Correlation Coefficient

The correlation coefficient, sometimes also called the cross-correlation coefficient, is a quantity that 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,

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These quantities are simply unnormalized forms of the variances and covariance of and given by

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For linear least squares fitting, the coefficient in

(16)

is given by

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and the coefficient in

(19)

is given by

(20)

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

			(21)
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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

(23)

and denote the "expected" value for as . Sums of are then

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The sum of squared errors is then

			(34)
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and the sum of squared residuals is

			(42)
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But

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so

			(49)
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and

(53)

The square of the correlation coefficient is therefore given by

			(54)
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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 (◇) for and equating to (◇) gives

(57)

Therefore, and , giving

(58)

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

(59)

where

			(60)
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