Quantitative Techniques

1. Course: MBA
Subject: Quantitative Techniques
Unit: 2.1

2. Ch 5_2

3. Ch 5_3
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…..

4. Ch 5_4
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

5. Ch 5_5
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.

6. Ch 5_6
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.

7. Ch 5_7
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.

8. Ch 5_8
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.

9. Ch 5_9
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.

10. Ch 5_10
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.

11. Ch 5_11
Example: Observe the following three series
Series A: 6, 46 46 46 46 46 46 46
Series B: 6 6 6 6 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.

12. Ch 5_12
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…..

13. Ch 5_13
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 inter-
quartile 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.

14. Ch 5_14
2
..
13
QQ
DQ


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.
The formula for computing inter-quartile deviation is
stated as under:

15. Ch 5_15
Co-efficient of Quartile deviation
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:
13
13
QQ
QQ



Coefficient of quartile deviation can be used to
compare the degree of variation in different
distributions.

16. Ch 5_16
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…………

17. Ch 5_17
Example:
Continued…………
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

18. Ch 5_18
Example:
Continued…………
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
Calculation of Quartile deviation

19. Ch 5_19
Example:
Continued…………
nobservatiothnobservatioth
N
146
2
292
2
Median = Size of
Median lies in the class 1410 - 1430
.14101390
73
4
292
4
333.141833381410
20
60
121146
1410
..
2
1
1









classtheinliesQ
nobservatiordnobservatioth
N
ofSizeQ
i
f
fcp
N
LMedain

20. Ch 5_20
Example:
.0200
4482842
55552
4482842
55255
44813931449
44813931449
..
144919143020
40
181219
1430
...
4
3
14491430219
4
2923
4
3
44813934483139020
58
6373
1390
...
4
13
13
3
3
3
1


























 QQ
QQ
DQofCoeffiecnt
i
f
fcp
N
LQ
classtheinliesQnobservatioth
nobservatioth
N
ofSizeQ
i
f
fcp
N
LQ

21. Ch 5_21
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.

22. Ch 5_22
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.
What are the limitations of quartile
deviation?

23. Ch 5_23
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…….

24. Ch 5_24
The formula for average deviation may be written as:
N
MedX
DA Med
 
.)(..
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.
Ungrouped data

25. Ch 5_25
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.
Ungrouped data
  Median
DA
DAoftCoefficien Med
..
.. . 
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.

26. Ch 5_26
Branch 1
Income (Tk)
Branch II
Income (Tk)
4,000 3,000
4,200 4,000
4,400 4,200
4,600 4,400
4,800 4,600
4,800
5,800
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:
Example:
Continued…..

27. Ch 5_27
Branch 1
│X- Med
Income (Tk) Med.=4,400
Branch II
│X- Med
Income (Tk) 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
Calculation of Average deviation
Continued…..

28. Ch 5_28
Brach I:
Brach II:
130
400,4
57143
...
43571
7
000,4.
..
0540
400,4
240..
...
240
5
1200.
..










DAofcoeff
N
MedX
DA
Median
DA
DAofCoeff
N
MedX
DA

29. Ch 5_29
Grouped data
In case of grouped data, the formula for calculating
average deviation is :
Continued………..
N
MedXf
DA Med
 
.)(..

30. Ch 5_30
Example:
Sales
(in thousand Tk)
No. of days
10 – 20 3
20 – 30 6
30 – 40 11
40 – 50 3
50 – 60 2
Continued……..
Calculation of Average Deviation from mean from the
following data:

31. Ch 5_31
Sales
(in thousand
Tk)
m.p
X
f
(=d)
fd
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
= 204
 
10
35X
XX  XX 
XX 
f
f
Calculation of Average deviation
Continued……

32. Ch 5_32
168
25
204
..
3323510
25
5
35
..






DA
i
N
fd
AX
N
XXf
DA
Thus the average sales are Tk. 33 thousand per day
and the average deviation of sales is Tk. 8.16
thousand.

33. Ch 5_33
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.

34. Ch 5_34
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.

35. Ch 5_35
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.

36. Ch 5_36
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.
Continued…….

37. Ch 5_37
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.
Continued…….

38. Ch 5_38
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 :
 
N
XX 

2


39. Ch 5_39
VarianceorVarianceHence   2
If we square standard deviation, we get what is called
Variance.
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.

40. Ch 5_40
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……..

41. Ch 5_41
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:
 
N
XX 

2

If we calculate standard deviation without taking
deviations, the above formula after simplification
(opening the brackets) can be used and is given by:
Continued……..

42. Ch 5_42
Formula:
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.
 2
22 2
X
N
X
or
N
X
N
X










 

43. Ch 5_43
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:
22










N
d
N
d

Where  AXd 

44. Ch 5_44
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

45. Ch 5_45
 2
XX  XX 
Calculations of Standard Deviation
Continued…….
 XX 
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  = 0  = 622 2
XX 

46. Ch 5_46
 
).....(..........
2
i
N
XX 
We know
Since
dAX
dAX
AXd



   ddXX 
getweXfromXgSubtractin ,
Continued…….

47. Ch 5_47
 XX Substituting the value of in (i), mentioned
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:
  222












N
d
N
d
N
dd

 
897
2.62
10
622
2





N
XX

1323.
10
13230
Tk
N
X
X 

Continued…….

48. Ch 5_48
Workers Weekly wages
(Tk) A = 1320
d2
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
N= 10 d=30  d2 =712
Calculation of standard deviation (assumed mean method)
Continued…….
  dAX 

49. Ch 5_49
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.
8972629271
10
30
10
712
222

















N
d
N
d


50. Ch 5_50
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.
Continued……..

51. Ch 5_51
Grouped data
Deviations taken from actual mean: When deviations
are taken from actual mean, the following formula is
used:
Continued……..
If we calculate standard deviation without taking
deviations, then this formula after simplification
(opening the brackets ) can be used and is given by
 2
222
X
N
fX
or
N
fX
N
fX










 
 
N
XXf 

2


52. Ch 5_52
Grouped data
Deviations taken from assumed mean: When
deviations are taken from assumed mean, the
following formula is applied :
Continued……..
,
22
i
N
fd
N
fd











i
Ax
d


where

53. Ch 5_53
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:
Example:
Length of life (in hours) Samples from
Company A Company B
1,700 and under 1,900 10 3
1,900 and under 2,100 16 40
2,100 and under 2,300 20 12
2,300 and under 2,500 8 3
2,500 and under 2,700 6 2
Continued……..

54. Ch 5_54
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?
Example:
Continued……..

55. Ch 5_55
Example:
Continued……..
Sample from Co. ALength of life
(in hours)
Midpoint Samples from Co. A
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=6
0
d=0 fd
= –39
fd2
= 63
Samples from Co. B
meanAssumedwhere
i
AX
d 

 ,
Here, A = 2200
i = 200

56. Ch 5_56
Example:
Continued……..
.11100
672146
4236
100..
4236200182120007104671
200
60
16
60
88
67146,2200
60
16
200,2
222
centper
X
VC
i
N
fd
N
fd
i
N
fd
AX










 















For Company A :
Here, N = 60
A = 2,200
fd = - 16
fd2 = 88

57. Ch 5_57
For Company B :
Continued……..
.677100
2070
8158
..
8158200794
20042051
200
60
39
60
63
20701302200200
60
39
2200
2
centperVC
X










 



2200
63
39
2





A
fd
fd

58. Ch 5_58
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
Illustration :18
You are given the data pertaining to kilowatt hours
electricity consumed by 100 persons in Deli.
Calculate the mean and the standard deviation.
Continued……..

59. Ch 5_59
Solution:
Calculation of Mean and standard Deviation (Taking
deviation from assumed mean)
 
10
25XConsumption
K. wait hours
m.p
(X)
No. of Users
(f)
d
fd fd2 c.f.
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 fd =9 fd2=121
Continued……..

60. Ch 5_60
hourswaitki
N
fd
AX .9.2510
100
9
25 

96.1010096.110008.21.1
10
100
9
100
121
222

















 i
N
fd
N
fd

(i)
(ii)

61. Ch 5_61
Calculation of Mean and the Standard Deviation
(Taking deviation from assumed mean)
 XX  2
XX   2
XXf 
   12019
2
XXf
Consumption
K. wait hours
Midpoi
nt
X
No. of
Users
f
fX
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
Continued……..

62. Ch 5_62
9.25
100
2590



N
fX
X
 
96.10
19.120
100
12019
2





N
XXf


63. Ch 5_63
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.

64. Ch 5_64
For two firms A and B belonging to same industry, the
following details are available:
Number of Employees
Average monthly wage:
Standard deviation:
Firm A
100
Tk. 4,800
Tk. 600
Firm B
200
Tk. 5,100
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.
Example:

65. Ch 5_65
i. For finding out which firm pays larger amount, we
have to find out X.
Firm A : N = 100,
Firm B : N = 200,
X
X
= 4800,  X=100×4800 =4,80,000
= 5100,  X=200×5100 =10,20,000
Hence firm B pays larger amount as monthly wages.

  XNXor
N
X
X

66. Ch 5_66
ii. For finding out which firm pays greater variability in
the distribution of wages, we have to calculate
coefficient of variation.
Since coefficient of variation is greater in case of
firm A, hence it shows greater variability in the
distribution of wages.
5012100
4800
600
100.. 
X
VC

Firm A :
5910100
5100
540
100.. 
X
VC
Firm B :

67. Ch 5_67
iii. Combined average weekly wage:
21
2211
12
NN
XNXN
X



= 4800,1N = 100, 1X 2N = 200,
2X = 5100,
     
000,5.
300
1020000480000
200100
51002004800100
12
TK
X







68. Ch 5_68
Combined Standard Deviation
540,200,600,100 2211   NN
d1= 1X 12X
d2 = = 5100 - 5000 =10012X
d1 = 1X = 4800 - 5000 =200
2X
Continued…….
21
2
2
2
2
1
1
2
2
2
2
1
1
12
NN
NNNN dd






69. Ch 5_69
       
25.578
300
000,320100
300
20000004000000583200036000000
200100
100200200100540200600100
2222
12








Hence the combined standard deviation is Tk. 578.25.

70. Ch 5_70
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

71. Ch 5_71
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.

72. Ch 5_72
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…..

73. Ch 5_73
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.

74. Ch 5_74
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.

75. Ch 5_75
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
Continued……

76. Ch 5_76
Continued……..
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
Calculation for drawing the Lorenz curve
Solution:

77. Ch 5_77
0.0
20.0
40.0
60.0
80.0
100.0
120.0
1
20
46
60
79
92
98
100
Lorenz curve
Per Cent of Companies
PerCentofProfits

78. References
Quantitative Techniques, by CR Kothari, Vikas publication
Fundamentals of Statistics by SC Guta Publisher Sultan
Chand
Quantitative Techniques in management by N.D. Vohra
Publisher: Tata Mcgraw hill