and business studies.
©. S. Bhattacharjee
Continued…….
Ch 5_37
What is meant by Standard Deviation?
Standard deviation is the square root of the squared
deviations of the scores around the mean divided by
N. S represents standard deviation of a sample; ∂, the
standard deviation of a population.
Standard deviation is also known as root mean square
deviation for the reason that it is the square root of the
means of square deviations from the arithmetic mean.
The formula for measuring standard deviation is as
follows :
 X  X 
2

©. S. Bhattacharjee
N
Ch 5_38
What is meant by Variance?
This refers to the squared deviations of the scores
around the mean divided by N. A measure of
dispersion is used primarily in inferential statistics and
also in correlation and regression techniques; S2
represents the variance of a sample ; ∂2 , the variance
of a population.
If we square standard deviation, we get what is called
Variance.
HenceVariance  or   Variance
2
©. S. Bhattacharjee
Ch 5_39
How is standard deviation calculated?
Ungrouped data
Standard deviation may be computed by applying
any of the following two methods:
By taking deviations from the actual mean
By taking deviations from an assumed mean
Continued……..
©. S. Bhattacharjee
Ch 5_40
How is standard deviation calculated?
Ungrouped data
By taking deviations from the actual mean:
When deviations are taken from the actual mean,
the following formula is applied:
 X  X 
2

N
If we calculate standard deviation without taking
deviations, the above formula after simplification
(opening the brackets) can be used and is given by:
Continued……..
©. S. Bhattacharjee
Ch 5_41
Formula:

X
N
2
X 


 N 


2
or  
2
X

N
 X 
2
By taking deviations from an assumed mean: When
the actual mean is in fractions, say 87.297, it would
be too cumbersome to take deviations from it and
then find squares of these deviations. In such a
case either the mean may be approximated or else
the deviations be taken from an assumed mean and
the necessary adjustment be made in the value of
standard deviation.
©. S. Bhattacharjee
Ch 5_42
How is standard deviation calculated ?
The former method of approximation is less accurate
and therefore, invariably in such a case deviations are
taken from assumed mean.
When deviations are taken from assumed mean the
following formula is applied:

Where
©. S. Bhattacharjee
d
 d 


N  N 
2
2
d   X  A
Ch 5_43
Example:
Find the standard deviation from the weekly wages of ten
workers working in a factory:
Workers
Weekly wages (Tk)
A
1320
B
1310
C
1315
D
1322
E
1326
F
1340
G
1325
H
1321
I
1320
j
1331
©. S. Bhattacharjee
Ch 5_44
Calculations of Standard Deviation
X  X 
X  X 
Workers
Weekly wages
(Tk)
A
1320
-3
9
B
1310
- 13
169
C
1315
-8
64
D
1322
-1
1
E
1326
+3
9
F
1340
+17
289
G
1325
+2
4
H
1321
-2
4
I
1320
-3
9
J
1331
+8
64
N= 10
x=13230
©. S. Bhattacharjee
 X  X = 0
Continued…….
2

 X X
2
= 622
Ch 5_45
 X  X 
2
We know
 
Since
N

...............(i )
dXA
X  Ad
X  Ad
Subtracting X from X , we get
X  X   d  d 
Continued…….
©. S. Bhattacharjee
Ch 5_46
Substituting the value of
X  X 
 d  d 
2

N
X

X
N
 

2


X

X

N
in (i), mentioned
d
2
N
 d 


 N 


2
13230

 Tk.1323
10

622
 62.2
10
 7  89
If, in the above question, deviations are taken from 1320
instead of the actual mean 1323, the assumed mean method
will be applied and the calculations would be as follows:
©. S. Bhattacharjee
Continued…….
Ch 5_47
Calculation of standard deviation (assumed mean method)
Workers
Weekly wages
(Tk)
 X  A  d
d2
A = 1320
A
1320
0
0
B
1310
-10
100
C
1315
-5
25
D
1322
+2
4
E
1326
+6
36
F
1340
+20
400
G
1325
+5
25
H
1321
+1
1
I
1320
0
0
j
1331
+11
121
d=30
 d2 =712
N= 10
©. S. Bhattacharjee
Continued…….
Ch 5_48

d
N
2
 d 


 N 


2
2
712  30 

    71 2  9  62  2  7  89
10  10 
Thus the answer remains the same by both the
methods. It should be noted that when actual mean is
not a whole number, the assumed mean method should
be preferred because it simplifies calculations.
©. S. Bhattacharjee
Ch 5_49
Grouped data
In grouped frequency distribution, standard deviation
can be calculated by applying any of the following two
methods:
By taking deviations from actual mean.
By taking deviations from assumed mean.
©. S. Bhattacharjee
Continued……..
Ch 5_50
Grouped data
Deviations taken from actual mean: When deviations
are taken from actual mean, the following formula is
used:
 f X  X 
2

N
If we calculate standard deviation without taking
deviations, then this formula after simplification
(opening the brackets ) can be used and is given by

 fX
©. S. Bhattacharjee
N
2
  fX

 N





2
or  
Continued……..
2
fX

N
 X 
2
Ch 5_51
Grouped data
Deviations taken from assumed mean: When
deviations are taken from assumed mean, the
following formula is applied :

 fd
N
xA
d
i
©. S. Bhattacharjee
2
  fd 
  i, where

 N 


2
Continued……..
Ch 5_52
Example:
A purchasing agent obtained samples of 60 watt bulbs
from two companies. He had the samples tested in his
own laboratory for length of life with the following
results:
Length of life (in hours)
1,700 and
1,900 and
2,100 and
2,300 and
2,500 and
under
under
under
under
under
©. S. Bhattacharjee
1,900
2,100
2,300
2,500
2,700
Samples from
Company A
Company B
10
3
16
20
8
40
12
3
6
2
Continued……..
Ch 5_53
Example:
1. Which Company’s bulbs do you think are better in
terms of average life?
2. If prices of both types are the same, which company’s
bulbs would you buy and why?
©. S. Bhattacharjee
Continued……..
Ch 5_54
Example:
Length of life
(in hours)
Sample from Co. A
Midpoint
Samples from Co. B
f
d
fd
fd2
f
d
fd
fd2
1,700– 1,900
1800
10
–2
–20
40
3
–2
–6
12
1,900–2,100
2000
16
–1
–16
16
40
–1
– 40
40
2,100–2,300
2200
20
0
0
0
12
0
0
0
2,300–2,500
2400
8
1
+8
8
3
1
+3
3
2,500–2,700
2600
6
2
+12
24
2
2
+4
8
N=60
d=0
fd
= –16
fd2
=88
N=60
d=0
fd
= –39
fd2
= 63
d
XA
, where  Assumed mean
i
Here, A = 2200
i = 200
©. S. Bhattacharjee
Continued……..
Ch 5_55
Example:
Here, N = 60
A = 2,200
For Company A :
fd = - 16
fd2 = 88
fd
16

 200  2,146 67
 i  2,200 
X  A
60
N
 

fd
N
2
  fd 
 i 

 N 


2
2
88   16 

  200
60  60 
 1  467  0  071  200  1  182 200  236 4

236 4
 100  11 per cent.
C.V .   100 
2146 67
X
©. S. Bhattacharjee
Continued……..
Ch 5_56
For Company B :
 fd  39
 fd  63
2
A  2200
39
X  2200  200  2200 130  2070
60
2
63   39 


  200
60  60 
 1  05  42  200
 794 200  158 8
158 8
C.V . 
 100  7  67 per cent.
2070
©. S. Bhattacharjee
Continued……..
Ch 5_57
Illustration :18
You are given the data pertaining to kilowatt hours
electricity consumed by 100 persons in Deli.
Consumption (K. Wait hours)
No. of users
0 but less than
10
6
10 but less than
20
25
20 but less than
30
36
30 but less than
40
20
40 but less than
50
13
Calculate the mean and the standard deviation.
©. S. Bhattacharjee
Continued……..
Ch 5_58
Solution:
Calculation of Mean and standard Deviation (Taking
deviation from assumed mean)
Consumption
K. wait hours
m.p
(X)
No. of Users
(f)
 X  25 
fd2
fd
c.f.
10
d
0–10
5
6
–2
–12
24
6
10–20
15
25
–1
–25
25
31
20–30
25
36
0
0
0
67
30–40
35
20
+1
+20
20
87
40–50
45
13
+2
+26
52
100
N=100
©. S. Bhattacharjee
fd =9
Continued……..
fd2=121
Ch 5_59
(i)
(ii)
fd

X  A
 i  25 
N

 fd
N
2
9
 10  25.9k. wait hours
100
2
  fd 
121  9 



i 
    10
 N 
100  100


2
 1.21  .008  10  1.096 10  10.96
©. S. Bhattacharjee
Ch 5_60
Calculation of Mean and the Standard Deviation
(Taking deviation from assumed mean)
Consumption Midpoi
K. wait hours
nt
X
No. of
Users
f
fX
X  X  X  X 
2
f X  X 
2
0–10
5
6
30
–2
436.81
2620.6
10–20
15
25
375
–10.9
118.81
2970.25
20–30
25
36
900
–0.9
0.81
29.16
30–40
35
20
700
9.1
82.81
1656.20
40–50
45
13
585
19.1
364.81
4742.53
N=100
fX=
2590
©. S. Bhattacharjee
Continued……..
 f X  X 
2
 12019
Ch 5_61
X
fX


N
2590
100
 25.9

f X  X 
N
2
  

12019
100
 120.19
 10.96
©. S. Bhattacharjee
Ch 5_62
1. Since average length of life is greater in case of
company A, hence bulbs of company A are
better.
2. Coefficient of variation is less for company B.
Hence if prices are same, we will prefer to buy
company B’s bulbs because their burning hours
are more uniform.
©. S. Bhattacharjee
Ch 5_63
Example:
For two firms A and B belonging to same industry, the
following details are available:
Firm B
Firm A
Number of Employees
100
200
Average monthly wage:
Tk. 4,800
Tk. 5,100
Standard deviation:
Tk. 600
Tk. 540
Find
i. Which firm pays out larger amount as wages?
ii. Which firm shows greater variability in the distribution of
wages?
iii. Find average monthly wage and the standard deviation
of the wages of all employees in both the firms.
©. S. Bhattacharjee
Ch 5_64
i. For finding out which firm pays larger amount, we
have to find out X.
X

X
or  X  NX
N
Firm A : N = 100,
X
= 4800,  X=100×4800 =4,80,000
Firm B : N = 200,
X
= 5100,  X=200×5100 =10,20,000
Hence firm B pays larger amount as monthly wages.
©. S. Bhattacharjee
Ch 5_65
ii. For finding out which firm pays greater variability in
the distribution of wages, we have to calculate
coefficient of variation.

Firm A :
600
C.V .   100 
 100  12  50
X
4800
Firm B :

540
C.V . 
 100 
 100  10  59
X
5100
Since coefficient of variation is greater in case of
firm A, hence it shows greater variability in the
distribution of wages.
©. S. Bhattacharjee
Ch 5_66
iii. Combined average weekly wage:
N1 X 1  N 2 X 2
X 12 
N1  N 2
N1 = 100, X 1 = 4800, N 2 = 200, X 2 = 5100,
X 12

1004800  2005100

100  200
480000 1020000

 TK .5,000
300
©. S. Bhattacharjee
Ch 5_67
Combined Standard Deviation
 12 
N1
1
2
 N2
2
2
 N1 d 2  N 2
1
d2
2
N1  N 2
N1  100,  1  600, N 2  200,  2  540
d1== X 1
d2 =
X2


X 12 = 4800 - 5000 =200
X 12
= 5100 - 5000 =100
Continued…….
©. S. Bhattacharjee
Ch 5_68
 12
100600  200 540  100 200  200 100

100  200
2
2
2
2
36000000 5832000 4000000 2000000

300
100 320,000

 578.25
300
Hence the combined standard deviation is Tk. 578.25.
©. S. Bhattacharjee
Ch 5_69
Which measure of variation to use?
The choice of a suitable measure depends on the
following two factors:
The type of data available
The purpose of investigation
©. S. Bhattacharjee
Ch 5_70
What is Lorenz curve?
It is cumulative percentage curve in which the
percentage of items is combined with the percentage of
other things as wealth, profit, turnover, etc. The Lorenz
curve is a graphic method of studying variation.
©. S. Bhattacharjee
Ch 5_71
What is the procedure of drawing the
Lorenz curve?
While drawing the Lorenz curve the following
procedure is used:
The size of items and frequencies are both
cumulated and the percentages are obtained for the
various commutative values.
On the X-axis, start from 0 to 100 and take the per
cent of variable.
On the Y-axis, start from 0 to 100 and take the per
cent of variable.
Continued…..
©. S. Bhattacharjee
Ch 5_72
What is the procedure of drawing the
Lorenz curve?
Draw a diagonal line joining 0 with 100. This is
known as line of equal distribution. Any point on this
line shows the same per cent on X as on Y.
Plot the various points corresponding to X and Y
and join them. The distribution so obtained, unless it
is exactly equal, will always curve below the diagonal
line.
©. S. Bhattacharjee
Ch 5_73
How is interpretation of the Lorenz
curve done?
If two curves of distribution are shown on the Lorenz
presentation, the curve that is farthest from the
diagonal line represents the greater inequality. Clearly
the line of actual distribution can never cross the line of
equal distribution.
©. S. Bhattacharjee
Ch 5_74
Example:
In the following table is given the number of companies
belonging to two areas A and B according to the amount of
profits earned by them. Draw in the same diagram their
Lorenz curves and interpret them.
Profits earned in Tk.'000
No. of companies
Area A
Area B
6
6
2
25
11
38
60
13
52
84
14
28
105
15
38
150
17
26
170
10
12
400
14
4
©. S. Bhattacharjee
Continued……
Ch 5_75
Solution:
Calculation for drawing the Lorenz curve
Profit
Profits
earned in
Tk. ‘000
Cumulativ
e
Profits
Cumulativ
e
Percentag
e
No. of
Companies
Cumulative
Number
Cumulative
Percentage
No. of
Companies
Cumulative
Number
Cumulative
Percentage
6
6
0.6
6
6
6
2
2
1
25
31
3.1
11
17
17
38
40
20
60
91
9.1
13
30
30
52
92
46
84
175
17.5
14
44
44
28
120
60
105
280
28.0
15
59
59
38
158
79
150
430
43.0
17
76
76
26
184
92
170
600
60.0
10
86
86
12
196
98
400
1000
100.0
14
100
100
4
200
100
©. S. Bhattacharjee
Continued……..
Ch 5_76
Lorenz curve
120.0
100.0
Per Cent of Profits
80.0
60.0
40.0
20.0
Per Cent of Companies
©. S. Bhattacharjee
100
98
92
79
60
46
20
1
0.0
Ch 5_77
©. S. Bhattacharjee
Ch 5_78