Probability and Statistics
BMAT202L
Module: 3,Correlation and Regression
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Correlation
Introduction
• Correlation Analysis attempts to determine the degree of relationship
between variables-Ya-Kun-Chou.
• Correlation is an analysis of the covariation between two or more
variables.- A.M.Tuttle.
Correlation expresses the inter-dependence of two sets of variables upon
each other. One variable may be called as (subject) independent and the
other relative variable (dependent). Relative variable is measured in terms
of subject.
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Correlation
Types of Correlation
Correlation is classified into various types. The most important ones are
• Positive and negative.
• Linear and non-linear.
• Partial and total.
• Simple and Multiple.
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Correlation
Positive Correlation
It depends upon the direction of change of the variables. If the two
variables tend to move together in the same direction (ie) an increase in
the value of one variable is accompanied by an increase in the value of the
other, (or) a decrease in the value of one variable is accompanied by a
decrease in the value of other, then the correlation is called positive or
direct correlation.
Examples: Price and supply, height and weight, yield and rainfall.
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Correlation
Negative Correlation
If the two variables tend to move together in opposite directions so that
increase (or) decrease in the value of one variable is accompanied by a
decrease or increase in the value of the other variable, then the correlation
is called negative (or) inverse correlation.
Examples: Price and demand, yield of crop and price.
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Correlation
Linear and Non-linear correlation
If the ratio of change between the two variables is a constant then there
will be linear correlation between them. Consider the following.
X 2 4 6 8 10 12
Y 3 6 9 12 15 18
Here the ratio of change between the two variables is the same. If we plot
these points on a graph we get a straight line. If the amount of change in
one variable does not bear a constant ratio of the amount of change in the
other. Then the relation is called Curvi-linear (or) non-linear correlation.
The graph will be a curve.
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Correlation
Simple and Multiple correlation
When we study only two variables, the relationship is simple correlation.
For example, quantity of money and price level, demand and price. But in
a multiple correlation we study more than two variables simultaneously.
The relationship of price, demand and supply of a commodity are an
example for multiple correlation.
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Correlation
Partial and total Correlation
The study of two variables excluding some other variable is called Partial
correlation. For example, we study price and demand eliminating supply
side. In total correlation all facts are taken into account.
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Correlation
Methods of Studying Correlation
• Scatter Diagram
• Graphical method
• Karl Pearson’s Method
• Spearman’s Rank Correlation Method
• Concurrent deviation Method
• Method of Least Squares
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Correlation
Computation of covariance
When there exists some relationship between two variables, we have to
measure the degree of relationship. This measure is called the measure of
correlation (or) correlation coefficient and it is denoted by ρ.
Co-variation
The covariation between the variables X and Y is defined as
P
(x − x)(y − y )
Cov (X , Y ) = ,
n
where x, y are respectively means of X and Y and n is the number of pairs
of observations.
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Correlation
Karl pearsons coefficient of correlation
Karl pearson, a great biometrician and statistician, suggested a
mathematical method for measuring the magnitude of linear relationship
between the two variables. It is most widely used method in practice and it
is known as pearson coefficient of correlation. It is denoted by ρ. The
formula for calculating ρ is
COV (X , Y )
ρ= ,
σx · σy
qP qP
(x−x)2 (y −y )2
where σx = n and σy = n
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Correlation
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Correlation
Properties of Correlation
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Correlation
Interpretation
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Correlation
Problem 1
Calculate the correlation co-efficient for the following heights (in inches) of
fathers X their sons Y .
X 65 66 67 67 68 69 70 72
Y 67 68 65 68 72 72 69 71
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Correlation
Solution
X Y X2 Y2 XY
65 67 4225 4489 4355
66 68 4356 4624 4488
67 65 4489 4225 4355
67 68 4489 4624 4556
68 72 4624 5184 4896
69 72 4761 5184 4968
70 69 4900 4761 4830
72 71 5184 5041 5112
x 2 =37028 y 2 =38132
P P P P P
x=544 y =552 xy =37560
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Correlation
Solution Cont...
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Correlation
Note
• If Z = aX + bY and ρ is the correlation coefficient between X and Y
then
σz2 = a2 σx2 + b 2 σy2 + 2abρσx σy .
• Correlation coefficient when a = 1, b = −1 and Z = X − Y
σx2 + σy2 − σx−y
2
ρ= ,
2(σx · σy )
where σx , σy and σx−y are standard deviation of X , Y and X − Y
respectively.
• Var (aX + bY ) = a2 Var (X ) + b 2 Var (Y ) + 2abCov (X , Y )
Var (aX − bY ) = a2 Var (X ) + b 2 Var (Y ) − 2abCov (X , Y )
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