April 8, 2023 1

CORRELATION(LINEAR)

April 8, 2023 2

CORRELATION If two quantities vary in such a way that

movements of one are accompanied by movements of others then these quantities are said to be correlated.

Ex: relationship between price of commodity and amount demanded, Increased in amount of the rainfall and the production of rice

The degree of relationship between variables under consideration is measured through the correlation analysis.

The measure of correlation is called the correlation coefficient or correlation index ( usually denoted by r or ρ )

The correlation analysis refers to the techniques used in measuring the closeness of the relationship between the variables.

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DEFINITIONS

• Correlation analysis deals with the association between two or more variables.

Simpson and Kafka

• Correlation is an analysis of co variation between two or more variables.

A.M.Tuttle

• If two or more quantities were in sympathy so that the movement of one tend to be accompanied by the corresponding movements in the other then they are said to be correlated

L.R.Conner

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ANALYSIS

• The problem of analyzing the relation between different series should be broken down in to three steps

1. Determining whether a relation exists

and if it does, measuring it. 2. Testing whether it is significant. 3. Establishing the cause and effect

relation if any.

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SIGNIFICANCE OF THE STUDY OF CORRELATION

Most of the variables show some kind of relationship

Once we know that two variables are closely related we can estimate the value of one variable given the value of another.

Correlation analysis contributes to the understanding of the economic behavior

The effect of correlation is to reduce the range of uncertainty

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CORRELATION AND CAUSATION

1. The correlation may be due to pure chance especially in a small sample.

The above data show a perfect positive

relationship between income and weight i.e., as the income is increasing the weight is increasing and the rate of change between two variables is the same.

Income(rs)

500 600 700 800 900

Weight(lbs)

120 140 160 180 200

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2. Both the correlated variables may be influenced by one or more other variables.

3. Both the variables may be mututally

influencing each other so that neither can be designated as the cause and the other the effect.

Correlation observed between variables that cannot conceivably be casually related is called spurious or nonsense correlation

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TYPES OF CORRELATION

Positive or negative Linear and non linear

correlation Simple , partial and multiple

correlation

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POSITIVE OR NEGATIVE CORRELATION

• Whether the correlation is positive or negative would depend up on the direction of the change of the variable.

• If both the variables are varying in the same direction , then the correlation is said to be positive.

• If the variables are varying in opposite direction the correlation is said to be negative

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Positive correlationX 10 12 15 18 20

Y 15 20 22 25 37

8 10 12 14 16 18 20 220

5

10

15

20

25

30

35

40

Y-Values

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Negative correlation

X 20 30 40 60 80

Y 40 30 22 15 10

10 20 30 40 50 60 70 80 9005

1015202530354045

Y-Values

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SIMPLE PARTIAL AND MULTIPLE CORRELATION

• The distinction between simple partial and multiple correlation is based up on the number of variables studied.

• When only two variables are studied it is a problem of simple correlation

• When three or more variable are studied it is problem of either multiple or partial correlation.

• In multiple correlation three or more variables are studied simultaneously.

• On the other hand in partial correlation we recognize more than two variables but consider only two variables to be influencing each other the effect of other influencing variable kept constant.

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LINEAR AND NONLINEAR(CURVILINEAR)

CORRELATION• Distinction between linear and non linear correlation is based up on the constancy of the ratio of change between the variables.

• If the amount of change in one variable tends to bear constant ratio to the amount of change in the other variable then the correlation is said to be linear.

It is clear that the ratio of change between the two variables is the same.

• If such variables are plotted on the graph paper all the plotted points would fall on a straight line.

X 10 20 30 40 50

Y 70 140 210 280 350

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5 10 15 20 25 30 35 40 45 50 550

50

100

150

200

250

300

350

400

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Correlation would be called non linear or curvilinear if the amount of change in one variable does not bear a constant ratio with the amount of change in the other variable.

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METHODS OF STUDYING CORRELATION

1. Scatter diagram2. Graphic method3. Karl Pearson’s coefficient

of correlation.4. Concurrent Deviation

Method5. Method of least squares

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SCATTER DIAGRAM METHOD

• The simplest device for ascertaining if the two variables are related is to prepare a dot chart called scatter diagram.

• When this method is used the given data are plotted on a graph paper in the form of dots. I.e., for each pair of X and Y values we put a dot and thus obtain as many points as the number of observations.

• By looking to the scatter of the various points we can form an idea as to whether the variables are related or not.

• The greater the scatter of the plotted points on the chart the lesser is the relationship between the two variables

• The more closely the points come to the straight line higher the degree of relationship.

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• If all the points lie on a straight line falling from the lower left hand corner to the upper right hand corner the correlation is said to be perfectly positive(r=1)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

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If all the points are lying on a straight line rising from the upper left hand corner to the lower right hand corner of the diagram correlation is said to be perfectly negative.

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

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• If the plotted points fall in a narrow band there would be a high degree of correlation between the variables.

• If the points are widely scattered over the diagram it indicates very little relation ship between the variables.

1 2 3 4 5 6 7 80

2

4

6

8

10

1 2 3 4 5 6 7 8 90

2

4

6

8

10

HIGH DEGREE OF POSITIVE CORRELATION

LOW DEGREE OF POSITIVE CORRELATION

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0 2 4 6 8 10 12 140

1

2

3

4

5

6

7

8

If the plotted points lie in a haphazard manner it shows the absence of any relationship between the variables

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EXAMPLE:X 2 3 5 6 8

Y 6 5 7 8 12

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

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• By looking at the scattered diagram we can say that the variables x and y are correlated. Further the correlation is positive because the trend of the points is upward rising from the lower left hand corner to the upper right hand corner of the diagram.

• It also indicates that the degree of relationship is higher because the plotted points are near to the line which shows perfect relationship between the variables.

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MERITS AND LIMITATIONS

• It is a simple and non mathematical method of studying correlation between variables.

• As such it can be easily understood and a rough idea can very quickly be formed as to whether or the variables are related.

• It is the first step in investigating relationship between 2 variables.

LIMITATIONS:• By applying this method we can get an idea

about the direction of correlation and also whether it is high or low

• But we cannot establish the exact degree of correlation between the variables as is possible by applying the mathematical methods.

MERITS

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GRAPHIC METHOD

• When this method is used the individual values of the two variables are plotted on the graph paper.

• We thus obtain 2 curves. One for x variable and another for y variable.

• By examining the direction and closeness of the two curves so drawn we can infer if the variables are related or not.

• If both the curves drawn on the graph are moving in the same direction (either upward or downward)then the correlation is said to be positive.

• On the other hand if the curves are moving in the opposite direction correlation is said to be negative.

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Year Average income

Average expenditure

1979 100 90

1980 102 91

1981 105 93

1982 105 95

1983 101 92

1984 112 94

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1979 1980 1981 1982 1983 19840

20

40

60

80

100

120

Series 1Series 2

EXPENDITURE

INCOME

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KARL PEARSON’S COEFFICIENT OF CORRELATION

• Among several mathematical methods of measuring correlation, the Karl Pearson’s method, popularly known as Pearson’s coefficient of correlation, is most widely used in practice

• It is denoted by the symbol ρ or r

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CORRELATION COEFFICIENT

• If [X,Y] is a two dimensional random variable, the correlation coefficient, denoted r, is

ρ=Cov(X,Y) ∕ Var(X) . Var(Y) = σXY ∕ σX σY

• This is also called as PEARSON CORRELATION COEFFICIENT

ρ= ∑xy ∕ √ (∑x2 * ∑y2) = ∑xy ∕ N σX σY , where

x=(X-X’) ; y=(Y-Y’) σX = Standard Deviation of

X and

σY = Standard Deviation of Y N = no of pairs of

observation ρ = correlation coefficient

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STEPS TO CALCULATE CORRELATION COEFFICIENT

• Take the deviations of X from the mean of X and denote by x

• Square these deviations and obtain the total i.e., Σx2

• Take the deviations of Y from the mean of Y and denote by y

• Square these deviations and obtain the total i.e., Σy2

• Multiply the deviations of X and Y and obtain the total i.e., Σxy

• Substitute the values in the formula

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EXAMPLE

• Calculate the Karl Pearson’s Correlation Coefficient from the following data and interpret it’s value

Roll no of students: 1 2 3 45

Marks in Accountancy : 48 35 1723 47

Marks in Statistics: 45 20 4025 45

SOLUTION: Let marks in Accountancy be denoted by

X and Statistics by Y

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Roll no

X (X-34) x

x2 Y (Y-35) y

y2

xy

1 48 14 196 45 10 100 140

2 35 1 1 20 -15 225 -15

3 17 -17 289

40 5 25 -85

4 23 -11 121

25 -10 100 110

5 47 13 169

45 10 100 130

∑X=170

∑x=0

∑x2=776

∑Y=175

∑y=0 ∑y2=550

∑xy=280

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• The Pearson’s coefficient of correlation is

ρ= ∑xy ∕ √(∑x2 *∑y2)

where x=(X-X’); y=(Y-Y’) , X'= ∑X ∕ N; Y’=∑Y ∕ N

∑xy=280 ∑x2=776 ∑y2=550

ρ = 280 ∕ √ (776 * 550) = 0.496

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DEGREE OF CORRELATION• The value of ρ always lies between -1 and 1.

• If ρ lies between 0 and 1, it is positive. Else, if it lies between -1 and 0, it is negative

• If ρ=1, then the two variables are said to be perfect positively correlated

• If ρ=-1, then the two variables are said to be perfect negatively correlated

• If ρ=0, then the two variables are not correlated

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THANK YOU!!!