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Linear Regression and Correlation: The Correlation Coefficient

Module by: Susan Dean, Dr. Barbara Illowsky

Summary: This module provides an overview of Linear Regression and Correlation: The Correlation Coefficient as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Besides looking at the scatter plot and seeing that a line seems reasonable, how can you tell if the line is a good predictor? Use the correlation coefficient as another indicator (besides the scatterplot) of the strength of the relationship between xx and yy. The correlation coefficient, rr, is defined as:

r = n ⋅ Σ x ⋅ y - ( Σ x ) ⋅ ( Σ y ) [ n ⋅ Σ x 2 - ( Σ x ) 2 ] ⋅ [ n ⋅ Σ y 2 - ( Σ y ) 2 ] r= n ⋅ Σ x ⋅ y - ( Σ x ) ⋅ ( Σ y ) [ n ⋅ Σ x 2 - ( Σ x ) 2 ] ⋅ [ n ⋅ Σ y 2 - ( Σ y ) 2 ]

where:

  • -1≤r≤1-1≤r≤1
  • nn = the number of data points

If you suspect a linear relationship between xx and yy, then rr can measure how strong it is.

If r=1r=1, there is perfect positive correlation. If r=-1r=-1, there is perfect negative correlation. In both these cases, the original data points lie on a straight line. Of course, in the real world, this will not generally happen.

The formula for rr looks formidable. However, many calculators and any regression and correlation computer program can calculate rr. The sign of rr is the same as the slope, bb, of the best fit line.

Glossary

Coefficient of Correlation:
A measure developed by Karl Pearson (early 1900s) that gives the strength of association between the independent variable and the dependent variable. The formula is:
r = n ∑ XY − ( ∑ X ) ( ∑ Y ) [ n ∑ X 2 − ( ∑ X ) 2 ] [ n ∑ Y 2 − ( ∑ Y ) 2 ] , r = n ∑ XY − ( ∑ X ) ( ∑ Y ) [ n ∑ X 2 − ( ∑ X ) 2 ] [ n ∑ Y 2 − ( ∑ Y ) 2 ] , size 12{r= { {n Sum { ital "XY"} - \( Sum {X \) \( Sum {Y \) } } } over { sqrt { \[ n Sum {X rSup { size 8{2} } - \( Sum {X \) rSup { size 8{2} } \] \[ n Sum {Y rSup { size 8{2} } - \( Sum {Y \) rSup { size 8{2} } \] } } } } } } } ,} {} (1)
where n is the number of data points. The coefficient cannot be more then 1 and less then -1. The closer the coefficient is to ±1±1 size 12{ +- 1} {}, the stronger the evidence of a significant linear relationship between XX size 12{X} {} and YY size 12{Y} {}.

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