session 11 correlation

7/30/2019 Session 11 Correlation

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CORRELATION

DMM-ISession-11th When something happens, we ask Why?

we want to know what cause the event.

But why are we interested in what caused the event?

knowing the causes provides understanding

knowing causes empowers us to intervene

These two tend to go together:

Why do more calls produce better sales?

learning the reason is more calls provides better

understanding and a procedure for making better sales

The roots of appeal of causation lie in ourdoingsomething to produce an effect:

We want to move a rock, so we push it.

Definition of Cause:

Independent of our own action, a cause is something

which brings about or increases the likelihood of an

effect.

A major logical fallacy with the people is that peoplebelieves that correlations might be indicative of

causation.

Correlations per se only allow you to predict: The correlation of frequent smoking with having cancer in 2 to 3

years time allows you to predict that if you engage in frequent

smoking , you are more likely to have a cancer 2 to 3 years later.

Causation tells you how to change the effect: Knowing that frequent smoking causes (increases the likelihood

of) having cancer in 2 to 3 years time allows you to take action to

have or not have a cancer.


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Correlation

Causation means cause & effect relation.

Correlation denotes the interdependency amongthe variables for correlating two phenomenon, it is

essential that the two phenomenon should have

cause-effect relationship,& if such relationship

does not exist then the two phenomenon can not

be correlated.

Correlation

If two variables vary in such a way thatmovement in one are accompanied by

movement in other, these variables are

called cause and effect relationship.

Causation always implies correlation butcorrelation does not necessarily implies

causation.

Dependent Variables(DVs) &

Independent Variables(IVs)

How to find DVs & IVs Relation

Direction

Origin

Results

Definition of DVs and IVs

Examples of DVs and IVs

Types of Correlation

Type I

Correlation

Positive Correlation Negative Correlation


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Types of Correlation Type I

Positive Correlation: The correlation is said to bepositive correlation if the values of two variableschanging with same direction.

Ex. Pub. Exp. & sales, Height & weight.

Negative Correlation: The correlation is said to benegative correlation when the values of variables

change with opposite direction.

Ex. Price & qty. demanded.

Direction of the Correlation

Positive relationshipVariables change in thesame direction.

As X is increasing, Y is increasing

As X is decreasing, Y is decreasing

E.g., As height increases, so does weight.

Negative relationshipVariables change inopposite directions.

As X is increasing, Y is decreasing

As X is decreasing, Y is increasing

E.g., As TV time increases, grades decrease

Indicated bysign; (+) or (-).

More examples

Positive relationships

water consumption andtemperature.

Stud y time and grades.

Negative relationships:

alcohol consumption anddriving ability.

Price & quantitydemanded


Type II

Correlation

Simple Multiple

Partial Total


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Types of Correlation Type IISimple correlation: Under simple correlation problem

there are only two variables are studied.

Multiple Correlation: Under Multiple Correlation three ormore than three variables are studied. Ex. Qd = f ( P,PC, PS,t, y )

Partial correlation: analysis recognizes more than twovariables but considers only two variables keeping the otherconstant.

Total correlation: is based on all the relevant variables,which is normally not feasible.


Type III

Correlation

LINEAR NON LINEAR

Types of Correlation Type III

Linear correlation: Correlation is said to belinear when the amount of change in one variabletends to bear a constant ratio to the amount ofchange in the other. The graph of the variableshaving a linear relationship will form a straightline.

Non Linear correlation: The correlation wouldbe non linear if the amount of change in onevariable does not bear a constant ratio to theamount of change in the other variable.

Methods of Studying Correlation

Scatter Diagram Method

Karl Pearsons Coefficient ofCorrelation


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Scatter Diagram Method

Scatter Diagram is a graph of observedplotted points where each points

represents the values of X & Y as a

coordinate. It portrays the relationship

between these two variables graphically.

A perfect positive correlation

Height

Weight

Heightof A

Weightof A

Heightof B

Weightof B

A linearrelationship

High Degree of positive correlation

Positive relationship

Height

Weight

r = +.80

Degree of correlation

Moderate Positive Correlation

Weight

ShoeSize

r = + 0.4


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Perfect Negative Correlation

Exam score

TVwatching

per

week

r = -1.0


Moderate Negative Correlation

Exam score

TVwatching

per

week

r = -.80


Weak negative Correlation

Weight

ShoeSize

r = - 0.2


No Correlation (horizontal line)

Height

IQ

r = 0.0


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Degree of correlation (r)

r = +.80 r = +.60

r = +.40 r = +.20

Advantages of ScatterDiagram

Simple & Non Mathematical method

Not influenced by the size of extremeitem

First step in investing the relationshipbetween two variables

Disadvantage of scatter diagram

Can not determine the an exact

degree of correlation

Number of Sales Calls and Copies Sold for 10 Salespeople

Sales Representative Number of Sales calls Number of copies sold

Tom 20 30

Jeff 40 60

Brian 20 40

Greg 30 60

Susan 10 30

Carlos 10 40

Rich 20 40

Mike 20 50

Mark 20 30

Soni 30 70

Mean 22 45


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Scatter Diagram Showing Sales Calls & Copies Sold

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50

CopiesSold(Y)

Sales Calls (X)

I

II

III

IV

Mean (X)

Mean(Y)

Product Moment Correlation

The product moment correlation,r, summarizes thestrength of association between two metric (interval orratio scaled) variables, sayXand Y.

It is an index used to determine whether a linear orstraight-line relationship exists betweenXand Y.

As it was originally proposed by Karl Pearson, it is alsoknown as thePearson correlation coefficient. It is alsoreferred to assimple correlation, bivariate correlation, ormerely the correlation coefficient.

Assumptions of Pearsons

Correlation Coefficient

There is linear relationship between two variables,i.e. when the two variables are plotted on a scatterdiagram a straight line will be formed by thepoints.

Cause and effect relation exists between differentforces operating on the item of the two variable

series.

Variables should be Interval scaled or ratio scaled

From a sample ofn observations,Xand Y, the product

moment correlation, r, can be calculated as:

r=

(Xi - X)(Yi - Y)i=1

n

(Xi - X)2

i=1

n

(Yi - Y)2

i=1

n

Division of the numerator and denominator b (n-1) gives

r=

(Xi - X)(Yi - Y)

n-1i=1

n

(Xi - X)2

n-1i=1

n (Yi - Y)2

n-1i=1

n

=COVxySxSy



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Procedure for computing the correlation coefficient

Calculate the mean of the two series x &y

Calculate the deviations x &y in two series from theirrespective mean.

Square each deviation of x &y then obtain the sum ofthe squared deviation i.e.x2 & .y2

Multiply each deviation under x with each deviation undery & obtain the product of xy.Then obtain the sum of the

product of x , y i.e. xy

Substitute the value in the formula.

Explaining Attitude Toward theCity of Residence

Respondent No Attitude Towardthe City

Duration ofResidence

ImportanceAttached to

Weather1 6 10 3

2 9 12 11

3 8 12 4

4 3 4 1

5 10 12 11

6 4 6 1

7 5 8 7

8 2 2 4

9 11 18 8

10 9 9 10

11 10 17 8

12 2 2 5


The correlation coefficient may be calculated as follows:

= (10 + 12 + 12 + 4 + 12 + 6 + 8 + 2 + 18 + 9 + 17 + 2)/12

= 9.333

Y= (6 + 9 + 8 + 3 + 10 + 4 + 5 + 2 + 11 + 9 + 10 + 2)/12

= 6.583

( i - )(Yi - Y)i=1

n= (10 -9.33)(6-6.58) + (12-9.33)(9-6.58)

+ (12-9.33)(8-6.58) + (4-9.33)(3-6.58)

+ (12-9.33)(10-6.58) + (6-9.33)(4-6.58)

+ (8-9.33)(5-6.58) + (2-9.33) (2-6.58)+ (18-9.33)(11-6.58) + (9-9.33)(9-6.58)

+ (17-9.33)(10-6.58) + (2-9.33)(2-6.58)

= -0.3886 + 6.4614 + 3.7914 + 19.0814

+ 9.1314 + 8.5914 + 2.1014 + 33.5714

+ 38.3214 - 0.7986 + 26.2314 + 33.5714

= 179.6668

Product Moment Correlation( i - )

2

i=1

n

= (10-9.33)2 + (12-9.33)2 + (12-9.33)2 + (4-9.33)2

+ (12-9.33)2 + (6-9.33)2 + (8-9.33)2 + (2-9.33)2

+ (18-9.33)2 + (9-9.33)2 + (17-9.33)2 + (2-9.33)2

= 0.4489 + 7.1289 + 7.1289 + 28.4089

+ 7.1289+ 11.0889 + 1.7689 + 53.7289

+ 75.1689 + 0.1089 + 58.8289 + 53.7289

= 304.6668

(Yi - Y)2

i=1

n

= (6-6.58)2 + (9-6.58)2 + (8-6.58)2 + (3-6.58)2

+ (10-6.58)2+ (4-6.58)2 + (5-6.58)2 + (2-6.58)2

+ (11-6.58)2

+ (9-6.58)2

+ (10-6.58)2

+ (2-6.58)2

= 0.3364 + 5.8564 + 2.0164 + 12.8164

+ 11.6964 + 6.6564 + 2.4964 + 20.9764

+ 19.5364 + 5.8564 + 11.6964 + 20.9764

= 120.9168

Thus, r= 179.6668

(304.6668) (120.9168)= 0.9361

session 11 correlation

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