ch 11 correlatiion analysis

8/14/2019 Ch 11 Correlatiion Analysis

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Correlation Analysis

Correlation is a statistical tool which studies therelationship between two variables and CorrelationAnalysis involves various methods and techniquesused for studying and measuring the extent of therelationship between two variables.

Correlation Analysis is a statistical procedure by

which we can determine the degree of association orrelationship between two or more variables.

Prof. Kuldeep Sharma, IIBSBengaluru

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Statistical Relationship


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Relation between height & weight; Price & demand,Age & Height; Radius & Area of a circle

Two variables are said to be correlated if a change

in the value of one variable is accompanied by achange in the value of another variable.

Such a relationship is called Statistical Relationship.

When both the variables in the bi-variate data arequantitative, we use the term Correlation analysis todescribe the methods to find out if relationshipexists or not?


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Croxton and Crowden defined thecorrelation as

The relationship of quantitative nature. The appropriate statisticaltool for discovering and measuring the relationship and expressingit in brief formula is known as Correlation.

According to the Statistician A. M. Tuttle

Correlation is an analysis of the covariation between two ormore variables.


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Sample Data for House Price Model

House Price in $1000s(Y)

Square Feet(X)

245 1400

312 1600

279 1700308 1875

199 1100

219 1550

405 2350324 2450

319 1425

255 1700

Prof Kuldeep Sharma, IIBSBengaluru


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0

50

100

150

200

250

300

350

400

450

0 500 1000 1500 2000 2500 3000

Square Fee

Hous

e

Price

($1000s)

Graphical Presentation

House price model: scatterplot

Prof Kuldeep Sharma, IIBS

Bengaluru


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Types of Relationships

Y

X

Y

X

Y

Y

X

X

Linear relationships Curvilinear relationships


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Y

X

Y

X

Y

Y

X

X

Strong relationships Weak relationships

(continued)



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Y

X

Y

X

No relationship

(continued)

Prof Kuldeep Sharma, IIBS

Bengaluru


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UNIVARIATE & BIVARIATE DISTRIBUTION

In a bivariate population we are interested to knowwhether there exists some sort of functionalrelationship between the two variables involved.

The change in one variable affects a change in theother variable or not?

If yes what is the nature of this relationship?


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COVARIANCE

Covariance is an absolute measure between twovariables X & Y, denoted by Cov. (X,Y) and definedas

Cov. (X,Y) = (x - )*(y - )/n

Cov. (X,Y) = 1/n*{ xy 1/n*( x)*( y)}

The covariance measures the strength of the linearrelationship between two variables


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SCATTER DIAGRAM OR DOT DIAGRAMMETHOD

Scatter diagram is a graphical method of showing thecorrelation between the two variables x & y.

The scatter diagram may indicate both degree andthe type of correlation.

From scatter diagram, we can form a fairly good,though rough idea about the relationship between thetwo variables.

PAGE11Prof. Kuldeep Sharma, IIBSBengaluru


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Scatter Plot

A scatter plot (or scatter diagram) can be used to show the

relationship between two variablesC os t pe r D a y v s . P rod uc tion

0

5 0

1 0 0

1 5 0

2 0 0

2 5 0

0 1 0 2 0 3 0 4 0 5 0 6 0 7

V o lu m e p e r

Cos

tperDay

Volumeper day Cost perday

23 125

26 140

29 146

33 16038 167

42 170

50 188

55 195

60 200


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Advantage & Disadvantage of Scatter Diagram

Readily comprehensible and enables us to form a rough idea ofthe nature of relationship between the two variables

Not affected by extreme observations Not influenced by extreme items

Not a suitable method if the number of observations is verylarge

Provides only rough measure of Correlation which can differ

from man to man


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Co-efficient of Correlation r

It gives the degree of association or relationship

correlation.

The relationship between two variables such that a

change in one variable results in a positive or

negative change in the other variable and also a

greater change in one variable results in

corresponding greater or smaller change in the othervariable is known as Correlation.


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Coefficient of Correlation

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( ) ( )

( ) ( )

1

2 2

1 1

n

i ii

n n

i i

i i

X X Y Y

r

X X Y Y

=

= =

=

Measures the strength of the linear relationshipbetween two quantitative variables



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Application of Correlation analysis

Correlation analysis is used to measurestrength of the association (linear

relationship) between two variables Correlation is only concerned with strength

of the relationship

No causal effect is implied with correlation


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Properties of Co-efficient of correlation

1. It is a measure of the closeness of a fit in a relativesense

2. R lies between -1 & +1

3. The correlation is perfect negative when r = -1

4. The correlation is perfect positive when r= +15. If r = 0 then there is no correlation, Thus Variables are

independent

6. R is a pure number and is not affected by a change oforigin & scale

7. Relative measure of association between two or morevariables


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Scatter Plots of Data with Various

Correlation Coefficients

Y

X

Y

X

Y

X

Y

X

Y

X

r = -1 r = -.6 r = 0

r

= .6

r = 1

Prof. Kuldeep Sharma, IIBSBen aluru


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Karl Pearsons Coefficient of Correlation

Karl Pearson (1857-1936) a great Statistician provided formula for measuringthe magnitude of linear correlation coefficient between two variables.

(X,Y) = rxy = Cov (x,y)

(VarX *VarY)

(X,Y) = rxy = (x - )(y - )

(x - )2* (y - ) 2


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Karl Pearsons Coefficient of Correlationcontd.

(X,Y) = rxy = n* x*y x* Y

{n* x2 ( x)2}*{n* y2 ( y)2}

Above formula saves a lot of computational labour.

Also It reduces the error due to computation & rounding off.

Other forms also can be used

(X,Y) = rxy = dx* dy where dx=(x- )

dx2 * dy2 where dx2=(x- )2

(X,Y) = rxy = dx* dy

n* x* y



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Another Formula called short cut method

(X,Y) = rxy = n* dx*dy dx* dY

{n* dx2 ( dx)2}*{n* dy2 ( dy)2}

where dx = (x - a) a is assumed mean for X where dx2= (x - a) 2

where dy= (y - b) b is assumed mean for Y

where dy2= (y - b) 2



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Prof. Kuldeep Sharma, IIBSBengaluruPAGE22

Nature of Relationship

Positive correlation means that low values of one variable areassociated with low values of the other, and high values of one

variable are associated with high values of the other.

Negative correlation means that low values of one variable areassociated with high values of the other, and high values of onevariable are associated with low values of the other.

The degree of correlation between two variables is measured by thePersonian ( Product moment) correlation coefficient. ( r )

The nearer r to +1 or 1. The stronger the relationship.


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Spearmans Rank Correlation Coefficient R

It is applied in the problems in which data cannot be measuredquantitatively but qualitatively assessment is possible such asbeauty, honesty etc.

In this case the best individual is given rank number1, next 2 andso on.

R = 1 - 6* (D)2

n(n2 1)

Where is the square of the difference of corresponding ranksand n is number of pairs of observations.



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Spearmans Rank Correlation CoefficientWhen Ranks are tied or Repeated ranks

R = 1 - 6[ (D)2+(p3p)/12+(q3q)/12]

n(n2 1)where p, q.. Are the number of times a value isrepeated



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Spearmans Rank Correlation Coefficient

It is simpler to understand and easy to calculate as compared toKarls Pearsons Method.

It is useful for qualitative data such as beauty, honesty,efficiency etc.

It is a useful method when the actual data is not given but onlyranks are given.

Limitation It cant be used for grouped frequency distribution

It is no as accurate as Pearsons coefficient. It cant be used in continuous series. When no of items is >30, and if ranks are not given; it takes more time and

therefore cant be used conveniently.



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Quiz

State the nature of the following correlation (positive, Negative or no correlation)

1. The amount of rainfall & Yield of crops

2. The colour of a saree and the intelligence of the girl wearing it

3. Age if life insurance & the premium of insurance

4. Demand for goods and their prices under normal time

5. Production of pig iron and soot contents in Durgapur

6. Unemployment index and the purchasing power of the commonman

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ch 11 correlatiion analysis

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