NCERT Textbook - Correlation Commerce Notes | EduRev

Statistics for Economics - Class XI

Created by: Pj Commerce Academy

Commerce : NCERT Textbook - Correlation Commerce Notes | EduRev

 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 a
princely 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
7
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
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
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 a
princely 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
7
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
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
92 STATISTICS FOR ECONOMICS
? If the value of one variable
changes, does the value of the
other also change?
• 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 read in class XII. Low
rainfall is related to low agricultural
productivity. Such examples of
relationship may be 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  can not 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
relationship 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
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 a
princely 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
7
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
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
92 STATISTICS FOR ECONOMICS
? If the value of one variable
changes, does the value of the
other also change?
• 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 read in class XII. Low
rainfall is related to low agricultural
productivity. Such examples of
relationship may be 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  can not 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
relationship 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
CORRELATION 93
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 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 study, chances of your failing
increase. These are instances of
negative correlation. The variables
move in opposite direction.
3. TECHNIQUES FOR MEASURING
CORRELATION
Widely used techniques for the study
of 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.  Another measure is
Spearman’s coefficient of correlation,
which 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. The cluster of points, so
plotted, is referred to  as a scatter
diagram. 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-
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 a
princely 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
7
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
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
92 STATISTICS FOR ECONOMICS
? If the value of one variable
changes, does the value of the
other also change?
• 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 read in class XII. Low
rainfall is related to low agricultural
productivity. Such examples of
relationship may be 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  can not 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
relationship 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
CORRELATION 93
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 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 study, chances of your failing
increase. These are instances of
negative correlation. The variables
move in opposite direction.
3. TECHNIQUES FOR MEASURING
CORRELATION
Widely used techniques for the study
of 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.  Another measure is
Spearman’s coefficient of correlation,
which 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. The cluster of points, so
plotted, is referred to  as a scatter
diagram. 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-
94 STATISTICS FOR ECONOMICS
ship. If all the points lie on a line, the
correlation is perfect and is said to be
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. 7.1 to Fig. 7.5  give us an idea of
the relationship between two
variables. Fig. 7.1 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. 7.2 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.7.3
there is no upward rising or downward
sloping line around which the  points
are scattered. This is an example of
no correlation. In Fig. 7.4 and Fig. 7.5
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
? 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?
Inspection 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 and simple correlation
coefficient. It gives a precise numerical
value of the degree of linear
relationship between two variables X
and Y. The linear relationship may be
given by
Y = a + bX
This type of relation may be
described by a straight line. The
intercept that the line makes on the
Y-axis is given by a and  the slope of
the line is given by  b. It gives the
change in   the value of Y for very small
change in the value of X. On the other
hand, if the relation cannot be
represented by a straight line as in
Y =  X
2
the  value of the coefficient will be zero.
It clearly shows that zero correlation
need not mean  absence of any type
of relation between the two variables.
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
X
N
Y
Y
N
==
SS
;
and their variances are as follows
s
2
22
2
x
XX
N
X
N
X =
-
=-
SS ()
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 a
princely 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
7
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
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
92 STATISTICS FOR ECONOMICS
? If the value of one variable
changes, does the value of the
other also change?
• 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 read in class XII. Low
rainfall is related to low agricultural
productivity. Such examples of
relationship may be 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  can not 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
relationship 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
CORRELATION 93
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 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 study, chances of your failing
increase. These are instances of
negative correlation. The variables
move in opposite direction.
3. TECHNIQUES FOR MEASURING
CORRELATION
Widely used techniques for the study
of 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.  Another measure is
Spearman’s coefficient of correlation,
which 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. The cluster of points, so
plotted, is referred to  as a scatter
diagram. 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-
94 STATISTICS FOR ECONOMICS
ship. If all the points lie on a line, the
correlation is perfect and is said to be
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. 7.1 to Fig. 7.5  give us an idea of
the relationship between two
variables. Fig. 7.1 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. 7.2 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.7.3
there is no upward rising or downward
sloping line around which the  points
are scattered. This is an example of
no correlation. In Fig. 7.4 and Fig. 7.5
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
? 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?
Inspection 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 and simple correlation
coefficient. It gives a precise numerical
value of the degree of linear
relationship between two variables X
and Y. The linear relationship may be
given by
Y = a + bX
This type of relation may be
described by a straight line. The
intercept that the line makes on the
Y-axis is given by a and  the slope of
the line is given by  b. It gives the
change in   the value of Y for very small
change in the value of X. On the other
hand, if the relation cannot be
represented by a straight line as in
Y =  X
2
the  value of the coefficient will be zero.
It clearly shows that zero correlation
need not mean  absence of any type
of relation between the two variables.
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
X
N
Y
Y
N
==
SS
;
and their variances are as follows
s
2
22
2
x
XX
N
X
N
X =
-
=-
SS ()
CORRELATION 95
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