NCERT Textbook - Measures of Dispersion | NCERT Books & Solutions for Commerce PDF Download

 Page 1


As the summer heat rises, hill
stations, are crowded with more and
more visitors. Ice-cream sales become
more brisk. Thus, the temperature is
related to number of visitors and sale
of ice-creams. Similarly, as the supply
of tomatoes increases in your local
mandi, its price drops. When the local
harvest starts reaching the market,
the price of tomatoes drops from Rs 40
per kg to Rs 4 per kg or even less. Thus
supply is related to price. Correlation
analysis is a means for examining such
relationships systematically. It deals
with questions such as:
• Is there any relationship between
two variables?
Correlation
1.  INTRODUCTION
In previous chapters you have learnt
how to construct summary measures
out of a mass of data and changes
among similar variables. Now you will
learn how to examine the relationship
between two variables.
Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should
enable you to: enable you to: enable you to: enable you to: enable you to:
• understand the meaning of the
term correlation;
• understand the nature of
relationship  between two
variables;
• calculate  the different measures
of correlation;
• analyse the degree and direction
of the relationships.
CHAPTER
2024-25
Page 2


As the summer heat rises, hill
stations, are crowded with more and
more visitors. Ice-cream sales become
more brisk. Thus, the temperature is
related to number of visitors and sale
of ice-creams. Similarly, as the supply
of tomatoes increases in your local
mandi, its price drops. When the local
harvest starts reaching the market,
the price of tomatoes drops from Rs 40
per kg to Rs 4 per kg or even less. Thus
supply is related to price. Correlation
analysis is a means for examining such
relationships systematically. It deals
with questions such as:
• Is there any relationship between
two variables?
Correlation
1.  INTRODUCTION
In previous chapters you have learnt
how to construct summary measures
out of a mass of data and changes
among similar variables. Now you will
learn how to examine the relationship
between two variables.
Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should
enable you to: enable you to: enable you to: enable you to: enable you to:
• understand the meaning of the
term correlation;
• understand the nature of
relationship  between two
variables;
• calculate  the different measures
of correlation;
• analyse the degree and direction
of the relationships.
CHAPTER
2024-25
CORRELATION 75
• It the value of one variable changes,
does the value of the other also
change?
given a cause and effect interpretation.
Others may be just coincidence.  The
relation between the  arrival of
migratory birds in a sanctuary and  the
birth rates in the locality  cannot be
given any cause and effect
interpretation. The relationships are
simple coincidence. The relationship
between size of the shoes and money
in your pocket is another such
example. Even if relationships exist,
they are difficult to explain it.
In another instance a third
variable’s impact on two variables
may give rise to a relation between the
two variables. Brisk sale of ice-creams
may be related to higher number of
deaths due to drowning. The victims
are not drowned due to eating of ice-
creams. Rising temperature leads to
brisk sale of ice-creams. Moreover, large
number of people start going to
swimming pools to beat the heat. This
might have raised the number of deaths
by drowning. Thus, temperature is
behind the  high correlation between
the sale of ice-creams and deaths due
to drowning.
What Does Correlation Measure?
Correlation studies and measures
the direction and intensity of
relationship among variables.
Correlation measures covariation, not
causation. Correlation should never be
interpreted as implying cause and
effect  relation. The  presence  of
correlation between two variables  X
and Y simply means that when the
value of one variable is found to change
in one direction, the value of the other
• Do both the variables move in the
same direction?
• How strong is the relationship?
2. TYPES OF RELATIONSHIP
Let us look at various types of
relationship. The relation between
movements  in quantity demanded and
the price of a commodity is an integral
part of the theory of demand, which you
will study in Class XII. Low agricultural
productivity is related to low rainfall.
Such examples of relationship may be
2024-25
Page 3


As the summer heat rises, hill
stations, are crowded with more and
more visitors. Ice-cream sales become
more brisk. Thus, the temperature is
related to number of visitors and sale
of ice-creams. Similarly, as the supply
of tomatoes increases in your local
mandi, its price drops. When the local
harvest starts reaching the market,
the price of tomatoes drops from Rs 40
per kg to Rs 4 per kg or even less. Thus
supply is related to price. Correlation
analysis is a means for examining such
relationships systematically. It deals
with questions such as:
• Is there any relationship between
two variables?
Correlation
1.  INTRODUCTION
In previous chapters you have learnt
how to construct summary measures
out of a mass of data and changes
among similar variables. Now you will
learn how to examine the relationship
between two variables.
Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should
enable you to: enable you to: enable you to: enable you to: enable you to:
• understand the meaning of the
term correlation;
• understand the nature of
relationship  between two
variables;
• calculate  the different measures
of correlation;
• analyse the degree and direction
of the relationships.
CHAPTER
2024-25
CORRELATION 75
• It the value of one variable changes,
does the value of the other also
change?
given a cause and effect interpretation.
Others may be just coincidence.  The
relation between the  arrival of
migratory birds in a sanctuary and  the
birth rates in the locality  cannot be
given any cause and effect
interpretation. The relationships are
simple coincidence. The relationship
between size of the shoes and money
in your pocket is another such
example. Even if relationships exist,
they are difficult to explain it.
In another instance a third
variable’s impact on two variables
may give rise to a relation between the
two variables. Brisk sale of ice-creams
may be related to higher number of
deaths due to drowning. The victims
are not drowned due to eating of ice-
creams. Rising temperature leads to
brisk sale of ice-creams. Moreover, large
number of people start going to
swimming pools to beat the heat. This
might have raised the number of deaths
by drowning. Thus, temperature is
behind the  high correlation between
the sale of ice-creams and deaths due
to drowning.
What Does Correlation Measure?
Correlation studies and measures
the direction and intensity of
relationship among variables.
Correlation measures covariation, not
causation. Correlation should never be
interpreted as implying cause and
effect  relation. The  presence  of
correlation between two variables  X
and Y simply means that when the
value of one variable is found to change
in one direction, the value of the other
• Do both the variables move in the
same direction?
• How strong is the relationship?
2. TYPES OF RELATIONSHIP
Let us look at various types of
relationship. The relation between
movements  in quantity demanded and
the price of a commodity is an integral
part of the theory of demand, which you
will study in Class XII. Low agricultural
productivity is related to low rainfall.
Such examples of relationship may be
2024-25
76 STATISTICS FOR ECONOMICS
variable is found to change either in the
same direction (i.e. positive change) or
in the opposite direction (i.e. negative
change), but in a definite way. For
simplicity we assume here that the
correlation, if it exists, is linear, i.e. the
relative movement of the two variables
can be represented by drawing a
straight line on graph paper.
Types of Correlation
Correlation is commonly classified
into negative and positive
correlation. The correlation is said to
be positive when the variables move
together in the same direction. When
the income rises, consumption also
rises. When income falls,
consumption also falls. Sale of ice-
cream and temperature move in the
same direction. The correlation is
negative when they move in opposite
directions. When the price of apples
falls its demand increases. When the
prices rise its demand decreases.
When you spend more time in
studying,  chances of your failing
decline. When you spend less hours
in your studies, chances of scoring
low marks/grades increase. These
are instances of negative correlation.
The variables move in opposite
direction.
3. TECHNIQUES FOR MEASURING
CORRELATION
Three important  tools used to study
correlation are scatter diagrams, Karl
Pearson’s coefficient of correlation and
Spearman’s rank correlation.
A scatter diagram visually presents
the nature of association without giving
any specific numerical value.  A
numerical measure of linear
relationship between two variables is
given by Karl Pearson’s coefficient of
correlation.  A relationship is said to
be linear if it can be represented
by a straight line. Spearman’s
coefficient of correlation measures the
linear association between ranks
assigned to indiviual items according
to their attributes. Attributes are those
variables which cannot be numerically
measured such as intelligence of
people, physical appearance, honesty,
etc.
Scatter Diagram
A scatter diagram is a useful
technique for visually examining the
form of relationship, without
calculating any numerical value. In
this technique, the values of the two
variables are plotted as points on a
graph paper. From a scatter diagram,
one can get a fairly good idea of the
nature of relationship. In a scatter
diagram the degree of closeness of the
scatter points and their overall direction
enable us to examine the relation-
ship. If all the points lie on a line, the
correlation is perfect and is said to be
in unity. If the scatter points are widely
dispersed around the line, the
correlation is low. The correlation is
said to be linear if the scatter points lie
near a line or on a line.
Scatter diagrams spanning over
Fig. 6.1 to Fig. 6.5  give us an idea of
2024-25
Page 4


As the summer heat rises, hill
stations, are crowded with more and
more visitors. Ice-cream sales become
more brisk. Thus, the temperature is
related to number of visitors and sale
of ice-creams. Similarly, as the supply
of tomatoes increases in your local
mandi, its price drops. When the local
harvest starts reaching the market,
the price of tomatoes drops from Rs 40
per kg to Rs 4 per kg or even less. Thus
supply is related to price. Correlation
analysis is a means for examining such
relationships systematically. It deals
with questions such as:
• Is there any relationship between
two variables?
Correlation
1.  INTRODUCTION
In previous chapters you have learnt
how to construct summary measures
out of a mass of data and changes
among similar variables. Now you will
learn how to examine the relationship
between two variables.
Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should
enable you to: enable you to: enable you to: enable you to: enable you to:
• understand the meaning of the
term correlation;
• understand the nature of
relationship  between two
variables;
• calculate  the different measures
of correlation;
• analyse the degree and direction
of the relationships.
CHAPTER
2024-25
CORRELATION 75
• It the value of one variable changes,
does the value of the other also
change?
given a cause and effect interpretation.
Others may be just coincidence.  The
relation between the  arrival of
migratory birds in a sanctuary and  the
birth rates in the locality  cannot be
given any cause and effect
interpretation. The relationships are
simple coincidence. The relationship
between size of the shoes and money
in your pocket is another such
example. Even if relationships exist,
they are difficult to explain it.
In another instance a third
variable’s impact on two variables
may give rise to a relation between the
two variables. Brisk sale of ice-creams
may be related to higher number of
deaths due to drowning. The victims
are not drowned due to eating of ice-
creams. Rising temperature leads to
brisk sale of ice-creams. Moreover, large
number of people start going to
swimming pools to beat the heat. This
might have raised the number of deaths
by drowning. Thus, temperature is
behind the  high correlation between
the sale of ice-creams and deaths due
to drowning.
What Does Correlation Measure?
Correlation studies and measures
the direction and intensity of
relationship among variables.
Correlation measures covariation, not
causation. Correlation should never be
interpreted as implying cause and
effect  relation. The  presence  of
correlation between two variables  X
and Y simply means that when the
value of one variable is found to change
in one direction, the value of the other
• Do both the variables move in the
same direction?
• How strong is the relationship?
2. TYPES OF RELATIONSHIP
Let us look at various types of
relationship. The relation between
movements  in quantity demanded and
the price of a commodity is an integral
part of the theory of demand, which you
will study in Class XII. Low agricultural
productivity is related to low rainfall.
Such examples of relationship may be
2024-25
76 STATISTICS FOR ECONOMICS
variable is found to change either in the
same direction (i.e. positive change) or
in the opposite direction (i.e. negative
change), but in a definite way. For
simplicity we assume here that the
correlation, if it exists, is linear, i.e. the
relative movement of the two variables
can be represented by drawing a
straight line on graph paper.
Types of Correlation
Correlation is commonly classified
into negative and positive
correlation. The correlation is said to
be positive when the variables move
together in the same direction. When
the income rises, consumption also
rises. When income falls,
consumption also falls. Sale of ice-
cream and temperature move in the
same direction. The correlation is
negative when they move in opposite
directions. When the price of apples
falls its demand increases. When the
prices rise its demand decreases.
When you spend more time in
studying,  chances of your failing
decline. When you spend less hours
in your studies, chances of scoring
low marks/grades increase. These
are instances of negative correlation.
The variables move in opposite
direction.
3. TECHNIQUES FOR MEASURING
CORRELATION
Three important  tools used to study
correlation are scatter diagrams, Karl
Pearson’s coefficient of correlation and
Spearman’s rank correlation.
A scatter diagram visually presents
the nature of association without giving
any specific numerical value.  A
numerical measure of linear
relationship between two variables is
given by Karl Pearson’s coefficient of
correlation.  A relationship is said to
be linear if it can be represented
by a straight line. Spearman’s
coefficient of correlation measures the
linear association between ranks
assigned to indiviual items according
to their attributes. Attributes are those
variables which cannot be numerically
measured such as intelligence of
people, physical appearance, honesty,
etc.
Scatter Diagram
A scatter diagram is a useful
technique for visually examining the
form of relationship, without
calculating any numerical value. In
this technique, the values of the two
variables are plotted as points on a
graph paper. From a scatter diagram,
one can get a fairly good idea of the
nature of relationship. In a scatter
diagram the degree of closeness of the
scatter points and their overall direction
enable us to examine the relation-
ship. If all the points lie on a line, the
correlation is perfect and is said to be
in unity. If the scatter points are widely
dispersed around the line, the
correlation is low. The correlation is
said to be linear if the scatter points lie
near a line or on a line.
Scatter diagrams spanning over
Fig. 6.1 to Fig. 6.5  give us an idea of
2024-25
CORRELATION 77
the relationship between two variables.
Fig. 6.1 shows a scatter around an
upward rising line indicating the
movement of the variables in the same
direction. When X rises Y will also rise.
This is positive correlation. In Fig. 6.2
the points are found to be scattered
around a  downward sloping line.  This
time the variables move in opposite
directions. When X rises Y falls and vice
versa. This is negative correlation.  In
Fig.6.3 there is no upward rising or
downward sloping line around which
the  points are scattered. This is an
example of no correlation. In Fig. 6.4
and Fig. 6.5, the points are no longer
scattered around an upward rising or
downward falling line. The points
themselves are on the lines. This is
referred to as perfect positive correlation
and perfect negative correlation
respectively.
Activity Activity Activity Activity Activity
• Collect data on height, weight
and marks scored by students
in your class in any two subjects
in class X. Draw  the scatter
diagram of these variables taking
two at a time. What type of
relationship do you find?
A careful observation of the scatter
diagram gives an idea of  the nature
and intensity of the relationship.
Karl Pearson’s Coefficient of
Correlation
This is also known as product moment
correlation coefficient or simple
correlation coefficient. It gives a precise
numerical value of the degree of linear
relationship between two variables X
and Y.
It is important to note that Karl
Pearson’s coefficient of correlation
should be used only when there is a
linear relation between the variables.
When there is a non-linear relation
between X and Y, then calculating the
Karl Pearson’s coefficient of correlation
can be misleading. Thus, if the true
relation is of the linear type as shown
by the scatter diagrams in figures 6.1,
6.2, 6.4 and 6.5, then the Karl
Pearson’s coefficient of correlation
should be calculated and it will tell us
the direction and intensity of the
relation between the variables. But if
the true relation is of the type shown in
the scatter diagrams in Figures 6.6 or
6.7, then it means there is a non-linear
relation between X and Y and we should
not try to use the Karl Pearson’s
coefficient of correlation.
It is, therefore, advisable to first
examine the scatter diagram of the
relation between the variables before
calculating the Karl Pearson’s
correlation coefficient.
Let X
1
, X
2
, ..., X
N
 be N values of X
and Y
1
, Y
2 
,..., Y
N 
 be the corresponding
values of Y. In the subsequent
presentations, the subscripts indicating
the unit are dropped for the sake of
simplicity. The arithmetic means of X
and Y are defined as
X Y
X ; Y
N N
 Â
= =
2024-25
Page 5


As the summer heat rises, hill
stations, are crowded with more and
more visitors. Ice-cream sales become
more brisk. Thus, the temperature is
related to number of visitors and sale
of ice-creams. Similarly, as the supply
of tomatoes increases in your local
mandi, its price drops. When the local
harvest starts reaching the market,
the price of tomatoes drops from Rs 40
per kg to Rs 4 per kg or even less. Thus
supply is related to price. Correlation
analysis is a means for examining such
relationships systematically. It deals
with questions such as:
• Is there any relationship between
two variables?
Correlation
1.  INTRODUCTION
In previous chapters you have learnt
how to construct summary measures
out of a mass of data and changes
among similar variables. Now you will
learn how to examine the relationship
between two variables.
Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should
enable you to: enable you to: enable you to: enable you to: enable you to:
• understand the meaning of the
term correlation;
• understand the nature of
relationship  between two
variables;
• calculate  the different measures
of correlation;
• analyse the degree and direction
of the relationships.
CHAPTER
2024-25
CORRELATION 75
• It the value of one variable changes,
does the value of the other also
change?
given a cause and effect interpretation.
Others may be just coincidence.  The
relation between the  arrival of
migratory birds in a sanctuary and  the
birth rates in the locality  cannot be
given any cause and effect
interpretation. The relationships are
simple coincidence. The relationship
between size of the shoes and money
in your pocket is another such
example. Even if relationships exist,
they are difficult to explain it.
In another instance a third
variable’s impact on two variables
may give rise to a relation between the
two variables. Brisk sale of ice-creams
may be related to higher number of
deaths due to drowning. The victims
are not drowned due to eating of ice-
creams. Rising temperature leads to
brisk sale of ice-creams. Moreover, large
number of people start going to
swimming pools to beat the heat. This
might have raised the number of deaths
by drowning. Thus, temperature is
behind the  high correlation between
the sale of ice-creams and deaths due
to drowning.
What Does Correlation Measure?
Correlation studies and measures
the direction and intensity of
relationship among variables.
Correlation measures covariation, not
causation. Correlation should never be
interpreted as implying cause and
effect  relation. The  presence  of
correlation between two variables  X
and Y simply means that when the
value of one variable is found to change
in one direction, the value of the other
• Do both the variables move in the
same direction?
• How strong is the relationship?
2. TYPES OF RELATIONSHIP
Let us look at various types of
relationship. The relation between
movements  in quantity demanded and
the price of a commodity is an integral
part of the theory of demand, which you
will study in Class XII. Low agricultural
productivity is related to low rainfall.
Such examples of relationship may be
2024-25
76 STATISTICS FOR ECONOMICS
variable is found to change either in the
same direction (i.e. positive change) or
in the opposite direction (i.e. negative
change), but in a definite way. For
simplicity we assume here that the
correlation, if it exists, is linear, i.e. the
relative movement of the two variables
can be represented by drawing a
straight line on graph paper.
Types of Correlation
Correlation is commonly classified
into negative and positive
correlation. The correlation is said to
be positive when the variables move
together in the same direction. When
the income rises, consumption also
rises. When income falls,
consumption also falls. Sale of ice-
cream and temperature move in the
same direction. The correlation is
negative when they move in opposite
directions. When the price of apples
falls its demand increases. When the
prices rise its demand decreases.
When you spend more time in
studying,  chances of your failing
decline. When you spend less hours
in your studies, chances of scoring
low marks/grades increase. These
are instances of negative correlation.
The variables move in opposite
direction.
3. TECHNIQUES FOR MEASURING
CORRELATION
Three important  tools used to study
correlation are scatter diagrams, Karl
Pearson’s coefficient of correlation and
Spearman’s rank correlation.
A scatter diagram visually presents
the nature of association without giving
any specific numerical value.  A
numerical measure of linear
relationship between two variables is
given by Karl Pearson’s coefficient of
correlation.  A relationship is said to
be linear if it can be represented
by a straight line. Spearman’s
coefficient of correlation measures the
linear association between ranks
assigned to indiviual items according
to their attributes. Attributes are those
variables which cannot be numerically
measured such as intelligence of
people, physical appearance, honesty,
etc.
Scatter Diagram
A scatter diagram is a useful
technique for visually examining the
form of relationship, without
calculating any numerical value. In
this technique, the values of the two
variables are plotted as points on a
graph paper. From a scatter diagram,
one can get a fairly good idea of the
nature of relationship. In a scatter
diagram the degree of closeness of the
scatter points and their overall direction
enable us to examine the relation-
ship. If all the points lie on a line, the
correlation is perfect and is said to be
in unity. If the scatter points are widely
dispersed around the line, the
correlation is low. The correlation is
said to be linear if the scatter points lie
near a line or on a line.
Scatter diagrams spanning over
Fig. 6.1 to Fig. 6.5  give us an idea of
2024-25
CORRELATION 77
the relationship between two variables.
Fig. 6.1 shows a scatter around an
upward rising line indicating the
movement of the variables in the same
direction. When X rises Y will also rise.
This is positive correlation. In Fig. 6.2
the points are found to be scattered
around a  downward sloping line.  This
time the variables move in opposite
directions. When X rises Y falls and vice
versa. This is negative correlation.  In
Fig.6.3 there is no upward rising or
downward sloping line around which
the  points are scattered. This is an
example of no correlation. In Fig. 6.4
and Fig. 6.5, the points are no longer
scattered around an upward rising or
downward falling line. The points
themselves are on the lines. This is
referred to as perfect positive correlation
and perfect negative correlation
respectively.
Activity Activity Activity Activity Activity
• Collect data on height, weight
and marks scored by students
in your class in any two subjects
in class X. Draw  the scatter
diagram of these variables taking
two at a time. What type of
relationship do you find?
A careful observation of the scatter
diagram gives an idea of  the nature
and intensity of the relationship.
Karl Pearson’s Coefficient of
Correlation
This is also known as product moment
correlation coefficient or simple
correlation coefficient. It gives a precise
numerical value of the degree of linear
relationship between two variables X
and Y.
It is important to note that Karl
Pearson’s coefficient of correlation
should be used only when there is a
linear relation between the variables.
When there is a non-linear relation
between X and Y, then calculating the
Karl Pearson’s coefficient of correlation
can be misleading. Thus, if the true
relation is of the linear type as shown
by the scatter diagrams in figures 6.1,
6.2, 6.4 and 6.5, then the Karl
Pearson’s coefficient of correlation
should be calculated and it will tell us
the direction and intensity of the
relation between the variables. But if
the true relation is of the type shown in
the scatter diagrams in Figures 6.6 or
6.7, then it means there is a non-linear
relation between X and Y and we should
not try to use the Karl Pearson’s
coefficient of correlation.
It is, therefore, advisable to first
examine the scatter diagram of the
relation between the variables before
calculating the Karl Pearson’s
correlation coefficient.
Let X
1
, X
2
, ..., X
N
 be N values of X
and Y
1
, Y
2 
,..., Y
N 
 be the corresponding
values of Y. In the subsequent
presentations, the subscripts indicating
the unit are dropped for the sake of
simplicity. The arithmetic means of X
and Y are defined as
X Y
X ; Y
N N
 Â
= =
2024-25
78 STATISTICS FOR ECONOMICS
Fig. 6.1: Positive Correlation
Fig. 6.2: Negative Correlation
Fig. 6.4: Perfect Positive Correlation
Fig. 6.6: Positive non-linear relation
Fig. 6.7: Negative non-linear relation
Fig. 6.5: Perfect Negative Correlation
Fig. 6.3: No Correlation
2024-25