Measures of Dispersion

H tudUing this chapter should en‹zble you to: Three friends, Ram, Rahim and
know the timitntions ofauerages,- Maria are chatting over a cup of tea.
appreciate the need/or measures During the course of their
of dispersion,- conversation, they start talking about
enumerate uct ’ ousmeasures of their family incomes. Ram tells them
dispersion;
that there are four members in his
• catcitlnte the meas iires and family and the average income per
compare them; member is Rs 15,000. Rahim says
• distinguish between absolute
that the average income is the same
and relative measures.
in his family, though the number of
1. INTRODUCTION members is six. Maria says that there
In the previous chapter, you have are five members in her family, out of
studied howto sum up the data into a which one is not working. She
single representative value. However, calculates that the average income in
that value does not reveal the her family too, is Rs 15,000. They are
variability present in the data. In this a little surprised since they know that
chapter you will study those Maria’s father is earning a huge
measures, which seek to quantify salary. They go into details and gather
variability of the data. the following data:

2020-21
 MEASURES OF DISPERSION 75

in values, your understanding of a

distribution
St. No. improves MariaFor example, per capita income gives only
Tom considerably.
Raliim
the 1.average income.
12,000 A7,000
measure 0of dispersion can tell you about income
inequalities,
2. thereby
14,000 improving
10,000 the
7,000
understanding
3. of 16,000
the relative standards
14,000 8,000
6.
4. 18,000 22,000
17,000 10,000 of living enjoyed by different strata of
5. income
Totot 60, 000 20,000 50,000
90,000 TH,000
Average income 1 IN,000 1 IN,000 1 IN,000
society.
Dispersion is the extent to which
Do you notice that although the values in a distribution d er om the
average is the same, there are average of the distribution.
considerable differences in individual To quantify the extent of the
incomes? variation, there are certain measures
It is quite obvious that averages try namely:
to tell only one aspect of a distribution (i) Range
i.e. a representative size of the values. (ii) Quartile Deviation
To understand it better, you need to (iii) Mean Deviation
know the spread of values also. (iv)Standard Deviation
You can see that in Ram’s family,
differences in incomes are Apart from these measures
comparatively lower. In Rahim’s which give a numerical value,
family, differences are higher and in there is a graphic method for
Maria’s family, the differences are estimating dispersion.
the highest. Knowledge of only Range and quartile deviation
average is insufficient. If you have measure the dispersion by calculating
another value which reflects the the spread within which the values
quantum of variation lie. Mean deviation and standard
deviation calculate the extent to
which the values differ from the
average.

2. MRASURES BASED UPOm SPRRAD

Range
Range (R) is the difference between the
largest (L) and the smallest value (S) in
a distribution. Thus,
R=L—S
Higher value of range implies
higher dispersion and vice-versa.
76 STATISTICS FOR ECONOMICS

Acttotttes e uartile Deviation

Look at the following values: The presence of even one extremely
20, 30, 40, 50, 200 high or low value in a distribution
• Calculate the Range.
• What is the Hange if the value can reduce the utility of range as a
200 is not present in the data measure of dispersion. Thus, you
set? may need a measure which is not
• If 50 is replaced by 150, what unduly affected by the outliers.
will be the Range? In such a situation, if the entire data
is divided into four equal parts, each
Range: Comments containing 25% of the values, we get
Range is unduly affected by the values of quartiles and median.
extreme values. It is not based on (You have already read about these in
all the values. As long as the Chapter 5).
minimum and maximum values The upper and lower quartiles (Q,
remain unaltered, any change in
and Q„ respectively) are used to
other values does not affect range.
It cannot be calculated for open- calculate inter-quartile range which is
ended frequency distribution. o z o-
Interquartile range is based upon
Notwithstanding some limitations, middle 50% of the values in a
range is understood and used distribution and is, therefore, not
frequently because of its simplicity. affected by extreme values. Half of
For example, we see the maximum the inter-quartile range is called
and minimum temperatures of quartile deviation (Q.D.). Thus:
different cities almost daily on our
TV screens and form judgments Q.D.=
about the temperature variations in 2
@.D. is therefore also called Derrii-
them.
Inter Quarttte Range.
Open-ended distributions are
those in which either the lower Calculation o/ ftnnpe rind Q.D. for
limit of the lowest class or the ungrouped dntn
upper limit of the highest class or Example 1
both are not specified.
Calculate range and Q.D. of the
following observations:
20, 25, 29, 30, 35, 39, 41,
• Collect data about 52—week
48, 51, 60 and 70
high/ low of shares of 10
companies from a newspaper. Range is clearly 70 — 20 = 50
Calculate the range of share For Q.D. , we need to calculate
prices. Which company’s share is values of Q, and Q,.
most volatile and which is the
most stable?
MEASURES OF DISPERSION
77

n+1 Range is just the difference between

Q, is the size of value. the upper limit of the highest class and
4 the lower limit of the lowest class. So
n being 11, Q, is the size of 3rd value.
range is 90 — 0 = 90. For Q.D., first
As the values are already arranged
calculate cumulative frequencies as
in ascending order, it can be seen that follows:
Q„ the 3rdvalue is 29. [What will
you
do if these values are not in an order?] Closs- Fr' equencies Cumulative
Interuols Mequencies
CI / c./
Similarly, Q, is size of 0—10 5 05
10—20 8 13
value; i.e. 9th value which is 51. Hence 20—40 16 29
40—60 7 36
60—90 4 40

.D.= 5 I — 29 =II n = 40
2 = 2
Do you notice that Q.D. is the Q, is the size of —th value in a
4
average difference of the Quartiles from continuous series. Thus, it is the size
the median. of the 10a value. The class containing
Rc tiuiw the 10a value is 10—20. Hence, Q, lies
• Calculate the median and
in class 10—20. Now, to calculate the
check whether the above exact value of Q„ the following formula
statement is correct. is used:
Calculation of Range and Q.D. for a n-d
frequency distribution.
=L + 4 g
Example 2 1
f

For the following distribution of marks Where L = 10 (lower limit of the

scored by a class of 40 students, relevant Quartile class)
calculate the Range and Q.D. c.f. = 5 (Value of c.f. for the class
preceding the quartile class)
TABLE 6. 1
i = 10 (interval of the quartile class),
Clnss internals No. of students and
CI If) f= 8 (frequency of the quartile class)
0—10 5 Thus,
10—20 8
20—40 16 10-5
40—60 7 ' '0’ g X '0 '16 25
60—90 4
40 3n
Similarly, Q, is the size of
4
 78
STATISTICS FOR ECONOMICS

value; i.e., 30th value, which lies in Quartile deviation can generally
class 40-60. Now using the formula be calculated for open-ended
for Q„ its value can be calculated as distributions and is not unduly
follows: affected by extreme values.
— c.f.
4 /. RASURES OF ISPRRSION FROM
Q=L+
f

@, = 40 + 30 — 29 x 20 Recall that dispersion was defined as

7 the extent to which values differ from
@, = 42.87 their average. Range and quartile
Q.D.= 42.87— 16.25 _ deviation are not useful in measuring,
2 i» »i
how far the values are, from their
average. Yet, by calculating the
In individual and discrete series, spread of values, they do give a good
n+1 idea about the dispersion. Two
Q, is the size of value, but measures which are based upon
4
in a continuous distribution, it is deviation of the values from their
average are Mean Deviation and
Standard Deviation.
the size of — thvalue. Similarly, Since the average is a central
4
for Q3 and median also, n is used in place of n+1. value, some deviations are positive
and some are negative. If these are
added as they are, the sum will not
reveal anything.
If the entire group is divided into In fact, the sum of deviations from
two equal halves and the median Arithmetic Mean is always zero. Look
calculated for each half, you will have at the following two sets of values.
the median of better students and the
median of we ah students. These Set A : 5, 9, 16
medians differ from the median of the Set B : 1, 9, 20
entire group by 13.31 on an average. You can see that values in Set B
Similarly, suppose you have data are farther from the average and
about incomes of people of a town. hence more dispersed than values in
Median income of all people can be Set A. Calculate the deviations from
calculated. Now, if all people are Arithmetic Mean and sum them up.
divided into two equal groups of rich What do you notice? Repeat the
and poor, medians of both groups can
be calculated. Quartile deviation will same with Median. Can you
tell you the average difference comment upon the quantum of
between medians of these two groups variation from the calculated values?
belonging to rich and poor, from the Mean Deviation tries to overcome
median of the entire group. this problem by ignoring the signs of
MEASURES OF DISPERSION
79

deviations, i.e., it considers all average. The average used is either

deviations positive. For standard the arithmetic mean or median.
deviation, the deviations are first (Since the mode is not a stable
squared and averaged and then square average, it is not used to calculate
root of the average is found. We shall mean deviation.)
now discuss them separately in
detail.
• Calculate the total distance to be
travelled by students if the college
is situated at town A, at town C,
Suppose a college is proposed for or town E and also if it is exactly
students of five towns A, B, C, D and E half way between A and E.
which lie in that order along a road. • Decide where, in your opinion,
Distances of towns in kilometres from the college should be establi-
town A and number of students in shed, if there is only one
these towns are given below: student in each town. Does it
change your answer?
Ton.›n Distance No.
com town A o/Stuctents Colculotion oJ3feon
A 0 90 DeriotionJrom Arithmetic Zfeon
B 2 150 Jor unprouped dots.
6 100
D 14 200 Direct Method
18 80
Steps:
6!2O
(i) The A.M. of the values is calculated
Now, if the college is situated in (ii) Difference between each value and
town A, 150 students from town B will the A.M. is calculated. All
have to travel 2 kilometers each (a total differences are considered positive.
of 300 kilometres) to reach the college. These are denoted as I d I
The objective is to find a location so (iii)The A.M. of these differences
that the average distance travelled by (called deviations) is the Mean
students is minimum. Deviation.
You may observe that the students
will have to travel more, on an average, 1d1
if the college is situated at town A or i.e. M.D. -
E. If on the other hand, it is somewhere
in the middle, they are likely to travel Example 3
less.
Mean deviation is the appropriate Calculate the mean deviation of the
statistical tool to estimate the average following values; 2, 4, 7, 8 and 9.
distance travelled by students. Mean
deviation is the arithmetic mean of the The A.M. Lx
differences of the values from their n =6
 80 STATISTICS FOR ECONOMICS

ldl Zfeon Dei›iotion from Zfeon Jor

Continuous Dist ribution
2
4 4
2 TABLE 6.2
7 1
g 2 Profits of Number of
9 companies Companies
(Rs in iakhl
12 Clnss internals
10—20 5
12 20-30 8
M.D. = = 2.4 30-50 16
5 50-70 8
70-80 3
Zfeon Detiiotion from medion for 40
unprouped dots.
Steps:
Method
(i) Calculate the mean of the
Using the values in Example 3, M.D. distribution.
from the Median can be calculated as
(ii) Calculate the absolute deviations
follows, I d I of the class midpoints from the
(i) Calculate the median which is 7. mean.
(ii) Calculate the absolute deviations (iii) Multiply each I d I value with its
from median, denote them as I d I . corresponding frequency to get f 1 d I
(iii)Find the average of these absolute values. Sum them up to get Zf I d I .
deviations. It is the Mean (iv)Apply the following formula,
Deviation.
Z f 1d 1
M.D.' X’ Zf
Example 5
Mean Deviation of the distribution
ct= IN-JUEDJAJYI
2 5 in Table 6.2 can be calculated as
4 follows:
7 0
8 1
9 2
Example 6
11 C.I. f p. Id 1 /l d 1
10-20 5 15 25.5 127.5
M. D. from Median is thus, 20-30 8 25 15.5 124.0
30-50 16 40 0.5 8.0
1d1 50-70 8 60 19.5 156.0
M.D. Me dianJ = =2.2 70-80 3 75 34.5 103.5
n 5 40 519.0
MEASURES OF DISPERSION 81

Zf1 d1 519
M.D. Zf 40 it ignores the signs of deviations
= 12.975
and cannot be calculated for open-
ended distributions.
Zfeon Detitotion Jrom Zfedion
TABLE 6.3 Standard Deviation
Clnss internals Standard Deviation is the positive
Weqiiencies
20—30 square root of the mean of squared
5
30—40 deviations from mean. So if there are
10
40—60 five values x„ , x„ x 4 and y, first
60—80 20
80-90 9 their
6 mean is calculated. Then deviations of
the values from mean are calculated.
These deviations are then squared. The
The procedure to calculate mean mean of these squared deviations is the
deviation from the median is the variance. Positive square root of the
same as it is in case of M.D. from o ‘ance is the standard deviation.
mean, except that deviations are to (Note that standard deviation is
be taken from the median as given calculated on the basis of the mean
below: only).

Example T Colcutotion oJ 6tondord Detiiotton

Jor unprouped dots
f p. Id 1 /l d 1
Four alternative methods are
20—30 5 25 25 125 available for the calculati on of
30—40 10 35 15 150
40—60 20 50 0 0 standard deviation of individual
60—80 9 70 20 180 values. All these methods result in
80-90 6 85 35 210 the same value of standard
ifi0 66ifi deviation. These are:
(i) Actual Mean Method
(ii) Assumed Mean Method
M.D.( p, d, j = (iii)Direct Method
(iv)Step-Deviation Method
Actual Menn Method:
Suppose you have to calculate the
Mean Deviation: Comments standard deviation of the following
Mean deviation is based on all values:
values. A change in even one value 5, 10, 25, 30, 50
will affect it. Mean deviation is the First step is to calculate
least when calculated from the
5+10+25+30+50 120
median i.e., it will be higher if X= = = 24
calculated from the mean. However 5 5
82
STATISTICS FOR ECONOMICS

Example 8 Formula for Standard Deviation

d x-x) d°
5 —19 361 n n
10 —14 196
25 +i 1
30 +6 36
1275—5 2
254 15.937
50 +26 676 ss
0 12 TO
Note that the sum of deviations
Then the following formula is used: from a value other than actual
mean will not be equal to zero.
I d2
Standard deviation is not affected
n by the value of the constant from
which deviations are calculated.
z xx 2 The value of the constant does not
a= figure in the standard deviation
n formula. Thus, Standard deviation
is independent o/ Oripin.
1270 = 4 = 15.937
5
Direct Method
Do you notice the value from which
deviations have been calculated in Standard Deviation can also be
the above example? Is it the Actual calculated from the values directly,
Mean? i.e., without taking deviations, as
shown below:
Assumed Menu Method
For the same values, deviations may be E xample 10
calculated from any arbitrary value
A x such that d = X — Ax . Taking Ax
= 25, the computation of the standard
5 25
deviation is shown below: 10 100
25 625
Example 9 30 900
50 2500
ct {x-A x ) d’ 1!2O 4150
5 —20 400
10 —15 225 (This amounts to taking deviations
25 0 0 from zero)
30 +5 25 Following formula is used.
50 +25 625
12TG
MEASURES OF DISPERSION
83

4150 50.80
or = — (24)2 a= 5 x5
5
or o-= = 15.937 cr= 0 6 x5
w = 15.937
Step-deviation Method
Alternatively, instead of dividing
If the values are divisible by a common the values by a common factor, the
factor, they can be so divided and deviations can be calculated and then
standard deviation can be calculated divided by a common factor.
from the resultant values as follows:
Standard deviation can be
Example 11 calculated as shown below:
Example 12
Since all the five values are divisible by
a common factor 5, we divide and get Ct =(x-25a Ct’ =(d/5J Ct’*
the following values: 5 —20 —4 16
x x' d’ = {x’-x’ ) ct” 10 —15 —3 9
25 0 0 0
5 1 —3.8 14.44 30 +5 +l 1
10 2 —2.8 7.84 50 +25 +5 25
25 5 +0.2 0.04
—1 ifi1
30 6 +1.2 1.44
50 10 +5.2 27.04 Deviations have been calculated
from an arbitrary value 25. Common
factor of 5 has been used to divide
In the above table, deviations.
X 2
X'= Z d“ 2 Z2d“
n n
where c = common
factor First step is to
51—1
calculate -— -
52
1+2+5+6+10 24 rr = 5
X= 5 = 5= 4.8
The following formula is used to 11
calculate standard deviation: °' x 5= 15.937
Standard deviation is not independent
Z d" 2 of scale. Thus, if the values or
= xc deviations are divided by a common
n factor, the value of the common
factor is used in the formula to get
Substituting the values, the value of standard deviation.
 84
STATISTICS FOR ECONOMICS

Standard Deviation in Continuous

frequency distribution: 5. Apply the formula as under:
Like ungrouped data, S.D. can be Zfd2 11790
=l7.168
calculated for grouped data by any of n 40
the following methods:
(i) Actual Mean Method Assumed Mean Method
(ii) Assumed Mean Method
(iii) Step-Deviation Method For the values in example 13,
standard deviation can be calculated
Actunl Mean Method by taking deviations from an assumed
mean (say
For the values in Table 6.2, Standard 40) as follows:
Example 14
Deviation can be calculated as follows:
(1) (2) (3) (4) (ifi) (6)
Example 13 CI m Ct /d /d*
10—20 5 15 -25 —125 3125
(1) (2) (3) (4) (5) (6) (T) 20—30 8 25 -15 —120 1800
C/ / m /m ct /r/ /r/* 30—50 16 40 0 0 0
10—20 5 15 75 —25.5 —127.5 3251.25 50—70 8 60 +20 160 3200
20—30 8 25 200 —15.5 —124.0 1922.00 70—80 3 75 +35 105 3675
30—50 16 40 640 —0.5 —8.0 4.00
50—70 8 60 480 +19.5 +156.0 3042.00 40 +2O 11800
70—80 3 75 225 +34.5 +103.5 3570.75
The following steps are required:
40 1 O!2O 0 1 1790.OO
1. Calculate mid-points of classes
Following steps are required: (Col. 3)
1. Calculate the mean of the 2. Calculate deviations of mid-points
distribution. from an assumed mean such that
Zfm 1620 d = m - A -(Col. 4). Assumed
x = Zf 40 = 40.5 Mean = 40.
2. Calculate deviations of mid-values 3. M ultiply values of ‘d’ with
from the mean so that d - - z corresponding frequencies to get
(Col. 5) ‘fd’ values (Col. 5). (Note that the
3. Multiply the deviations with their total of this column is not zero
corresponding frequencies to get since deviations have been taken
‘fd’ values (Col. 6) [Note that y fd from assumed mean).
= 0] 4. Multiply ‘fd’ values (Col. 5) with ‘d’
4. Calculate ‘fd2’ values by values (col. 4) to get fd2values (Col.
multiplying ‘fd’ values with ‘d’ 6). Find y fd2.
values. (Col. 7). Sum up these to 5. Standard Deviation can be
get y fd 2. calculated by the following formula.
MEASURES OF DISPERSION
85

4. Multiply ‘fd” values with ‘d” values

p _ Zfd2Zfd 2
nn to get ‘fd' 2’ values (Col. 7)
5. Sum up values in Col. 6 and Col. 7
i 180020 2 to get y fd' and y fd'2 values.
or a =
4040 6. Apply the following formula.
or a = 294 75 = 17. 168 Zfd’Zfd’ 2
= — xC
Zf Zf
Step-deviation Method
In case the values of deviations are 2
or o = 472 4
divisible by a common factor, the X *
40 40
calculations can be simplified by the
step-deviation method as in the or w = 11 8 0 01 x 5
following example. or 1 1 79 X@
"
Example 15 o=l7.168

(1) (2) (3) (4) (ifi) (6) (T) Standard Deviation: Comments
CI m Ct Ct’ /d’ /d“ Standard Deviation, the most
10—20 5 15 —25 —5 —25 125 widely used measure of
20—30 8 25 —15 —3 —24 72 dispersion, is based on all values.
30—50 16 40 0 0 0 0 Therefore a change in even one
50—70 8 60 +20 +4 +32 128 value affects the value of standard
70—80 3 75 +35 +7 +2 1 147 deviation. It is independent of
40 +4 4T2 origin but not of scale. It is also
useful in certain advanced
Steps required: statistical problems.

1. Calculate class mid-points (Col. 4. Aasoavzs aero Rsw'rnm Mzasvnss

3) and deviations from an OF DISPERSION
arbitrarily chosen value, just like All the measures, described so far,
in the assumed mean method. In are absolute measures of dispersion.
this example, deviations have They calculate a value which, at
been taken from the value 40. times, is difficult to interpret. For
(Col. 4) example, consider the following two
2. Divide the deviations by a common data sets:
factor denoted as ‘c’. c = 5 in the SetA SOO 700 1000
above example. The values so SetB 1,00,000 1,20,000 1,30,000
obtained are ‘d” values (Col. 5).
Suppose the values in Set A are
3. Multiply ‘d” values with the daily sales recorded by an ice-
corresponding ‘f’ values (Col. 2) cream vendor, while Set B has the
to obtain ‘fd” values (Col. 6). daily sales of a big departmental store.
Range for Set A is 500 whereas for Set
B, it is
86 STATISTICS FOR ECONOMICS

30,000. The value of Range is much For Mean Deviation, it is Coefficient

higher in Set B. Can you say that of Mean Deviation.
the variation in sales is higher for the Coefficient of Mean Deviation =
departmental store? It can be easily
observed that the highest value in Set M.D.(x) r
M.D.(Median)
A is double the smallest value, x Median
whereas for the Set B, it is only 30% Thus, if Mean Deviation is
higher. Thus, absolute measures may calculated on the basis of the Mean, it
give misleading ideas about the is divided by the Mean. If Median is
extent of variation specially when the used to calculate Mean Deviation, it
averages differ significantly. is divided by the Median.
Another weakness of absolute For Standard Deviation, the
measures is that they give the answer relative measure is called Coefficient
in the units in which original values of Variation, calculated as below:
are expressed. Consequently, if the Coefficient of Variation
values are expressed in kilometers, _ Standard Deviation
the
dispersion will also be in kilometers. 10 a
However, if the same values are Arithmetic Mean
expressed in meters, an absolute It is usually expressed in
measure will give the answer in percentage terms and is the most
meters and the value of dispersion commonly used relative measure of
will appear to be 1000 times. dispersion. Since relative measures
To overcome these problems, are free from the units in which the
relative measures of dispersion can values have been expressed, they can
be used. Each absolute measure has be compared even across different
a relative counterpart. Thus, for groups having different units of
range, there is coefficient of range measurement.
which is calculated as follows:
L—
Coefficient of Range — S The measures of dispersion discussed
L+ so far give a numerical value of
where L = Largest value dispersion. A raphical measure called
S = Smallest value Lorenz Curve lS available for
estimating inequalities in distribution.
Similarly, for Quartile Deviation,
You may have heard of statements
it is Coefficient of Quartile Deviation
which can be calculated as follows: like ‘top 10% of the people of a
Coefficient of Quartile Deviation country earn 50% of the national
income while top 20% account for
80%’. An idea about income
disparities is given by such
Q3— Q
Q3+ Q where Q,=3' d Quartile figures. Lorenz Curve uses the
information expressed in a
@, = l *t Quartile cumulative manner to indicate the
degree of
MEASURES OF DISPERSION
87
inequality. For example, Lorenz Curve
as a percentage (%) of the grand
of income gives a relationship between
total income of all classes together.
percentage of population and its share
of income in total income. It is Thus obatain Col. (6) of Table 6.4.
specially useful in comparing the 5. Prepare less than cumulative
variability of two or more distributions frequency and Cumulative
by drawing two or more Lorenz curves income Table 6.5.
on the same axis. 6. Col. (2) of Table 6.5 shows the
cumulative frequency of
Construction of the Lorenz curve
empolyees.
Follou›ing steps are required. 7. Col. (3) of Table 6.5 shows the
1. Calculate class Midpoints to obtain cumulative income going to these
Col.2 of Table 6.4. persons.
2. Calculate the estmated total 8. Draw a line joining Co-ordinate
income of employees in each class (0,0) with (100, 100). This is
by multiplying the midpoint of the called the line of equal
class by the frequency in the class. distribution shown as line ‘OE’ in
Thus obtain Col. (4) of Table 6.4.
figure 6. 1.
3. Express frequency in each class
as a percentage (%) of total 9. Plot the cumulative percentages of
frequency. Thus, obtain Col. (5) of empolyees on the horizontal axis
Table 6.4. and cumulative income on the
4. Express total income of each class vertical axis. We will the thus gate
the line.

Given below are the monthly incomes of employees of a company:

TABLE 6.4
Income Midpoint (X) Frequency (f) Total income % of frequency % of Total
class of class (FX) income
(1) (2) (3) (4) (6)
0—5000 2500 5 12500 10 1.29
5000—10000 7500 10 75000 20 7.71
10000-20000 15000 18 270000 36 27.76
20000-40000 30000 10 300000 20 30.85
40000-50000 45000 7 315000 14 32.39
50 972500 100
 88 STATISTICS FOR ECONOMICS

TABLE 6. 5
20% of total income and top 60% earn
‘Less Than’ Cumulative Cumulative
60% of the total income. The farther
frequency Income the curve OABCDE from this line, the
greater is the inequality present in the
5,000 10 1.29 distribution. If there are two or more
10,000 3 9.00 curves on the same axes, the one
20,000 66 36.76 is the farthest from line OE has the
which
40,000 86 67.61 highest inequality.
50,000 100 100.00

Studying the Morenz Curve 8. CONCLUSION

OE is called the line of equ a1 Although Range is the simplest to
distribution, since it would imply a calculate and understand, it is unduly
situation like, top 20% people earn affected by extreme values. QD is not
affected by extreme values as it is
based on only middle 50% of the
data. However, it is more difficult to
interpret
D
M.D. and S.D. Both are based upon
deviations of values from their average.
M.D. calculates average of deviations
from the average but ignores signs
of deviations and therefore appears
to be unmathematical. Standard
deviation attempts to calculate
„” B average deviation from mean. Life
M.D., it is based on all values and is
Cvimvilallv< Percentage of Emp s also applied in more advanced
statistical problems. It is the most
widely used measure of dispersion.

recap
• A measure of dispersion improves our understanding about the
behaviour of an economic variable.
• Range and Quartile Deviation are based upon the spread of values.
• M.D. and S.D. are based upon deviations of values from the average.
• Measures of dispersion could be Absolute or Relative.
• Absolute measures give the answer in the units in which data are
expressed.
• Relative measures are free from these units, and consequently
can be used to compare different variables.
• A graphic method, which estimates the dispersion from shape
of a curve, is called Lorenz Curve.
MEASURES OF DISPERSION 89

EXERCISES
1. A measure of dispersion is a good supplement to the central value in
understanding a frequency distribution. Comment.
2. Which measure of dispersion is the best and how?
3. Some measures of dispersion depend upon the spread of values whereas
some are estimated on the basis of the variation of values from a central
value. Do you agree?
4. In a town, 25% of the persons earned more than Rs 45,000 whereas
7i5% earned more than 18,000. Calculate the absolute and relative values
of dispersion.
5. The yield of wheat and rice per acre for 10 districts of a state is as
under:
District 1 2 3 4 5 6 7 8 9 10
Wheat 12 10 15 19 21 16 18 9 25 10
Rice 22 29 12 23 18 15 12 34 18 12
Calculate for each crop,
(i) Range
(ii .D.
(iii) Mean deviation about Mean
(iv) Mean deviation about Median
(v) Standard deviation
(vi) Which crop has greater variation?
(vii) Compare the values of different measures for each crop.
6. In the previous question, calculate the relative measures of variation
and indicate the value which, in your opinion, is more reliable.
7. A batsman is to be selected for a cricket team. The choice is between X
and Y on the basis of their scores in five previous tests which are:
X 25 85 40 80 120
Y TO 70 65 45 80
Which batsman should be selected if we want,
(i) a higher run getter, or
(ii) a more reliable batsman in the team?
8. To check the quality of two brands of lightbulbs, their life in burning hours
was estimated as under for 100 bulbs of each brand.
L.ife No. oJ bittbs
(in hrs) Brnnd A Brnnd B
0-50 15 2
50-lOO 20 8
100-150 18 60
150-200 25 25
200-250 22 5
1OO 1OO
90 STATISTICS FOR ECONOMICS

(i) Which brand gives higher life?

(ii) Which brand is more dependable?
9. Averge daily wage of 50 workers of a factory was Rs 200 with a standard
deviation of Rs 40. Each worker is given a raise of Rs 20. What is the
new average daily wage and standard deviation? Have the wages become
more or less uniform?
10. If in the previous question, each worker is given a hike of 10 % in wages,
how are the mean and standard deviation values affected?
11. Calculate the mean deviation using mean and Standard Deviation for
the following distribution.
Classes Frequencies
20—40 3
40—80 6
80—100 20
100—120 12
120—140 9
50
12. The sum of 10 values is 100 and the sum of their squares is 1090. Find
out the coefficient of variation.