SLMNA1-11 EcoB 07 Correlation Goutam

8/14/2019 SLMNA1-11 EcoB 07 Correlation Goutam

1/50

CLASS XI

ECONOMICS

Basic Concepts

CORRELATION


2/50

Introduction

Many times, we come across problems which involve

two or more variables.

Example: Rainfall and production of rice

Road accidents and number of cars

Sales and profit etc.


3/50

We find that there is always some relationship between

the two variables.

When one variable changes, the other also changes in

the same or in the opposite direction, we say that the

two variables are correlated.


4/50

A single number that describes the degree of relationship

between two variables.

Correlation

Relation between income and

consumption. With rise in

income consumption increasesand vice-versa


5/50

Determines the degree of relationship between

variables. By knowing one variable other

variables can be chalked out.

Helps in measuring the relationship between

the two variables

Significance


6/50

Helps in formation of laws and concepts in economictheory

Economists establish relationship between the

variables like demand and supply, price level etc.


7/50

Helps in framing policies

Helps in business activities to take fruitful decisions


8/50

REMEMBER

Correlationdoe

sn'tmeasurec

auseandeffect

relation.

Itmeasuresonly

degreeandinte

nsityof

relationship.Exampl

e:Lowrainfallisrelated

tolow

agriculturalprod

uctivity.Butlow

production

maybeduetoo

therreasonssu

chaspoor

qualityofseeds,

traditionalmetho

dsof

agricultureetc.


9/50

Negative correlationPositive correlation

Types

OX

Y Y

OX


10/50

Positive correlation: Variables move together insame direction

Example: Advertising and sales

Negative correlation: Variables move in opposite

direction

Example: Higher the price of petrol less will be its

demand


11/50

The value of correlation (r) always lies between1 to + 1 ( 1 < r < + 1).

Value of r lies between 0 and 1 Positive correlation

Value of r lies between 0 and -1 Negative

correlation

r = 0 No correlation

Properties

-1 0 +1

Perfect

Negative

Correlation

No Correlation Perfect

Positive

Correlation


12/50

Methods of Studying Correlation

Scatter Diagram Spearmans Rank

Correlation

Karl Pearsons

Coefficient of

Correlation


13/50

Simplest way of determining the relationship between two

variable in a special type ofdotted chart

Scatter Diagram

Y

XO


14/50

1

2

3

4

5

Positive Correlation

Negative Correlation

Perfect Negative Correlation

Perfect Positive Correlation

No Correlation

Types of Scatter Diagram


15/50

As the value of one variable increases, the

value of other variable also increases.

Example: Temperature and sale of cold drinks.

Positive Correlation


16/50

Example: Watching TV and marks scored.

Students spending more time watching TV tend to

score less marks in class and vice-versa.

As the value of one variable increases, the value ofother variable also decreases.

or

Two variables move in different direction

Negative Correlation


17/50

Change in two variables in equal proportion in

the same direction

Y

XO

Perfect Positive Correlation


18/50

Change in two variables in equal proportion

in an inverse direction

Perfect Negative Correlation

Y

XO


19/50

The points are randomly scattered on graph

Example: Rainfall in India and production of cars in

Germany

Ra

infa

ll

Production of cars

No Correlation


20/50

Your Turn

Following are the details of heights and weights of 5students of a class, draw a scatter diagram and

determine the form of association.

Weight(Kg)

50 65 60 50

Height(inches)

62 72 70 58


21/50

A mathematical method for

measuring the linear relationship

between the variable X and Y

It indicates the quantitative

relationship between two

variables.Karl Pearson, a pioneer of statistics,

developed ideas of correlation and

regression that have been widely

applied across different branches of

science.

Karl Pearsons Coefficient of

Correlation


22/50

1. Direct method

2. Indirect method

2 2

=

xyr

x y

( ) ( )

( ) ( )2 2

2 2

=

dx dy dxdy

Nr

dx dy dx dy

N N

Methods of Calculation


23/50

Example: Calculate the correlation between the

weights and heights of 9 students by directmethod.

Weight(Kg)

48 49 50 51 52 53 54 55 56

Height(cm)

100 105 105 104 110 115 125 130 132


24/50

Weight (X) Height (Y)

48

49

50

51

52

53

54

55

56

100

105

105

104

110

115

125

130

132

Calculate mean for X and Y

468

9

52

XX

N

=

=

=

1026

9

114

YY

N

=

=

=

468X= 1026Y=


25/50

X Y

48

49

50

51

52

53

54

55

56

100

105

105

104

110

115

125

130

132

- 4

- 3

- 2

- 1

0

1

2

3

4

- 14

- 9

- 9

- 10

- 4

1

11

16

18

56

27

18

10

0

1

22

48

72

16

9

4

1

0

1

4

9

16

196

81

81

100

16

1

121

256

324

x y

254xy=

114

,

X = 52

=

Here

Y

= x X X = y Y Y2y2x

21176y =2 60x =


26/50

2 2

254

60 1176

95

r

.

xy

x y

=

=

=

Positive correlation between weight and height


27/50

Example: Calculate Karl Pearsons correlation

between price and demand by indirect method.

Price (Rs) 14 16 17 18 19 20 21 22 23

Demand(Quantity)

84 78 70 75 66 67 62 58 56


28/50

Price(X)

Demand(Y)

14

16

17

18

19

20

21

22

23

84

78

70

75

66

67

62

58

56

- 6

- 4

- 3

- 2

-1

0

1

2

3

14

8

0

5

- 4

-3

-8

-12

-14

20( )

dx X A

A

=

= 70( )

dy Y A

A

=

=

12d x=

14d y=


29/50

- 6

- 4

- 3

- 2

-1

0

1

2

3

14

8

0

-5

- 4

-3

-8

-12

-14

36

16

9

4

1

0

1

4

9

196

64

0

25

16

9

64

144

196

-84

-32

0

10

4

0

-8

-24

-42

2

8 0d x=2

714d y = 176.d yd y=

dx X A= dy Y A=

14d y= 12d x=

2d x 2d y .dx dy


30/50

( ) ( )

( ) ( )

2 2

2 2

2 2

12 14176

9

12 1480 7149 9

. -

( ) ( )( )

dx dy

dx dy Nr

dx dy dx dy

N N

r

=

=


31/50

176 (18.66)

144 19680 714

9 9

176 (18.66)

64 692.33

194.66

8 26.31

194.66210.48

.92

.

r

r

r

r

r

Itisacaseofstrongnegativecorrelation

=

=

=

=

=


32/50

Calculate correlation of coefficient between

variable X and Y.

Your Turn

Variable (X) 6 2 4 9 1 3 5 8

Variable (Y) 13 8 12 15 9 10 11 16


33/50

Charles Edward Spearman, an

English psychologist Known for

work in statistics and for

Spearman's rank correlation

coefficient.

Helps in calculating the correlationofqualitative variables

Based on ranks of items rather than

their actual values

Can be used even when actual

values are unknown

Spearmans Rank Correlation

Example: To know the correlation

between honesty and wisdom, one

can use this method by assigning

ranks to items.


34/50

When ranks are given

When ranks are not given

2

3

61k Dr

N N=

Formula for Different Cases

2

N Number of pairs of observations

D Total of squares of the

differences of corresponding ranks

=

=


35/50

2 3 3

1 1 2 2

3

1 1

612 12

1

( ) ( ) ....

k

D m m m mr

N N

m Number items of equal ranks

+ + + =

=

When ranks are equal or repeated

Formula for Different Cases


36/50

In a singing competition, two judges accorded following

ranks to 10 contestant.

Judge A 10 8 5 3 6 1 2 9 7 4

Judge B 10 6 5 4 7 9 8 2 1 3

When ranks are given

JUDGE AJUDGE BD =R1 R2 D2


37/50

1 2

10

8

5

3

6

1

2

9

7

4

10

6

5

4

7

9

8

2

1

3

0

2

0

-1

-1

-8

-6

7

6

1

0

4

0

1

1

64

36

49

36

1

2

192D =

Here,

R1=Row 1

R2= Row 2

2


38/50

2

3

3

61

6 1921

10 10

11521

990

1 1 16

0 16

( )

.

.

k

Dr

N N=

=

=

=

=


39/50

Calculate Spearmans coefficient of correlation between

marks assigned to 7 students by tow judges in a poem

competition.

Judge A 25 12 40 20 8 15 10

Judge B 12 10 18 16 6 25 15

When ranks are not given


40/50

Judge A Rank

(R1)

JudgeB

Rank

(R2) D = R1- R2 D2

25

12

40

20

8

15

10

2

5

1

3

7

4

6

12

10

18

16

6

25

15

5

6

2

3

7

1

4

-3

-1

-1

0

0

3

2

9

1

1

0

0

9

4

224D =


41/50

3

3

3

61

6(24)1

7 7

1441

7 7

1441

336

k

Dr

N N=

=

=

=


42/50

k

1 0.43

r 0.57

Positive correlation

=

=


43/50

Your Turn

Marks of 10 students in Hindi and English are given below.

Find the correlation between the two subjects.

Hindi 80 38 95 30 74 84 91 60 66 40

English 85 50 92 58 70 65 88 56 52 46


44/50

Marks of 7 students in accounts and statistics out

of 50 marks are given.

Subject Marks

Accounts 40 42 35 40 47 42 30

Statistics 38 45 42 35 30 40 35

When ranks are equal or repeated


45/50

Accounts Statistics R1 R2 D = R1- R2 D2

37

42

35

40

47

42

30

38

45

42

35

30

40

35

3

5.5

2

4

7

5.5

1

4

7

6

2.5

1

5

2.5

-1

-1.5

-4

1.5

6

0.5

-1.5

1

2.25

16

2.25

36

0.25

2.25

260D =


46/50

2 3 3

1 1 2 2

3

3 3

3

3

1 16

12 121

1 16 60 2 2 2 2

12 121

7 7

1 16 60 6 6

12 121

7 7

( ) ( ) ....

,

( ) ( ) ....

( ) ( ) ....

k

D m m m m

r

N N

Here m number of items of equal ranks

+ + + =

=

+ + + =

+ + + =


47/50

[ ]

1 16 60

2 21

336

6 611

336

3661

336

1 1 08

0 08

.

.r

+ + =

=

=

=

=

Negative correlation

N FINGER TIPS


48/50

Now, on your FINGER TIPS

Qualitative Variables:Those variables which cannot

be measured such as bravery, wisdom, beauty etc.

Correlation: A single number that describes the

degree of relationship between two variables. When

both the variables move in same direction they are

said to the positively correlated and when move in

opposite direction, it is called negative correlation.

Scatter Diagram: It is a graphic method of studying

correlation.


49/50

Ranking: Allotment of rank on the basis of ascending

or descending order

Negative correlation:When the two variables movein opposite direction, it is called negative correlation.With an increase in the value of one variable there is

a decrease in value of other.


50/50

A presentation byA presentation by

Documents

SLMNA1-11 EcoB 07 Correlation Goutam