Linear Regression and Correlation: The Correlation Coefficient 1.5 2008/06/23 15:59:41 GMT-5 2008/10/27 18:13:32.044 GMT-5 Susan Dean deansusan@deanza.edu Barbara Illowsky illowskybarbara@deanza.edu Susan Dean deansusan@deanza.edu Barbara Illowsky illowskybarbara@deanza.edu Connexions cnx@cnx.org elementary statistics 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 x and y. The correlation coefficient, r, 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 ] where: -1≤r≤1 n = the number of data points If you suspect a linear relationship between x and y, then r can measure how strong it is.If r=1, there is perfect positive correlation. If r=-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 r looks formidable. However, many calculators and any regression and correlation computer program can calculate r. The sign of r is the same as the slope, b, of the best fit line. 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 ] , 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} } \] } } } } } } } ,} {} 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 size 12{ +- 1} {}, the stronger the evidence of a significant linear relationship between X size 12{X} {} and Y size 12{Y} {}.