Pearson Correlation Calculator — Test & CI

The Pearson Correlation Test. Two numeric variables can move together, yet the shape of that movement matters. Pearson correlation measures how tightly paired observations follow one straight-line direction. Pairing is part of the data. Entry i in X must belong to entry i in Y, and the two fields must have equal lengths. Plot every pair before calculating. A straight rising cloud supports positive r, a falling cloud supports negative r, while curvature and clusters demand caution. Center both variables at their sample means. Multiply each X deviation by its matched Y deviation, then sum those cross-products to capture shared direction. Normalize that cross-product sum by the square root of both centered square sums. The resulting r stays between minus one and one and has no units. For inference, transform r to t using n minus two degrees of freedom. Stronger absolute correlation moves farther from the zero-correlation center. The selected tail states the alternative in advance. Two-tailed means nonzero, right-tailed means positive, and left-tailed means negative population correlation.

The exact t reference requires independent pairs and approximate bivariate normality. The calculator performs the arithmetic, but it cannot verify either design condition. Linearity is an interpretation condition. A perfectly ordered curve may have r below one, and a U-shaped relationship may produce r near zero. One influential pair can manufacture or erase a large coefficient. Inspect the point cloud and investigate leverage before treating r as a stable summary. Pooled clusters can also mislead. A strong overall line may disappear inside each subgroup, so preserve known groups and avoid an ecological shortcut. With more than three pairs, Fisher's z transform builds the confidence interval using standard error one over square root of n minus three. Transforming that z interval back makes it asymmetric around r, especially near minus or plus one. With exactly three pairs, this page shows no interval. The page also reports r squared, a simple linear association summary. It is not a causal fraction, regression slope, adjusted effect, or prediction guarantee.

Now follow fourteen paired readings with a two-sided alpha of point zero five. The scatter plot forms a narrow rising band without a lone remote point. The sample means are three point five for X and five point five seven nine for Y. Keep every centered deviation attached to its original partner. The centered cross-product sum is twenty-three point five. The X and Y centered square sums are eighteen point eight four and twenty-nine point eight six four. Divide twenty-three point five by the square root of those two square sums. Pearson r is zero point nine nine zero seven, a very strong positive linear association. The t transform gives twenty-five point two six eight with twelve degrees of freedom, far beyond the two-sided critical magnitude of two point one seven nine. The two-sided p-value is about eight point nine six times ten to the minus twelve. Reject zero population linear correlation at alpha point zero five. The ninety-five percent Fisher interval runs from zero point nine seven zero one to zero point nine nine seven one, and r squared is zero point nine eight one six.

Negate Y and r becomes negative zero point nine nine zero seven. The two-sided p-value stays fixed, while the left and right one-sided probabilities swap. Shift Y by three and multiply it by positive two. Every Pearson result stays unchanged because origin and positive measurement units cancel during normalization. For X from one through seven and Y equal to X squared, Spearman rank correlation is one but Pearson r is only zero point nine seven seven four. Seven balanced points can give r equal to zero. Add one remote pair at ten, ten, and r jumps to zero point eight three seven with p below point zero one. Validation stops unequal lengths, fewer than three pairs, a constant column, or invalid alpha. Three valid pairs permit a test but not the Fisher interval. Report the pairing design, scatter shape, n, r, t, degrees of freedom, chosen tail, p, alpha, Fisher interval, r squared, and influential-point checks. Use Spearman for monotone curvature, Kendall for ordinal concordance, and regression for slopes or prediction. Run the Pearson Correlation Test free at Distri Scope dot com.