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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3. 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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4. 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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5. 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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6. CORRELATION AND CAUSATION
1. The correlation may be due to pure chance especially in a small sample.
Income(rs) 500 600 700 800 900
Weight(lbs) 120 140 160 180 200
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.
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7. 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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8. TYPES OF CORRELATION
Positive or negative
Linear and non linear correlation
Simple , partial and multiple correlation
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9. 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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12. 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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13. 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.
X 10 20 30 40 50
Y 70 140 210 280 350
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.
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15. 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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16. METHODS OF STUDYING CORRELATION
1. Scatter diagram
2. Graphic method
3. Karl Pearson’s coefficient of correlation.
4. Concurrent Deviation Method
5. Method of least squares
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17. 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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18. • 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)
8
7
6
5
4
3
2
1
0
0 2 4 6 8
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19. 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.
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8
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20. • 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.
10 10
8 8
6 6
4 4
2 2
0 0
0 5 10 0 5 10
HIGH DEGREE OF POSITIVE CORRELATION LOW DEGREE OF POSITIVE CORRELATION
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21. If the plotted points lie in a haphazard manner it shows the absence of any
relationship between the variables
8
7
6
5
4
3
2
1
0
0 2 4 6 8 10 12 14
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23. • 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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24. MERITS AND LIMITATIONS
MERITS
• 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.
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25. 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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26. 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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27. 120
INCOME
100
EXPENDITURE
80
60 Series 1
Series 2
40
20
0
1979 1980 1981 1982 1983 1984
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28. 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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29. 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
3 September 2012 ρ = correlation coefficient 29
30. 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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31. EXAMPLE
• Calculate the Karl Pearson’s Correlation Coefficient from the following
data and interpret it’s value
Roll no of students: 1 2 3 4 5
Marks in Accountancy : 48 35 17 23 47
Marks in Statistics: 45 20 40 25 45
SOLUTION:
Let marks in Accountancy be denoted by X and Statistics by Y
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32. Roll no X (X-34) x2 Y (Y-35) y2 xy
x y
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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33. • 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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34. 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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