Download - 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
Types of Correlation
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.
Types of Correlation
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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Degree of correlation
Perfect Negative Correlation
Exam score
TVwatching
per
week
r = -1.0
Degree of correlation
Moderate Negative Correlation
Exam score
TVwatching
per
week
r = -.80
Degree of correlation
Weak negative Correlation
Weight
ShoeSize
r = - 0.2
Degree of correlation
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
Product Moment Correlation
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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
Product Moment Correlation
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