a5.presentation correlaton

8/6/2019 A5.Presentation Correlaton

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CorrelationThe term correlation indicates the relationship

between two such variables in which the change

in value of one variable affects the value of the

other variable


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Correlation may be classified according to following criteria

The number of variable

Degree of correlation

The direction of change


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When the change in one variable are accomplished by change in

another variable in the same direction ,it is called as +ve correlation.

If the change in one variable accomplished by change in another

variable in opposite direction it is called as -ve correlation

The correlation coefficient lies between +/- 1

In case where the correlation coefficient is +/- 1 , the correlation is

called perfect correlation

In case where the correlation coefficient is reaching to +/- 1 ,then that

is high degree imperfect correlation

In case where the correlation coefficient is reaching to `0 ` from

+ ve or ve side of the number but not exactly zero , then it is

low degree imperfect correlation


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Methods of determining correlation

i. Scatter diagram

ii. Karl pearson`s product moment coefficient

iii. Spearman`s rank correlation coefficient


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Scatter diagram

This is a graphical method for studying correlation. This

method may not give us any mathematical relation between

the two variables , it certainly helps us in visualizing the

behaviour pattern of the two variables.

The pairs of values of X and Y are represented by a dot ,

plotted on a graph paper . The graph is called scatter

diagram.


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r =1

0 < r < 1

r = - 1

-1 < r < 0


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r = 0


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0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25 30 35 40 45 50

Y

X 15 20 33 25 25 35 36 40 18 22

Y 5 15 23 15 20 28 30 40 10 15

Draw a scatter diagram for the following data and comment on it.

On observation we

can conclude that

there is a positive

correlation between

X and Y


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Karl pearson`s product moment coefficient

The coefficient gives numerical measure of nature

and extent of correlation.

It is a pure number independent of the units of

measurement of X and Y .

It always lies between -1 & +1 .

It is independent of change of origin and scale.

It is defined as

r = Cov ( x, y) /( S.D of X * S.D of Y)


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The following data represents the time in weeks (X) and out put of units (Y) of a

factory. Fine the coefficient of correlation and interpret it.

X 7 5 4 11 10 12 14 9

Y 14 8 8 19 16 19 20 16

Find the S.D of X

Find the S.D of Y

Apply in the formula


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X Y X-mean (x) y-mean (y)(X-mean (x)) *(

y-mean (y))

2

X-mean (x)

2

y-mean (y)

7 14 -2 -1 2 4 1

5 8 -4 -7 28 16 49

4 8 -5 -7 35 25 49

11 19 2 4 8 4 16

10 16 1 1 1 1 1

12 19 3 4 12 9 16

14 20 5 5 25 25 25

9 16 0 1 0 0 1

72 120 0 0 111 84 158

n=8

mean= 9 15

The following data represents the time in weeks (X) and out put of units (Y) of a

factory. Fine the coefficient of correlation and interpret it.

X 7 5 4 11 10 12 14 9

Y 14 8 8 19 16 19 20 16

Find the S.D of X

Find the S.D of Y

Apply in the formula


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X Y2

X

2

YXY

7 14 49 196 98

5 8 25 64 40

4 8 16 64 32

11 19 121 361 209

10 16 100 256 160

12 19 144 361 228

14 20 196 400 280

9 16 81 256 144

72 120 732 1958 1191

Substitute the values

in the formula


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some times there are certain characteristics which are qualitative in

nature and they cannot be measured numerically. We can rank the

individual according to these characteristic in ascending or descending

order, and these ranks provide the data to calculate spearman`s rank

correlation coefficient which is derived from karl pearson`s coefficent.

where d represents difference

between ranks i.e. d = R1 R2,

(R1 & R2 are ranks assigned for

characteristics )

The formula forrank correlation coefficient is

R =6 d

2

n (n 1)21 -

& n = no of pairs of observations

The values of X and Y can be ranked as first, second ,third

and so on and then the formula can be applied

Spearman`s rank correlation


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Iftwo or more observations have the same value then common

rank by considering the average can be given to all repeated values.

A correction factor is to be added while calculating the rank

correlation coefficient

m ( m 1)2

12C.F = Where m is number of times a rank is repeated

So rank correlation coefficient is given by

R =6 ( d + C.F )

2

n (n 1)2

1 -


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Ten girls participate for a beauty pageant. Their rank for beauty and

intelligence are as follows:

(1,7),(2,9),(3,2),(4,10),(5,1),(6,4),(7,8),(8,5)(9,3),(10,6)

Find their rank correlation coefficient.

Rank in

beauty

Rank in

intelligence

d = 2

d

R 1 R 2 R1 R2

1 7 -6 36

2 9 -7 49

3 2 1 1

4 10 -6 36

5 1 4 16

6 4 2 4

7 8 -1 1

8 5 3 9

9 3 6 36

10 6 4 16

0 204

n =10

Apply the values in the formula

R =6 d2

n (n 1)21 -


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Find the spearman`s rank correlation coefficient for the following data

exports( in lacks Rs.) x 12 15 13 20 15 14 19 13 21 18

local sales (in lakhs ofRs.) y 25 21 15 18 20 17 20 16 20 22

Let R1 , R2 be the ranks assigned to X & Y .

Say for R1 (x) : assign 1 to the highest value i.e. 21 and 2,3, & 4 tovalues 20,19, & 18 respectively .

The next value 15 is repeated twice , and the ranks are (5,6 )/2 i.e 5.5

to both repetitions .

Like this the last value 12 is given rank 10

Do the ranking R2 for (Y)


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X Y R1 R2 d= R1- R2

2

d

12 25 10 1 9 8115 21 5.5 3 2.5 6.25

13 15 8.5 10 -1.5 2.25

20 18 2 7 -5 25

15 20 5.5 5 0.5 0.25

14 17 7 8 -1 1

19 20 3 5 -2 413 16 8.5 9 -0.5 0.25

21 20 1 5 -4 16

18 22 4 2 2 4

140

C.F for R1 ( FOR RANK 5.5 & 8.5

BOTH REPEATED 2 TIMES )

C.F for R2 ( FOR RANK 5

REPEATED 3 TIMES)

Total C.FApply in the formula

R =6 ( d + C.F )

2

n (n 1)2

1 -

n = 10

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