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Correlation Quantitative A Aptitude & & Business S Statistics Correlation Correlation is the relationship that exists betw een tw o or more variables. If tw o variables are related to each other in such a w ay that change

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Correlation

Quantitative A Aptitude & & Business S Statistics

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Quantitative Aptitude & Business Statistics: Correlation 2

Correlation

  • Correlation is the relationship that

exists betw een tw o or more variables.

  • If tw o variables are related to

each other in such a w ay that change increases a corresponding change in other, then variables are said to be correlated.

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Quantitative Aptitude & Business Statistics: Correlation 3

Examples

  • Relationship between the heights

and weights.

  • Relationship between the quantum
  • f rainfall and the yield of wheat.
  • Relationship between the Price and

demand of commodity.

  • Relationship between the dose of

insulin and blood sugar.

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Quantitative Aptitude & Business Statistics: Correlation 4

Uses of Correlation

  • Economic theory and business

studies relationship between variables like price and quantity demand.

  • Correlation analysis helps in

deriving precisely the degree and the direction of such relationships.

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Quantitative Aptitude & Business Statistics: Correlation 5

  • The effect of correlation is to

reduce the range of uncertainty

  • f our prediction .
  • The prediction based on

correlation analysis will more reliable and near to reality.

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Quantitative Aptitude & Business Statistics: Correlation 6

Positive correlation

  • If both the variables are vary in

the same direction ,correlation is said to be positive .

  • If one variable increases ,the
  • ther also increases or ,if one

variable decreases ,the other also decreases ,then the tw o variables are said to be positive.

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Quantitative Aptitude & Business Statistics: Correlation 7

Negative correlation

  • If both the variables are vary in the
  • pposite direction ,correlation is

said to be Negative.

  • If one variable increases ,the other

decrease or ,if one variable decreases ,the other also increases ,then the tw o variables are said to be Negative .

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Quantitative Aptitude & Business Statistics: Correlation 8

Types of Correlation

  • Simple correlation
  • Multiple correlation
  • Partial Multiple correlation
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Quantitative Aptitude & Business Statistics: Correlation 9

Methods of studying correlation

Method of studying Correlation Graphic Algebraic

1.Karl Pearson 2.Rank method 3.Concurrent Deviation Scatter Diagram Method

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Quantitative Aptitude & Business Statistics: Correlation 10

Scatter Diagram Method

  • Scatter diagrams are used to

demonstrate correlation betw een tw o quantitative variables.

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Quantitative Aptitude & Business Statistics: Correlation 11

Scatter Plots of Data w ith Various Correlation Coefficients

Y X Y X Y X Y X Y X

r = -1 r = -Ve r = 0 r = +Ve r = 1

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Quantitative Aptitude & Business Statistics: Correlation 12

Features of Correlation Coefficient

  • Ranges betw een –1 and 1
  • The closer to –1, the stronger the

negative linear relationship

  • The closer to 1, the stronger the

positive linear relationship

  • The closer to 0, the w eaker any

positive linear relationship

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Quantitative Aptitude & Business Statistics: Correlation 13

The value of r lies betw een - 1 and +1

  • If r=0 There exists no relationship

between the variables

  • If +0.75 ≤r ≤ +1 There exists high

positive relationship between the variables .

  • If -0.75 ≥ r ≥ -1 There exists high

negative relationship between the variables

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Quantitative Aptitude & Business Statistics: Correlation 14

  • If +0.5 ≤r ≤ 0.75 There exists Moderate

positive relationship between the variables .

  • If -0.50 ≥ r >-0.75 There exists moderate

negative relationship between the variables.

  • If r > -0.50 There exists low negative

relationship between the variables

  • If r <0.5 There exists low positive

relationship between the variables .

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Quantitative Aptitude & Business Statistics: Correlation 15

Covariance

  • Definition : Given a n pairs of
  • bservations (X 1,Y 1),(X 2,Y 2) .,,,,,,

(X n,Y n) relating to tw o variables X and Y ,the Covariance of X and Y is usually represented by Cov(X,Y)

( )( )

N xy N Y Y X X Y X Cov

∑ ∑

= − − = . ) , (

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Quantitative Aptitude & Business Statistics: Correlation 16

Properties of Co-Variance

  • Independent of Choice of origin
  • not Independent of Choice of

Scale.

  • Co-variance lies betw een negative

infinity to positive infinity.

  • In other w ords co-variance may

be positive or negative or Zero.

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Quantitative Aptitude & Business Statistics: Correlation 17

From the follow ing Data Calculate Co-Variance

X 1 2 3 4 5 Y 10 20 30 50 40

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Quantitative Aptitude & Business Statistics: Correlation 18

Calculation of Covariance

X X-X=x Y Y-Y=y x.y 1 2 3 4 5

  • 2
  • 1

1 2 10 20 30 50 40

  • 20
  • 10

20 10 40 10 20 20 =15 =0 =150 =0 =90

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Quantitative Aptitude & Business Statistics: Correlation 19

  • N= number of pairs =5

3 5 15 = = = ∑ N X X

30 5 150 = = = ∑ N Y Y

( )( )

18 5 90 . ) , ( = = = − − =

∑ ∑

N xy N Y Y X X Y X Cov

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Quantitative Aptitude & Business Statistics: Correlation 20

Karl Pearson's Correlation

  • The most w idely used

mathematical method for measuring the intensity or the magnitude of linear relationship betw een tw o variables w as suggested by Karl Pearson's

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Quantitative Aptitude & Business Statistics: Correlation 21

Coefficient of Correlation

  • Measures the strength of the

linear relationship betw een tw o quantitative variables

( )( ) ( ) ( )

1 2 2 1 1 n i i i n n i i i i

X X Y Y r X X Y Y

= = =

− − = − −

∑ ∑ ∑

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Quantitative Aptitude & Business Statistics: Correlation 22

Properties of KralPear son’s Coefficient of Correlation

  • Independent of choice of origin
  • Independent of Choice Scale
  • Independent of units of

Measurement

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Quantitative Aptitude & Business Statistics: Correlation 23

Assumptions of Karl Pearson’s Coefficient of Correlation

  • Linear relationship between

variables.

  • Cause and effect relationship.
  • Normality.
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Quantitative Aptitude & Business Statistics: Correlation 24

  • The correlation coefficient

lies betw een -1 and +1

  • The coefficient of correlation

is the geometric mean of tw o regression coefficients.

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Quantitative Aptitude & Business Statistics: Correlation 25

Merits of Karl Pear son’s Coefficient of Correlation

  • Coefficient of Correlation gives

direction as well as degree of relationship between variables

  • Coefficient of Correlation along

with other information helps in estimating the value of the dependent variable from the known value of independent variable.

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Quantitative Aptitude & Business Statistics: Correlation 26

Limitations of KralPear son’s Coefficient of Correlation

  • Assumptions of Linear

Relationship

  • Time consuming
  • Affected by extreme values
  • Requires careful Interpretation
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Quantitative Aptitude & Business Statistics: Correlation 27

From the follow ing Data Calculate Coefficient of correlation

X 1 2 3 4 5 Y 10 20 30 50 40

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Quantitative Aptitude & Business Statistics: Correlation 28

X X-X=x x2 1 2 3 4 5

  • 2
  • 1

1 2 4 1 1 4 =15 =0 =10

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Quantitative Aptitude & Business Statistics: Correlation 29

Y Y-Y=y y2 x.y 10 20 30 50 40

  • 20
  • 10

20 10 400 100 400 100 40 10 20 20 =150 =0 =1000 =90

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Quantitative Aptitude & Business Statistics: Correlation 30

  • N= number of pairs =5
  • r=0.9 there exists high degree of positive

correlation

3 5 15 = = = ∑ N X X

30 5 150 = = = ∑ N Y Y

9 . 100 90 10000 90

2 2

+ = = = × =

∑ ∑ ∑

y x xy r

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Quantitative Aptitude & Business Statistics: Correlation 31

Correlation for Bivariate analysis ( )(

) ( ) ( )

∑ ∑ ∑ ∑ ∑ ∑ ∑

− − − = N dx f d f N dx f d f N d f d f d fd r

y x y x y x 2 2 2 2

. . . . . . .

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Quantitative Aptitude & Business Statistics: Correlation 32

Standard error

  • Standard error of co efficient of

correlation is used foe ascertaining the probable error of coefficient of correlation

  • Where r=Coefficient of correlation
  • N= No. of Pairs of observations

N r SE

2

1− =

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Quantitative Aptitude & Business Statistics: Correlation 33

Probable Error

  • The Probable error of coefficient
  • f correlation is an amount w hich

if added to and subtracted from value of r gives the upper and low er limits w ith in w hich coefficients of correlation in the population can be expected to lie. It is 0.6745 times of standard error.

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Quantitative Aptitude & Business Statistics: Correlation 34

N r r

  • bableErro

2

1 . 6745 . Pr − =

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Quantitative Aptitude & Business Statistics: Correlation 35

Uses of Probable Error

  • PE is used to for determining

reliability of the value of r in so far as it depends on the condition of random sampling.

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Quantitative Aptitude & Business Statistics: Correlation 36

Case Interpretation 1.If |r |< 6 PE

  • 2. 1.If |r | >6 PE

The value of r is not at all significant. There is no evidence of correlation. The value of r is

  • significant. There is

evidence of correlation

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Quantitative Aptitude & Business Statistics: Correlation 37

Example

  • If r=-0.8 and N=36 ,Calculate a) Standard

Error ,b) Probable Error and C) Limits of Population correlation .Also State whether r is significant

  • Solution
  • A)

06 . 6 36 . 6 64 . 1 36 ) 8 . ( 1 1

2 2

= = − = − − = − = N r SE

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Quantitative Aptitude & Business Statistics: Correlation 38

  • b) Probable

Error=0.6745.SE=0.6745* 0.06=0.04

  • c) Limits of Population Correlation
  • =r± PE (r)= -0.8±0.04
  • =-0.84 to -0.76
  • d) Ratio of r to PE of r =
  • |r |/PE( r)=0.8/0.04=20times
  • Since the value of r is more than 6

times the Probable error ,the value of r is significant .Hence the existence of correlation

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Quantitative Aptitude & Business Statistics: Correlation 39

Coefficient of determination

  • The coefficient of determination

is defined as the ratio of the explained variance to the total variance

  • Calculation: The coefficient

determination is calculated by squaring the coefficient of correlation

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Quantitative Aptitude & Business Statistics: Correlation 40

Example

  • If r=0.8 ,w hat is the proportion of

variation in the dependent variable w hich is explained the independent variable?

  • Solution :
  • If r=0.8 ,r2=0.64,
  • It means 64% variation in the

dependent variable explained by independent variable.

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Quantitative Aptitude & Business Statistics: Correlation 41

Coefficient of non-determination

  • The coefficient of non

determination is defined as the ratio of the unexplained variance to the total variance

  • Calculation: The coefficient non

determination is calculated by subtracting the Coefficient of determination from one.

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Quantitative Aptitude & Business Statistics: Correlation 42

Example

  • If r=0.8 ,w hat is the proportion of

variation in the dependent variable w hich is not explained the independent variable?

  • Solution; Coefficient of determination

=r2=0.64

  • Coefficient of non-determination
  • =1-r2=0.36,It means 36% variation in

the dependent variable not explained by independent variable.

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Quantitative Aptitude & Business Statistics: Correlation 43

Spearman’s Rank Correlation

Spearman’s Rank Correlation uses

ranks than actual observations and make no assumptions about the population from which actual

  • bservations are drawn.

( )

1 6 1

2 2

− − =

∑

n n d r

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Quantitative Aptitude & Business Statistics: Correlation 44

Spearman’s Rank Correlation for repeated ranks

  • Where m=the no of times ranks

are repeated

  • n=No of observations
  • r= Correlation Coefficient

( )

1 ..... 12 6 1

2 3 2

−       + − + − =

∑

n n m m D r

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Quantitative Aptitude & Business Statistics: Correlation 45

Calculation of Rank Correlation

  • Tw o judges in a beauty

contest ranked the entries as follow s

X 1 2 3 4 5 Y 5 4 3 2 1

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Quantitative Aptitude & Business Statistics: Correlation 46

X Y d=r1-r2 1 5

  • 4

16 2 4

  • 2

4 3 3 4 2 2 4 5 1 4 16 n=5 =40

2

d

∑

2

d

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Quantitative Aptitude & Business Statistics: Correlation 47

( ) ( )

1 1 5 5 40 6 1 1 6 1

2 2 2

− = − × − = − − =

∑

n n d r

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Quantitative Aptitude & Business Statistics: Correlation 48

Features of Spearman’s Rank Correlation

  • Spearman’s Correlation

coefficient is based on ranks rather than actual observations .

  • Spearman’s Correlation

coefficient is distribution –free and non-parametric because no strict assumptions are made about the form of population from w hich sample observation are draw n.

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Quantitative Aptitude & Business Statistics: Correlation 49

Features of Spearman’s Rank Correlation

  • The sum of the differences of

ranks betw een tw o variables shall be Zero

  • It can be interpreted like Karl

Pearson’s Coefficient of Correlation.

  • It lies betw een -1 and +1
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Quantitative Aptitude & Business Statistics: Correlation 50

Merits of Spearman’s Rank Correlation

  • Simple to understand and

easy to apply

  • Suitable for Qualitative Data
  • Suitable for abnormal data.
  • Only method for ranks
  • Appliacble even for actual

data.

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Quantitative Aptitude & Business Statistics: Correlation 51

Limitations of Spearman’s Rank Correlation

  • Unsuitable data
  • Tedious calculations
  • Approximation
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Quantitative Aptitude & Business Statistics: Correlation 52

When is used Spearman’s Rank Correlation method

  • The distribution is not normal
  • The behavior of distribution is

not know n

  • only qualitative data are given
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Quantitative Aptitude & Business Statistics: Correlation 53

Meaning of Concurrent Deviation Method

  • Concurrent Deviation Method is

based on the direction of change in the two paired variables .The coefficient of Concurrent Deviation between two series of direction of change is called coefficient of Concurrent Deviation .

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Quantitative Aptitude & Business Statistics: Correlation 54

  • rc=Coefficient of Concurrent deviation
  • C= no of positive signs after multiplying

the change direction of change of X- series and Y-Series

  • n=no. of pairs of observations computed

n n c r

c

− ± ± = 2

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Quantitative Aptitude & Business Statistics: Correlation 55

Limitations of Concurrent Deviation Method

  • This method does not

differentiate betw een small and big changes .

  • Approximation
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Quantitative Aptitude & Business Statistics: Correlation 56

Merits of Concurrent Deviation

  • Simple to understand and easy to

calculate.

  • Suitable for large N
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Quantitative Aptitude & Business Statistics: Correlation 57

Calculation of coefficient of concurrent deviation

X 59 69 39 49 29 Y 79 69 59 49 39

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Quantitative Aptitude & Business Statistics: Correlation 58

X Direction

  • f Change
  • f X (Dx)

Y Direction

  • f

Change

  • f X (Dy)

Dx* Dy 59 69 39 49 29 +

  • +
  • 79

69 59 49 39

  • +
  • +

n=4 C=2

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Quantitative Aptitude & Business Statistics: Correlation 59

2 = − ± ± = n n c r

c

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Quantitative Aptitude & Business Statistics: Correlation 60

  • 1___ is a relative measure of

association between two or more variables (a) coefficient of correlation (b) coefficient of regression (c) both (d) none of these

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Quantitative Aptitude & Business Statistics: Correlation 61

  • 1___ is a relative measure of

association between two or more variables (a) coefficient of correlation (b) coefficient of regression (c) both (d) none of these

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Quantitative Aptitude & Business Statistics: Correlation 62

  • 2.The correlation coefficient lies

between (a) –1 and +1 (b) 0 and +1 (c) –1 and 0 (d) none of these

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Quantitative Aptitude & Business Statistics: Correlation 63

  • 2.The correlation coefficient lies

between (a) –1 and +1 (b) 0 and +1 (c) –1 and 0 (d) none of these

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Quantitative Aptitude & Business Statistics: Correlation 64

  • 3. r is independent of __

(a) choice of origin and not of choice of scale (b) choice of scale and not of choice of

  • rigin

(c) both choice of origin and choice of scale (d) none of these

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Quantitative Aptitude & Business Statistics: Correlation 65

  • 3. r is independent of __

(a) choice of origin and not of choice of scale (b) choice of scale and not of choice of

  • rigin

(c) both choice of origin and choice of scale (d) none of these

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Quantitative Aptitude & Business Statistics: Correlation 66

  • 4.Probable error is ___

(a) 0.6475 standard error (b) 0.6745 standard error (c) 0.6457 standard error (d) 0.6547 standard error

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Quantitative Aptitude & Business Statistics: Correlation 67

  • 4.Probable error is ___

(a) 0.6475 standard error (b) 0.6745 standard error (c) 0.6457 standard error (d) 0.6547 standard error

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Quantitative Aptitude & Business Statistics: Correlation 68

  • 5.The product moment correlation coefficient

is obtained by the formula (a) r = (b) r = (c) r = (d) r =

Y X N XY σ σ

∑

y x N xy σ σ

∑

y x N xy σ σ

∑

y x N xy σ σ

∑

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Quantitative Aptitude & Business Statistics: Correlation 69

  • 5.The product moment correlation

coefficient is obtained by the formula (a) r = (b) r = (c) r = (d) r =

Y X N XY σ σ

∑

y x N xy σ σ

∑

y x N xy σ σ

∑

y x N xy σ σ

∑

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Quantitative Aptitude & Business Statistics: Correlation 70

  • 6. Correlation between

Temperature and Sale of Woolen Garments.

  • A) Positive
  • B) 0
  • C) negative
  • D) none of these
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Quantitative Aptitude & Business Statistics: Correlation 71

  • 6. Correlation between

Temperature and Sale of Woolen Garments.

  • A) Positive
  • B) 0
  • C) negative
  • D) none of these
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Quantitative Aptitude & Business Statistics: Correlation 72

  • 7.Covarince can vary from
  • A)-1 to +1
  • B)- infinity to + infinity
  • C)-1 to 0
  • D) 0 to +1
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Quantitative Aptitude & Business Statistics: Correlation 73

  • 7.Covarince can vary from
  • A)-1 to +1
  • B)- infinity to + infinity
  • C)-1 to 0
  • D) 0 to +1
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Quantitative Aptitude & Business Statistics: Correlation 74

  • 8.Karl Pearson’ s coefficient is

defined from

  • A) Ungrouped data
  • B) grouped data
  • C) Both
  • D) none
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Quantitative Aptitude & Business Statistics: Correlation 75

  • 8.Karl Pearson’ s coefficient is

defined from

  • A) Ungrouped data
  • B) grouped data
  • C) Both
  • D) none
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Quantitative Aptitude & Business Statistics: Correlation 76

  • 9. The coefficient of non determination is

0.36 ,the value of r will be

  • A)0.64
  • B)0.60
  • C)0.80
  • D)0.08
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Quantitative Aptitude & Business Statistics: Correlation 77

  • 9. The coefficient of non determination is

0.36 ,the value of r will be

  • A)0.64
  • B)0.60
  • C)0.80
  • D)0.08
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Quantitative Aptitude & Business Statistics: Correlation 78

  • 10.What is Spurious correlation
  • A) It is bad relation between

variables

  • B) It is low correlation between

variables

  • C) It is the correlation between two

variables having no causal relation

  • D) It is a negative correlation
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Quantitative Aptitude & Business Statistics: Correlation 79

  • 10.What is Spurious correlation
  • A) It is bad relation between

variables

  • B) It is low correlation between

variables

  • C) It is the correlation between two

variables having no causal relation

  • D) It is a negative correlation
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Quantitative Aptitude & Business Statistics: Correlation 80

  • 11.Rank coefficient correlation was

developed by

  • A) Karl Pearson
  • B) R.A.Fisher
  • C) Spearman
  • D) Bowley
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Quantitative Aptitude & Business Statistics: Correlation 81

  • 11.Rank coefficient correlation was

developed by

  • A) Karl Pearson
  • B) R.A.Fisher
  • C) Spearman
  • D) Bowley
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Quantitative Aptitude & Business Statistics: Correlation 82

  • 12. If r=0.9 probable error = 0.032 ,
  • Value of N will be
  • A)14
  • B)15
  • C)16
  • D)17
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Quantitative Aptitude & Business Statistics: Correlation 83

  • 12. If r=0.9 probable error = 0.032 ,
  • Value of N will be
  • A)14
  • B)15
  • C)16
  • D)17
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Quantitative Aptitude & Business Statistics: Correlation 84

  • 13.If the value of r2for a particular

situation is 0.49.what is the coefficient

  • f correlation
  • A)0.49
  • B)0.7
  • C)0.07
  • D) cannot be determined
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Quantitative Aptitude & Business Statistics: Correlation 85

  • 13.If the value of r2 for a particular

situation is 0.49.what is the coefficient

  • f correlation
  • A)0.49
  • B)0.7
  • C)0.07
  • D) cannot be determined
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Quantitative Aptitude & Business Statistics: Correlation 86

  • 14.What is the Quickest method to find

correlation between variables .

  • A) Scatter method
  • B) Method of Concurrent Deviation
  • C) Method of Rank correlation
  • D) Method of Product moment

correlation

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SLIDE 87

Quantitative Aptitude & Business Statistics: Correlation 87

  • 14.What is the Quickest method to find

correlation between variables .

  • A) Scatter method
  • B) Method of Concurrent Deviation
  • C) Method of Rank correlation
  • D) Method of Product moment

correlation

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Quantitative Aptitude & Business Statistics: Correlation 88

  • 15 If r=0.6 ,then the coefficient of non

determination is

  • A)0.4
  • B)-0.6
  • C)0.36
  • D)0.64
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Quantitative Aptitude & Business Statistics: Correlation 89

  • 15 If r=0.6 ,then the coefficient of non

determination is

  • A)0.4
  • B)-0.6
  • C)0.36
  • D)0.64
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Quantitative Aptitude & Business Statistics: Correlation 90

  • 17. If the relationship between two

variables x and y is given by 2x + 3y + 4 = 0, then the value of the correlation coefficient between x and y is

  • A) 0
  • B) 1
  • C) –1
  • D) Negative
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Quantitative Aptitude & Business Statistics: Correlation 91

  • 17. If the relationship between two

variables x and y is given by 2x + 3y + 4 = 0, then the value of the correlation coefficient between x and y is

  • A) 0
  • B) 1
  • C) –1
  • D) Negative
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Quantitative Aptitude & Business Statistics: Correlation 92

  • 18 When r = 0 then cov(x,y) is equal to
  • A) + 1
  • B) – 1
  • C) 0
  • D) None of these.
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Quantitative Aptitude & Business Statistics: Correlation 93

  • 18 When r = 0 then cov(x,y) is equal to
  • A) + 1
  • B) – 1
  • C) 0
  • D) None of these.
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Quantitative Aptitude & Business Statistics: Correlation 94

  • 19. For finding the degree of agreement

about beauty between two Judges in a Beauty Contest, we use______ .

  • A) Scatter diagram
  • B) Coefficient of rank correlation
  • C) Coefficient of correlation
  • D) Coefficient of concurrent deviation
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Quantitative Aptitude & Business Statistics: Correlation 95

  • 19. For finding the degree of agreement

about beauty between two Judges in a Beauty Contest, we use______ .

  • A) Scatter diagram
  • B) Coefficient of rank correlation
  • C) Coefficient of correlation
  • D) Coefficient of concurrent deviation
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Quantitative Aptitude & Business Statistics: Correlation 96

  • 20. Coefficient of determination is

defined as

  • A) r3
  • B) 1–r2
  • C) 1+r2
  • D) r2
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Quantitative Aptitude & Business Statistics: Correlation 97

  • 20. Coefficient of determination is

defined as

  • A) r3
  • B) 1–r2
  • C) 1+r2
  • D) r2
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SLIDE 98

THE END

Correla latio ion