## Correlation Coefficient

The correlation coefficient is a quantity which gives the quality of a Least Squares Fitting to the original data. To define the correlation coefficient, first consider the sum of squared values , , and of a set of data points about their respective means,

 (1) (2) (3)

For linear Least Squares Fitting, the Coefficient in
 (4)

is given by
 (5)

and the Coefficient in
 (6)

is given by
 (7)

The correlation coefficient (sometimes also denoted ) is then defined by

 (8)

which can be written more simply as
 (9)

The correlation coefficient is also known as the Product-Moment Coefficient of Correlation or Pearson's Correlation. The correlation coefficients for linear fits to increasingly noisy data are shown above.

The correlation coefficient has an important physical interpretation. To see this, define

 (10)

and denote the expected'' value for as . Sums of are then
 (11) (12) (13) (14)

The sum of squared residuals is then
 (15)

and the sum of squared errors is
 (16)

But
 (17) (18)

so
 (19) (20)

and
 (21)

The square of the correlation coefficient is therefore given by

 (22)

In other words, is the proportion of which is accounted for by the regression.

If there is complete correlation, then the lines obtained by solving for best-fit and coincide (since all data points lie on them), so solving (6) for and equating to (4) gives

 (23)

Therefore, and , giving
 (24)

The correlation coefficient is independent of both origin and scale, so

 (25)

where
 (26) (27)

See also Correlation Index, Correlation Coefficient--Gaussian Bivariate Distribution, Correlation Ratio, Least Squares Fitting, Regression Coefficient

References

Acton, F. S. Analysis of Straight-Line Data. New York: Dover, 1966.

Kenney, J. F. and Keeping, E. S. Linear Regression and Correlation.'' Ch. 15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 252-285, 1962.

Gonick, L. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, 1993.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Ninear Correlation.'' §14.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 630-633, 1992.