slmna2-11 ecob 07 correlation goutam

8/14/2019 SLMNA2-11 EcoB 07 Correlation Goutam

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CLASS XI

ECONOMICS

Understanding Concepts

CORRELATION


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One important job of statistics is to compare differentsets of things to see if there is a possible link between

the two.

Example

To compare:

people's income with their education prices and demand of goods

weight of person with height etc.

Introduction


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These sorts of studies involve comparison betweentwo variables to see the connection.

With implementation of mid-day meal scheme,

does the frequency of attending schools increases? As price decreases, does demand for the good

increases?

Does weight increases with height?

What we are looking for, with such questions

statistically is called correlation.


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Determines the degree of relationship between

variables.

By knowing one variable other variables can be

known.

Example:If price of wheat decreases, demand for

wheat will increase.

Significance


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Helps in formation of laws and concepts in

economic theory.

Example:Law of supply, law of demand etc.

Y

XO

PriceP

Q

Quantity

Demand

Supply


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Helps in framing policies

Example:If there is positive correlation between

investment policy and development, then government

would increase investment.


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Economists establish relationship between the

variables like demand and supply, price level etc.

Helps in business activities to take profitable decisions

Example:If producer is earning profit, he will increaseproduction.


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

Positive relationship: Value of r lies between 0 and 1

Example:When income increases demand also

increases.

Negative relationship: Value of r lies between 0 and -1

Example: Price increases demand decreases and price

decreases demand increases

Properties


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No linear relation: Ifr = 0, there is no correlationbetween the two variables

Example:Size of shoes and number of children born

Perfect correlation: If the value of r = +1 or-1

-1 0 +1

Perfect

Negative

Correlation

No Correlation Perfect

Positive

Correlation


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As correlation gets close to -1, it gets stronger

Example: A correlation of - .9 is stronger than - .5

If the value of r is close to 1, it gets stronger.

Example: A correlation of .6 is stronger than .3


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REMEMBER

Negativeorpos

itivesigndoes

not

indicateanythin

gaboutstrengt

h.Itisa

symbolthatindicatesdi

rection.

Whilejudgingst

rength,justlookatthe

numberandign

oresign.


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Correlation does not mean causation

It does not measure cause and effect relationship.

It measures only degree and intensity of relationship.


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Example:Correlation between number of hours studentsdevote on study with computers and the result achieved.

It is not necessary that computer users score more marks.

Other factors, like socioeconomic status might also play avital role.

Thus there is no cause and effect relationship.


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In some cases we may be interested not just in whether

there is a correlation or not, but how strong that

correlation might be.


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Negative correlationPositive correlation

Types


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Both Increasing

Positive correlation

Variables move together in same direction i.e., if one

increases the other also increases, or vice-versa

Both Decreasing

Temperature(C0)

Demand for ice-cream

12

8

3

18

12

5

Price of petrol(Rs)

Taxi fare (Rs)

12

14

16

8

10

12


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Negative correlation

When variables move in the opposite direction i.e,

if one increases, the other decreases and vice-

versa

Price (Rs) Demand (Quantity)

5

8

10

20

18

15

Price (Rs) Demand (Quantity)

10

8

5

15

18

20


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A statistical tool for analyzing graphically therelationship between two variables.

By looking at points we obtain an estimation whether the

variables are related or not.

Scatter Diagram


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1

2

3

4

5

Positive Correlation

Negative Correlation

Perfect Negative Correlation

Perfect Positive Correlation

No Correlation

Types of Scatter

Diagram


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Variables change in the same direction.

or

Relationship between two variables thatvary together in the same direction

Example:

More education, more salary

Positive

Correlation

Education

OSalary


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Variables move in opposite direction.

When one variable increases, other

decreases and vice-versa.

Example:

Decrease in price will lead to an increasein quantity demanded.

Negative

Correlation

Pric

e

OQuantity demanded


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Proportionate change of the

variables in same direction

Example:

Amount of money collected by movie

tickets with the number of sale of tickets.

Perfect Positive

Correlation

O

TicketsSo

ld

Money collected


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Proportionate change in the variables

in opposite direction.

Example:

Speed of a car and the time it takes

to reach destination. As the speed

increases, the total time taken

decreases.

Perfect Negative

Correlation

Speed

Time

O


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When no relationship is found

between the two variables

Example:

High score in exam and weather

conditions.

No

Correlation

Marksin

exam

Temperature


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1 Simplest method of studying the relationship

between the two variables

Shows whether the relationship is positive or

negative

One can know the result in seconds after looking at

the graph

2

3

Benefits


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1 Gives an idea but not exact answer.

Shows only quantitative relationship not

qualitative

Does not measure the precise extent ofcorrelation

2

3

Drawback

s


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Let us see a better method to measurethe degree of correlation.


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Gives an exact idea about the degree of linear

relationship between the two variables

It is also known as coefficient of correlation or

product moment correlation coefficient.

Karl Pearsons Coefficient

of Correlation


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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


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Coefficient of correlation=

= =

r

x X X y Y Y

Direct method

2 2

=

xyr

x y


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It can be written as:

X

y

r = Coeficient of correlation

x = X - X

y = Y - Y

= Standard Deviation of x series = Standard Deviation of y series

N = Number of observations

r =

x y

xy

N


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Example: Calculate the correlation between

production of bread and demand for flour.

Bread 9 11 13 12 10 9 6

Flour 4 8 13 11 9 6 5


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Bread (X) Flour (Y)

9

11

13

12

10

9

6

4

8

13

11

9

6

5

7010

7

= = =

XX

N

Calculate arithmetic meanStep 1

70= X 56=Y

568

7

= = =

YY

N


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Bread (X) Wheat (Y)

9

11

13

12

10

9

6

4

8

13

11

9

6

5

70= X 56=Y

Take deviations of both the series with their

corresponding meanStep 2

( )1 0= = x X X X ( )8= =y Y Y Y

0= x 0=y

-1

1

3

2

0

-1

-4

-4

0

5

3

1

-2

-3

and sum up these deviations.


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Step 3 Square these deviations to get and2x 2 y

Bread(X)

Wheat(Y)

9

11

13

12

10

9

6

4

8

13

11

9

6

5

-1

1

3

2

0

-1

-4

-4

0

5

3

1

-2

-3

2y2x

1

1

9

4

0

1

16

16

0

25

9

1

4

9

264=

y2 32

= x

= x X X = y Y Y


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Step 4 Multiply both these deviations to get x yBread

(X)Wheat

(Y)

9

11

13

12

10

9

6

4

8

13

11

9

6

5

-1

1

3

2

0

-1

-4

-4

0

5

3

1

-2

-3

1

1

9

4

0

1

16

16

0

25

9

1

4

9

x X X = y Y Y= 2y2x x y

4

0

15

6

0

2

12

39x y=


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Step 5 Put values in theformula

39

32 64

0 86.

=

=

0.86 means positive and high degree of correlation

2 2

=

xyr

x y


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Indirect method

( ) ( )

( ) ( )2 2

2 2

dx dy dxdy

Nr

dx dy dx dy

N N

N Number of observations

=

=


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Example:Calculate the correlation between death

rate and birth rate from the following hypotheticaldata:

Year 1941 1951 1961 1971 1981 1991

Birth rate 24 26 32 33 35 30

Death rate 15 20 22 24 27 24


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Step 1Take any arbitrary value in the X series and Y

series as assumed mean (A).

Birth rate (X) Death rate (Y)

24

26

32

33

35

30

15

20

22

24

27

24

(A) (A)


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Step 2Take the deviations of both series from assumed

mean and add to get and dx dy

X Y

24

26

32

33

35

30

15

20

22

24

27

24

= dx X A = dy Y A

(A) (A)

- 9

- 6

- 1

0

2

-3

- 9

- 4

- 2

0

3

0

17= d x 12= dy


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Step 3Square the deviations and sum up to get

and2

dy

2

dx

X Y

24

26

32

33

35

30

15

20

22

24

27

24

- 9

- 6

- 1

0

2

-3

- 9

- 4

- 2

0

3

0

dx X A= dy Y A=

2

131dx =2

110dy =

2dx 2dy81

36

1

0

4

9

81

16

4

0

9

0

17d x= 12d y=


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X Y

24

26

32

33

35

30

15

20

22

24

27

24

- 9

- 6

- 1

0

2

-3

- 9

- 4

- 2

0

3

0

81

36

1

0

4

9

81

16

4

0

9

0

= dx X A = dy Y A

17= dx 12= dy2

131= dx2

110= dy

2dx 2dy .dx dy

113. = dx dy

Multiply both deviations to getStep 4 . dx dy

81

24

2

0

6

0


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( ) ( )

( ) ( )

2 2

2 2

=

dx dy dxdy

Nr

dx dy dx dy N N

Put values in the formulaStep 5

2 2

17 121136

17 12131 110

6 6

( ) ( )

( ) ( )

=

r


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204113

6289 144

131 1106 6

=

113 34

131 48 16 110 24

=

.

79

82 84 86.=


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799 10 9 27

79

84 35

0 93

. .

.

.

=

=

=

There is high degree of correlation


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Follow same steps as in case of indirect method.

The difference is that, in this method we divide all

deviations by some common value.

Step deviationmethod

( ) ( )2 2

2 2

.

=

dx dy

dx dy Nr

dx dy dx dy

N N


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Sometimes definite measurement of variables is not

possible.

Example: Variables such as leadership ability,

intelligence, beauty etc. cannot be measured inquantitative terms.

Such variables are Known as qualitative variables.

Spearmans Rank Correlation


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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


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Example:In a dancing competition, two judges gave the

following ranks to 9 contestants.

When ranks are

given

Rank

Judge A 8 7 6 3 9 2 1 5 4

Judge B 7 5 4 1 9 3 2 6 8


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JUDGE A JUDGE B8

7

6

3

9

2

1

5

4

7

5

4

1

9

3

2

6

8

Findrank differences of corresponding variablesStep 1

R1 = Row 1

R2 = Row 2

D = Rank difference of

corresponding

variables

D = R1 R2

1

2

2

2

0

-1

-1

-1

-4


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JUDGE A JUDGE B D = R1 R2

8

7

6

3

9

2

1

5

4

7

5

4

1

9

3

2

6

8

1

2

2

2

0

-1

-1

-1

-4

Step 2 Square differences (D) and add to get2

D

2

32D =

2D

1

4

4

4

0

1

1

1

16


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Step 3 Put values in theformula

2

3

61=

k

Dr

N N

3

6 32

19 9

192 1921 1

729 9 720

0 74

( )

.kr

=

= =

=


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Step 1 Assign ranks to each series by taking eitheracsending or decsending order.

English

16

10

20

30

14

Economics

25

15

10

12

16

Rank (R1 )

3

1

4

5

2

Rank (R2)

5

3

1

2

4


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Step 2 Findrank differences of corresponding variables

English R1 Economics R2

16

10

20

30

14

3

1

4

5

2

25

15

10

12

16

5

3

1

2

4

D= R1 - R2

-2

-2

3

3

-2


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English R1 Economics R2 D

16

10

20

30

14

3

1

4

5

2

25

15

10

12

16

5

3

1

2

4

-2

-2

3

3

-2

Step 3 Square the differences (D) and add to get2

D

D2

4

4

9

9

42

30=D


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Step 4 Put values in the formula

2

3

61k

Dr

N N=

3

6 301

5 5

1801

120

k

( )r =

=


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1 1 5

0 5

.

.kr

It impliesanegativecorrelation

=

=

Y T


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Calculate the Spearman's Rank Correlation of

Coefficient for from the set of data given below.

Your Turn

Height(cm)

145 183 175 168 169 170

Weight(Kg)

45 82 89 65 66 70


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Sometimes, more than one item has equal rank.

In that condition, averages of repeated ranks to eachvalues are assigned.

When ranks are equal or

repeated


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Example:Mr. Sam and Mr. Ajit after tasting 10 different

Indian food rank it as follows

Rank

Mr. Sam 10 18 14 5 7 12 6 3 10 4

Mr. Ajit 12 20 14 10 20 17 3 20 18 5

Step 1 There are two 10s.Take the mean of their ranks.


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Mr. Sam Rank (R1) Mr. Ajit Rank (R

2)

10

18

14

5

7

12

63

10

4

12

20

14

10

20

17

320

18

5

Step 1

Same with 3rd column.

2

4.5

4.5

2

2

Note:

Total allotted rankshould be equal to

total number of

items.

1

2

8

6

3

7

10

9

The mean of 4+5/2 = 4.5. Assign rank 4.5 to

both10s.

7

6

8

5

10

4

9

Step 2 Find rank differences of corresponding variables


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Step 2 Findrank differences of corresponding variables

Mr. Sam Rank (R1) Mr. Ajit Rank (R2)

10

18

14

5

7

12

6

3

10

4

4.5

1

2

8

6

3

7

10

4.5

9

12

20

14

10

20

17

3

20

18

5

7

2

6

8

2

5

10

2

4

9

D =R1 - R2

-2.5

-1

-4

0

4

-2

-3

8

0.5

0

Step 3 Square differences (D) and add to get2

D


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Mr. Sam Rank(R1)

Mr. Ajit Rank(R2)

D

10

18

14

5

7

12

6

3

10

4

4.5

1

2

8

6

3

7

10

4.5

9

12

20

14

10

20

17

3

20

18

5

7

2

6

8

2

5

10

2

4

9

-2.5

-1

-4

0

4

-2

-3

8

0.5

0

D2

Step 3

6.25

1

16

0

16

4

9

64

.25

0

2 116 5.D =


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Step 4 Put values in formula

2 3 3

1 1 2 2

3

1 1612 12

1

( ) ( ) ....

,

+ + + =

=

k

D m m m m

rN N

Here m n umber of items of equal ranks

3 3

3

1 1116 5 2 2 3 3

12 121

10 10

1 1116 5 6 24

12 121

1000 10

. ( ) ( )

. ( ) ( )

kr

+ + =

+ + =


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Now, on your FINGER TIPS


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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.


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Ranking: Allotment of rank on the basis of ascending

or descending order.

Negative correlation:When the two variables movein opposite direction then it is called negative

correlation. With an increase in the value of one

variable there is a decrease in value of other.


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