NCERT Textbook - Measures of Dispersion Commerce Notes | EduRev

Statistics for Economics - Class XI

Created by: Pj Commerce Academy

Commerce : NCERT Textbook - Measures of Dispersion Commerce Notes | EduRev

 Page 1


1. INTRODUCTION
In the previous chapter, you have
studied how to sum up the data into
a single representative value. However,
that value does not reveal the
variability present in the data. In this
chapter you will study those
measures, which seek to quantify
variability of the data.
Three friends, Ram, Rahim and
Maria are chatting over a cup of tea.
During the course of their
conversation, they start talking about
their family incomes. Ram tells them
that there are four members in his
family and the average income per
member is Rs 15,000.  Rahim says that
the average income is the same in his
family, though the number of members
is six. Maria says that there are five
members in her family, out of which
one is not working.  She calculates that
the average income in her family too,
is Rs 15,000.  They are a little surprised
since they know that Maria’s father is
earning a huge salary.  They go into
details and gather the following data:
Measures of Dispersion
Studying this chapter should
enable you to:
• know the limitations of averages;
? appreciate the need of measures
of dispersion;
? enumerate various measures of
dispersion;
? calculate the measures and
compare them;
? distinguish between absolute
and relative measures.
CHAPTER
Page 2


1. INTRODUCTION
In the previous chapter, you have
studied how to sum up the data into
a single representative value. However,
that value does not reveal the
variability present in the data. In this
chapter you will study those
measures, which seek to quantify
variability of the data.
Three friends, Ram, Rahim and
Maria are chatting over a cup of tea.
During the course of their
conversation, they start talking about
their family incomes. Ram tells them
that there are four members in his
family and the average income per
member is Rs 15,000.  Rahim says that
the average income is the same in his
family, though the number of members
is six. Maria says that there are five
members in her family, out of which
one is not working.  She calculates that
the average income in her family too,
is Rs 15,000.  They are a little surprised
since they know that Maria’s father is
earning a huge salary.  They go into
details and gather the following data:
Measures of Dispersion
Studying this chapter should
enable you to:
• know the limitations of averages;
? appreciate the need of measures
of dispersion;
? enumerate various measures of
dispersion;
? calculate the measures and
compare them;
? distinguish between absolute
and relative measures.
CHAPTER
MEASURES OF DISPERSION 75
Family Incomes
Sl. No. Ram Rahim Maria
1. 12,000 7,000 0
2. 14,000 10,000 7,000
3. 16,000 14,000 8,000
4. 18,000 17,000 10,000
5. ----- 20,000 50,000
6. ----- 22,000 ------
Total income 60,000 90,000 75,000
Average income 15,000 15,000 15,000
Do you notice that although the
average is the same, there are
considerable differences in individual
incomes?
It is quite obvious that averages
try to tell only one aspect of a
distribution i.e. a representative size
of the values.  To understand it better,
you need to know the spread of values
also.
You  can see that in Ram’s family.,
differences in incomes are
comparatively lower. In Rahim’s
family, differences are higher and in
Maria’s family are the highest.
Knowledge of only average is
insufficient.  If you have another value
which reflects the quantum of
variation in values, your understan-
ding of a distribution improves
considerably.  For example, per capita
income gives only the average income.
A measure of dispersion can tell you
about income inequalities, thereby
improving the understanding of the
relative standards of living enjoyed by
different strata of society.
Dispersion is the extent to which
values in a distribution differ from the
average of the distribution.
To quantify the extent of the
variation, there are certain measures
namely:
(i) Range
(ii) Quartile Deviation
(iii)Mean Deviation
(iv)Standard Deviation
Apart from these measures which
give a numerical value, there is a
graphic method for estimating
dispersion.
Range and Quartile Deviation
measure the dispersion by calculating
the spread within which the values lie.
Mean Deviation and Standard
Deviation calculate the extent to
which the values differ from the
average.
2. MEASURES BASED UPON SPREAD OF
VALUES
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.
Page 3


1. INTRODUCTION
In the previous chapter, you have
studied how to sum up the data into
a single representative value. However,
that value does not reveal the
variability present in the data. In this
chapter you will study those
measures, which seek to quantify
variability of the data.
Three friends, Ram, Rahim and
Maria are chatting over a cup of tea.
During the course of their
conversation, they start talking about
their family incomes. Ram tells them
that there are four members in his
family and the average income per
member is Rs 15,000.  Rahim says that
the average income is the same in his
family, though the number of members
is six. Maria says that there are five
members in her family, out of which
one is not working.  She calculates that
the average income in her family too,
is Rs 15,000.  They are a little surprised
since they know that Maria’s father is
earning a huge salary.  They go into
details and gather the following data:
Measures of Dispersion
Studying this chapter should
enable you to:
• know the limitations of averages;
? appreciate the need of measures
of dispersion;
? enumerate various measures of
dispersion;
? calculate the measures and
compare them;
? distinguish between absolute
and relative measures.
CHAPTER
MEASURES OF DISPERSION 75
Family Incomes
Sl. No. Ram Rahim Maria
1. 12,000 7,000 0
2. 14,000 10,000 7,000
3. 16,000 14,000 8,000
4. 18,000 17,000 10,000
5. ----- 20,000 50,000
6. ----- 22,000 ------
Total income 60,000 90,000 75,000
Average income 15,000 15,000 15,000
Do you notice that although the
average is the same, there are
considerable differences in individual
incomes?
It is quite obvious that averages
try to tell only one aspect of a
distribution i.e. a representative size
of the values.  To understand it better,
you need to know the spread of values
also.
You  can see that in Ram’s family.,
differences in incomes are
comparatively lower. In Rahim’s
family, differences are higher and in
Maria’s family are the highest.
Knowledge of only average is
insufficient.  If you have another value
which reflects the quantum of
variation in values, your understan-
ding of a distribution improves
considerably.  For example, per capita
income gives only the average income.
A measure of dispersion can tell you
about income inequalities, thereby
improving the understanding of the
relative standards of living enjoyed by
different strata of society.
Dispersion is the extent to which
values in a distribution differ from the
average of the distribution.
To quantify the extent of the
variation, there are certain measures
namely:
(i) Range
(ii) Quartile Deviation
(iii)Mean Deviation
(iv)Standard Deviation
Apart from these measures which
give a numerical value, there is a
graphic method for estimating
dispersion.
Range and Quartile Deviation
measure the dispersion by calculating
the spread within which the values lie.
Mean Deviation and Standard
Deviation calculate the extent to
which the values differ from the
average.
2. MEASURES BASED UPON SPREAD OF
VALUES
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
Activities
Look at the following values:
20, 30, 40, 50, 200
? Calculate the Range.
? What is the Range if the value
200 is not present in the data
set?
? If 50 is replaced by 150, what
will be the Range?
Range:  Comments
Range is unduly affected by extreme
values. It is not based on all the
values. As long as the minimum and
maximum values remain unaltered,
any change in other values does not
affect range. It can not be calculated
for open-ended frequency distri-
bution.
Notwithstanding some limitations,
Range is understood and used
frequently because of its simplicity.
For example, we see the maximum
and minimum temperatures of
different cities almost daily on our TV
screens and form judgments about the
temperature variations in them.
Open-ended distributions are those
in which either the lower limit of the
lowest class or the upper limit of the
highest class or both are not
specified.
Activity
? Collect data about 52-week
high/low of 10 shares from a
newspaper. Calculate the range
of share prices.  Which stock is
most volatile and which is the
most stable?
Quartile Deviation
The presence of even one extremely
high or low value in a distribution can
reduce the utility of range as a
measure of dispersion. Thus, you may
need a measure which is not unduly
affected by the outliers.
In such a situation, if the entire
data is divided into four equal parts,
each containing 25% of the values, we
get the values of Quartiles and
Median.  (You have already read about
these in Chapter 5).
The upper and lower quartiles (Q
3
and Q
1
, respectively) are used to
calculate Inter Quartile Range which
is Q
3 
– Q
1
.
Inter-Quartile Range is based
upon middle 50% of the values in a
distribution and is, therefore, not
affected by extreme values.  Half of
the Inter-Quartile Range is called
Quartile Deviation. Thus:
Q.D . = 
Q - Q
2
31
Q.D. is therefore also called Semi-
Inter Quartile Range.
Calculation of Range and Q.D. for
ungrouped data
Example 1
Calculate Range and Q.D. of the
following observations:
20, 25, 29, 30, 35, 39, 41,
48, 51, 60 and 70
Range is clearly 70 – 20 = 50
For Q.D., we need to calculate
values of Q
3
 and Q
1
.
Page 4


1. INTRODUCTION
In the previous chapter, you have
studied how to sum up the data into
a single representative value. However,
that value does not reveal the
variability present in the data. In this
chapter you will study those
measures, which seek to quantify
variability of the data.
Three friends, Ram, Rahim and
Maria are chatting over a cup of tea.
During the course of their
conversation, they start talking about
their family incomes. Ram tells them
that there are four members in his
family and the average income per
member is Rs 15,000.  Rahim says that
the average income is the same in his
family, though the number of members
is six. Maria says that there are five
members in her family, out of which
one is not working.  She calculates that
the average income in her family too,
is Rs 15,000.  They are a little surprised
since they know that Maria’s father is
earning a huge salary.  They go into
details and gather the following data:
Measures of Dispersion
Studying this chapter should
enable you to:
• know the limitations of averages;
? appreciate the need of measures
of dispersion;
? enumerate various measures of
dispersion;
? calculate the measures and
compare them;
? distinguish between absolute
and relative measures.
CHAPTER
MEASURES OF DISPERSION 75
Family Incomes
Sl. No. Ram Rahim Maria
1. 12,000 7,000 0
2. 14,000 10,000 7,000
3. 16,000 14,000 8,000
4. 18,000 17,000 10,000
5. ----- 20,000 50,000
6. ----- 22,000 ------
Total income 60,000 90,000 75,000
Average income 15,000 15,000 15,000
Do you notice that although the
average is the same, there are
considerable differences in individual
incomes?
It is quite obvious that averages
try to tell only one aspect of a
distribution i.e. a representative size
of the values.  To understand it better,
you need to know the spread of values
also.
You  can see that in Ram’s family.,
differences in incomes are
comparatively lower. In Rahim’s
family, differences are higher and in
Maria’s family are the highest.
Knowledge of only average is
insufficient.  If you have another value
which reflects the quantum of
variation in values, your understan-
ding of a distribution improves
considerably.  For example, per capita
income gives only the average income.
A measure of dispersion can tell you
about income inequalities, thereby
improving the understanding of the
relative standards of living enjoyed by
different strata of society.
Dispersion is the extent to which
values in a distribution differ from the
average of the distribution.
To quantify the extent of the
variation, there are certain measures
namely:
(i) Range
(ii) Quartile Deviation
(iii)Mean Deviation
(iv)Standard Deviation
Apart from these measures which
give a numerical value, there is a
graphic method for estimating
dispersion.
Range and Quartile Deviation
measure the dispersion by calculating
the spread within which the values lie.
Mean Deviation and Standard
Deviation calculate the extent to
which the values differ from the
average.
2. MEASURES BASED UPON SPREAD OF
VALUES
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
Activities
Look at the following values:
20, 30, 40, 50, 200
? Calculate the Range.
? What is the Range if the value
200 is not present in the data
set?
? If 50 is replaced by 150, what
will be the Range?
Range:  Comments
Range is unduly affected by extreme
values. It is not based on all the
values. As long as the minimum and
maximum values remain unaltered,
any change in other values does not
affect range. It can not be calculated
for open-ended frequency distri-
bution.
Notwithstanding some limitations,
Range is understood and used
frequently because of its simplicity.
For example, we see the maximum
and minimum temperatures of
different cities almost daily on our TV
screens and form judgments about the
temperature variations in them.
Open-ended distributions are those
in which either the lower limit of the
lowest class or the upper limit of the
highest class or both are not
specified.
Activity
? Collect data about 52-week
high/low of 10 shares from a
newspaper. Calculate the range
of share prices.  Which stock is
most volatile and which is the
most stable?
Quartile Deviation
The presence of even one extremely
high or low value in a distribution can
reduce the utility of range as a
measure of dispersion. Thus, you may
need a measure which is not unduly
affected by the outliers.
In such a situation, if the entire
data is divided into four equal parts,
each containing 25% of the values, we
get the values of Quartiles and
Median.  (You have already read about
these in Chapter 5).
The upper and lower quartiles (Q
3
and Q
1
, respectively) are used to
calculate Inter Quartile Range which
is Q
3 
– Q
1
.
Inter-Quartile Range is based
upon middle 50% of the values in a
distribution and is, therefore, not
affected by extreme values.  Half of
the Inter-Quartile Range is called
Quartile Deviation. Thus:
Q.D . = 
Q - Q
2
31
Q.D. is therefore also called Semi-
Inter Quartile Range.
Calculation of Range and Q.D. for
ungrouped data
Example 1
Calculate Range and Q.D. of the
following observations:
20, 25, 29, 30, 35, 39, 41,
48, 51, 60 and 70
Range is clearly 70 – 20 = 50
For Q.D., we need to calculate
values of Q
3
 and Q
1
.
MEASURES OF DISPERSION 77
Q
1
 is the size of   
n
th
+1
4
 value.
n being 11, Q
1
 is the size of 3rd
value.
As the values are already arranged
in ascending order, it can be seen that
Q
1
, the 3rd value is 29. [What will you
do if these values are not in an order?]
Similarly, Q
3
 is size of    
31
4
() n
th
+
value; i.e. 9th value which is 51. Hence
Q
3    
= 51
Q.D . = 
Q - Q
2
31
 = 
51 29
2
11
-
=
Do you notice that Q.D. is the
average difference of the Quartiles
from the median.
Activity
? Calculate the median and check
whether the above statement is
correct.
Calculation of Range and Q.D. for a
frequency distribution.
Example 2
For the following distribution of marks
scored by a class of 40 students,
calculate the Range and Q.D.
TABLE 6.1
Class intervals No. of students
 C I (f)
 0–10 5
10–20 8
20–40 16
40–60 7
60–90 4
40
Range is just the difference
between the upper limit of the highest
class and the lower limit of the lowest
class. So Range is 90 – 0 = 90. For
Q.D., first calculate cumulative
frequencies as follows:
Class- Frequencies Cumulative
Intervals Frequencies
CI f c. f.
  0–10 5 05
10–20 8 13
20–40 16 29
40–60 7 36
60–90 4 40
       n = 40
Q
1
 is the size of 
nth
4
 value in a
continuous series. Thus it is the size
of the 10th value. The class containing
the 10th value is 10–20. Hence Q
1 
lies
in class 10–20. Now, to calculate the
exact value of Q
1
, the following
formula is used:
QL
n
cf
f
i
1
4
=+ ·
Where L = 10  (lower limit of the
relevant Quartile class)
c.f. = 5 (Value of c.f. for the class
preceding the Quartile class)
i = 10 (interval of the Quartile
class), and
f = 8 (frequency of the Quartile
class) Thus,
Q
1
10
10 5
8
10 16 25 =+
-
·= .
Similarly, Q
3
 is the size of 
3
4
nth
Page 5


1. INTRODUCTION
In the previous chapter, you have
studied how to sum up the data into
a single representative value. However,
that value does not reveal the
variability present in the data. In this
chapter you will study those
measures, which seek to quantify
variability of the data.
Three friends, Ram, Rahim and
Maria are chatting over a cup of tea.
During the course of their
conversation, they start talking about
their family incomes. Ram tells them
that there are four members in his
family and the average income per
member is Rs 15,000.  Rahim says that
the average income is the same in his
family, though the number of members
is six. Maria says that there are five
members in her family, out of which
one is not working.  She calculates that
the average income in her family too,
is Rs 15,000.  They are a little surprised
since they know that Maria’s father is
earning a huge salary.  They go into
details and gather the following data:
Measures of Dispersion
Studying this chapter should
enable you to:
• know the limitations of averages;
? appreciate the need of measures
of dispersion;
? enumerate various measures of
dispersion;
? calculate the measures and
compare them;
? distinguish between absolute
and relative measures.
CHAPTER
MEASURES OF DISPERSION 75
Family Incomes
Sl. No. Ram Rahim Maria
1. 12,000 7,000 0
2. 14,000 10,000 7,000
3. 16,000 14,000 8,000
4. 18,000 17,000 10,000
5. ----- 20,000 50,000
6. ----- 22,000 ------
Total income 60,000 90,000 75,000
Average income 15,000 15,000 15,000
Do you notice that although the
average is the same, there are
considerable differences in individual
incomes?
It is quite obvious that averages
try to tell only one aspect of a
distribution i.e. a representative size
of the values.  To understand it better,
you need to know the spread of values
also.
You  can see that in Ram’s family.,
differences in incomes are
comparatively lower. In Rahim’s
family, differences are higher and in
Maria’s family are the highest.
Knowledge of only average is
insufficient.  If you have another value
which reflects the quantum of
variation in values, your understan-
ding of a distribution improves
considerably.  For example, per capita
income gives only the average income.
A measure of dispersion can tell you
about income inequalities, thereby
improving the understanding of the
relative standards of living enjoyed by
different strata of society.
Dispersion is the extent to which
values in a distribution differ from the
average of the distribution.
To quantify the extent of the
variation, there are certain measures
namely:
(i) Range
(ii) Quartile Deviation
(iii)Mean Deviation
(iv)Standard Deviation
Apart from these measures which
give a numerical value, there is a
graphic method for estimating
dispersion.
Range and Quartile Deviation
measure the dispersion by calculating
the spread within which the values lie.
Mean Deviation and Standard
Deviation calculate the extent to
which the values differ from the
average.
2. MEASURES BASED UPON SPREAD OF
VALUES
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
Activities
Look at the following values:
20, 30, 40, 50, 200
? Calculate the Range.
? What is the Range if the value
200 is not present in the data
set?
? If 50 is replaced by 150, what
will be the Range?
Range:  Comments
Range is unduly affected by extreme
values. It is not based on all the
values. As long as the minimum and
maximum values remain unaltered,
any change in other values does not
affect range. It can not be calculated
for open-ended frequency distri-
bution.
Notwithstanding some limitations,
Range is understood and used
frequently because of its simplicity.
For example, we see the maximum
and minimum temperatures of
different cities almost daily on our TV
screens and form judgments about the
temperature variations in them.
Open-ended distributions are those
in which either the lower limit of the
lowest class or the upper limit of the
highest class or both are not
specified.
Activity
? Collect data about 52-week
high/low of 10 shares from a
newspaper. Calculate the range
of share prices.  Which stock is
most volatile and which is the
most stable?
Quartile Deviation
The presence of even one extremely
high or low value in a distribution can
reduce the utility of range as a
measure of dispersion. Thus, you may
need a measure which is not unduly
affected by the outliers.
In such a situation, if the entire
data is divided into four equal parts,
each containing 25% of the values, we
get the values of Quartiles and
Median.  (You have already read about
these in Chapter 5).
The upper and lower quartiles (Q
3
and Q
1
, respectively) are used to
calculate Inter Quartile Range which
is Q
3 
– Q
1
.
Inter-Quartile Range is based
upon middle 50% of the values in a
distribution and is, therefore, not
affected by extreme values.  Half of
the Inter-Quartile Range is called
Quartile Deviation. Thus:
Q.D . = 
Q - Q
2
31
Q.D. is therefore also called Semi-
Inter Quartile Range.
Calculation of Range and Q.D. for
ungrouped data
Example 1
Calculate Range and Q.D. of the
following observations:
20, 25, 29, 30, 35, 39, 41,
48, 51, 60 and 70
Range is clearly 70 – 20 = 50
For Q.D., we need to calculate
values of Q
3
 and Q
1
.
MEASURES OF DISPERSION 77
Q
1
 is the size of   
n
th
+1
4
 value.
n being 11, Q
1
 is the size of 3rd
value.
As the values are already arranged
in ascending order, it can be seen that
Q
1
, the 3rd value is 29. [What will you
do if these values are not in an order?]
Similarly, Q
3
 is size of    
31
4
() n
th
+
value; i.e. 9th value which is 51. Hence
Q
3    
= 51
Q.D . = 
Q - Q
2
31
 = 
51 29
2
11
-
=
Do you notice that Q.D. is the
average difference of the Quartiles
from the median.
Activity
? Calculate the median and check
whether the above statement is
correct.
Calculation of Range and Q.D. for a
frequency distribution.
Example 2
For the following distribution of marks
scored by a class of 40 students,
calculate the Range and Q.D.
TABLE 6.1
Class intervals No. of students
 C I (f)
 0–10 5
10–20 8
20–40 16
40–60 7
60–90 4
40
Range is just the difference
between the upper limit of the highest
class and the lower limit of the lowest
class. So Range is 90 – 0 = 90. For
Q.D., first calculate cumulative
frequencies as follows:
Class- Frequencies Cumulative
Intervals Frequencies
CI f c. f.
  0–10 5 05
10–20 8 13
20–40 16 29
40–60 7 36
60–90 4 40
       n = 40
Q
1
 is the size of 
nth
4
 value in a
continuous series. Thus it is the size
of the 10th value. The class containing
the 10th value is 10–20. Hence Q
1 
lies
in class 10–20. Now, to calculate the
exact value of Q
1
, the following
formula is used:
QL
n
cf
f
i
1
4
=+ ·
Where L = 10  (lower limit of the
relevant Quartile class)
c.f. = 5 (Value of c.f. for the class
preceding the Quartile class)
i = 10 (interval of the Quartile
class), and
f = 8 (frequency of the Quartile
class) Thus,
Q
1
10
10 5
8
10 16 25 =+
-
·= .
Similarly, Q
3
 is the size of 
3
4
nth
78 STATISTICS FOR ECONOMICS
value; i.e., 30th value, which lies in
class  40–60. Now using the formula
for Q
3
, its value can be calculated as
follows:
Q = L + 
3n
4
 - c.f.
f
  i
3
Q = 40 + 
30 - 29
7
  20
3
Q = 42.87
Q.D. = 
42.87 - 16.25
2
 = 13.31
3
In individual and discrete series,  Q
1
is the size of 
nth +1
4
value, but in a
continuous distribution, it is the size
of 
nth
4
 value.  Similarly, for Q
3 
and
median also, n is used in place of
n+1.
If the entire group is divided into
two equal halves and the median
calculated for each half, you will have
the median of better students and the
median of weak students.  These
medians differ from the median of the
entire group by 13.31 on an average.
Similarly, suppose you have data
about incomes of people of a town.
Median income of all people can be
calculated.  Now if all people are
divided into two equal groups of rich
and poor, medians of both groups can
be calculated.  Quartile Deviation will
tell you the average difference between
medians of these two groups belonging
to rich and poor, from the median of
the entire group.
Quartile Deviation can generally be
calculated for open-ended distribu-
tions and is not unduly affected by
extreme values.
3. MEASURES OF DISPERSION FROM
AVERAGE
Recall that dispersion was defined as
the extent to which values differ from
their average. Range and Quartile
Deviation do not attempt to calculate,
how far the values are, from their
average. Yet, by calculating the spread
of values, they do give a good idea
about the dispersion. Two measures
which are based upon deviation of the
values from their average are Mean
Deviation and Standard Deviation.
Since the average is a central
value, some deviations are positive
and some are negative. If these are
added as they are, the sum will not
reveal anything. In fact, the sum of
deviations from Arithmetic Mean is
always zero. Look at the following two
sets of values.
Set A : 5, 9, 16
Set B : 1, 9, 20
You can see that values in Set B
are farther from the average and hence
more dispersed than values in Set A.
Calculate the  deviations from
Arithmetic Mean amd sum them up.
What do you notice? Repeat the same
with Median.  Can you comment upon
the quantum of variation from the
calculated values?
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