Correlation coefficient calculator (Pearson and Spearman)

A fairly strong positive linear association: as x rises, y tends to rise, and r² = 0.49 says a straight line accounts for 49% of the variation in y. Cohen (1988) called r = 0.1 small, 0.3 medium and 0.5 large, and the Strength output uses those cut-offs. They are rough conventions from the behavioural sciences, so what counts as strong still depends on the field.

Pearson's r measures linear association using the values themselves; Spearman's ρ applies the same formula to their ranks, so it measures any monotonic trend. For y = x³ with x from 1 to 10, Spearman gives exactly 1 but Pearson gives 0.928. Spearman also resists outliers and suits ordinal data such as ratings, while Pearson's r is the one that matches a least-squares line.

No. A correlation shows that two variables move together, not why. A third variable can drive both, as hot weather raises both ice-cream sales and drownings. The cause can also run the other way, or the pattern can be chance: test 20 unrelated pairs at α = 0.05 and about one will look significant. Showing cause takes a randomized experiment or a careful causal design.

Convert r to t = r√((n − 2)/(1 − r²)) and compare it with a t distribution with n − 2 degrees of freedom. The default data give r = 0.9816 with n = 10, so t = 14.56 on 8 df and p = 4.9 × 10⁻⁷. Significance depends heavily on n: with 1,000 pairs, r = 0.07 already gives p = 0.027, although it explains under 0.5% of the variation.

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 4 worked examples whose answers come from independent sources; for example, “Hours studied vs score (default)” is checked against Python fractions for Sxx, Syy, Sxy with a decimal square root; p from the closed-form Student t CDF for integer df (Abramowitz & Stegun 26.7.4).

NIST/SEMATECH e-Handbook of Statistical Methods, §7.1.3 / Dataplot CORRELATION (Pearson and rank correlation); Spearman (1904), The proof and measurement of association between two things, American Journal of Psychology 15; Cohen (1988), Statistical Power Analysis for the Behavioral Sciences, 2nd ed., §3.2 (r = 0.1, 0.3, 0.5).