Correlation Analysis
Muhammad Irfan Malik
PhD (Econometrics)
School of Social Sciences and Humanities,
National University of Sciences and Technology, Islamabad
Email: irfanmalik@s3h.nust.edu.pk
Outline
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Correlation
Correlation Coefficient
Application
Interpretation of correlation coefficient
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What is Correlation?
Correlation is a statistical measure that expresses the extent to
which two variables are linearly related.
It’s a common tool for describing simple relationships without
making a statement about cause and effect.
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Correlation Coefficient
The correlation coefficient is a descriptive measure of the
strength and direction of the linear (straight-line) relationship
between two variables.
There are several statistics that can be used to measure the
correlation between the variables. The most commonly used
measure is Karl Pearson Product Moment correlation
coefficient.
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Pearson Correlation Coefficient
𝑟=
(𝑥 − 𝑥)(𝑦 − 𝑦)
(𝑥 − 𝑥)2 (𝑦 − 𝑦)2
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Pearson Correlation Coefficient
𝑟=
(𝑥 − 𝑥)(𝑦 − 𝑦)
− 𝑥)2
(𝑦
𝑥𝑦 −
𝑥 𝑦
𝑛
(𝑥
=
𝑥2 −
𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 (𝑥, 𝑦)=
𝑥
𝑛
2
1
𝑆𝑥𝑦 =𝑛−1
=
− 𝑦)2
𝑦2 −
𝑥𝑦 −
𝑥𝑦 − 𝑛𝑥𝑦
𝑦
𝑛
𝑥 𝑦
𝑛
[ 𝑥 2 − 𝑛𝑥 2 ] [ 𝑦 2 − 𝑛𝑦 2 ]
2
𝑆𝑥𝑦
𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 (𝑥, 𝑦)
=
=
𝑆𝑥 𝑆𝑦
𝑆𝐷 𝑥 𝑆𝐷(𝑦)
𝑆𝐷 𝑥 = 𝑆𝑥 =
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(𝑥−𝑥)2
𝑛−1
𝑆𝐷 𝑦 = 𝑆𝑦 =
(𝑦−𝑦)2
𝑛−1
Understanding the Linear Correlation Coefficient
 r reflects the slope of the scatterplot.
 The magnitude of r indicates the strength of the
linear relationship.
 The sign of r suggests the type of linear relationship
 The sign of r and the sign of the slope of the
regression line are identical
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