Pearson Correlation Coefficient Calculator

The Pearson product-moment coefficient is the covariance of two variables divided by the product of their standard deviations, which cancels all units and confines the answer to the closed interval -1 to +1. The computational form shown here, using n times the sum of XY minus the product of the two sums over a square-root denominator, is algebraically identical to the covariance definition but requires only one pass through the data. Squaring the result gives the coefficient of determination: r = 0.60 means 36 percent of the variance in Y is linearly accounted for by X. Significance is tested with t = r times sqrt((n-2)/(1-r squared)) on n-2 degrees of freedom. The coefficient only detects straight-line association. A perfect parabola centred on the mean of X returns r near zero, and Anscombe's four datasets from 1973 all yield r = 0.816 with completely different shapes, which is why a scatterplot is mandatory before quoting a number. Pearson r is also sensitive to single outliers and is attenuated by range restriction, so a coefficient computed on a narrow slice of dosages will understate the true association. Because the sampling distribution of r is skewed near the limits, confidence intervals should be built on Fisher's arctanh transform rather than on r directly. With n below about 10 the estimate is very unstable, and no value of r establishes causation.

Auguste Bravais worked out the mathematics of the bivariate normal error surface in 1846, but he did not treat the cross-product term as a measure of association. Francis Galton reached that idea empirically while analysing anthropometric data at his London laboratory, and his 1888 Royal Society paper Co-relations and their Measurement introduced the index he called r, obtained by plotting standardised deviations of forearm length against height. Karl Pearson, working from Galton's material at University College London, published the modern product-moment formula in 1896 in the third of his Mathematical Contributions to the Theory of Evolution, in the Philosophical Transactions of the Royal Society. The coefficient was descriptive until Ronald A. Fisher derived the exact sampling distribution of r under bivariate normality in 1915, and in 1921 introduced the z transformation, z = arctanh r, whose variance is approximately 1/(n-3) and which is still the basis of confidence intervals and meta-analytic pooling. Distribution through Biometrika and through Fisher's 1925 Statistical Methods for Research Workers made r the default measure of association in genetics, agriculture and psychometrics. Later commentary, particularly Frank Anscombe's 1973 note in The American Statistician, pushed the discipline back toward plotting the data alongside the coefficient.