Karl Pearson's coefficient of correlation

Indian Economy glossary

Also called: Product moment correlation, Simple correlation coefficient, r · Topic: Economic Data: Census, NSS, Surveys and Statistical Tools · NCERT: Class 11, Ch 6 "Correlation"

Meaning

Karl Pearson's coefficient of correlation, written r, is a number that shows how strongly two variables move together in a straight-line pattern, and in which direction. It divides the covariance by the product of the two standard deviations. Covariance measures how the two variables move together. Standard deviation measures how spread out each variable is.

r = Cov(X,Y)/(σx·σy) = Σxy/(N·σx·σy), where Cov(X,Y) = Σ(X − X̄)(Y − Ȳ)/N

The result has no units. It always lies between −1 and +1. The sign of the covariance gives the sign of r. Changing the origin or the scale of the data (for example, subtracting a constant, or dividing by a constant of the same sign) does not change r. The step-deviation shortcut uses this property.

Example

NCERT compares farmers' years of schooling with their yield per acre and gets r = 42/(√112·√38) = 0.644. This is a fairly strong positive link. A price index compared with money supply gives r = 0.98.

Don't confuse with

  • Spearman's rank correlation: it works on ranks rather than actual values. It suits attributes such as beauty or honesty, and data with outliers.
  • Causation: a high r only shows that two variables move together. It never proves that one causes the other.

Related concepts

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