Correlation Coefficient Calculator: Pearson, Spearman and Kendall

Computes all three correlation coefficients—Pearson, Spearman and Kendall—in a single step, returning their p-values, the Pearson 95% confidence interval, and r².

The three coefficients measure different things

Pearson r measures the strength of a linear relationship. It uses the raw values, so it is sensitive to outliers and captures only straight-line relationships.

Spearman ρ converts the data to ranks and then applies Pearson’s formula. It measures monotonic relationships—as long as one variable consistently rises with the other, direction alone matters, not linearity.

Kendall τ directly counts concordant and discordant pairs. It is more robust than Spearman and has better distributional properties at small sample sizes, though computation scales as n².

When ties are present, this tool reports τb. The concordant-minus-discordant sum S can be divided two ways: by the total number of pairs n₀ = n(n−1)/2 to give τa, or by √[(n₀−n₁)(n₀−n₂)] to give τb, where n₁ and n₂ are the tie-pair counts for x and y respectively. Without ties the two are identical; with ties, τa is systematically smaller—for example, x = 1,2,2,3,5,5,5 and y = 2,3,4,5,5,6,8 give τa = 0.762 and τb = 0.868. R, scipy, and SPSS all report τb, so this tool does too and labels the output “τb” as a reminder. The p-value also incorporates the tie correction (Kendall 1945 variance formula); without it the test is conservative.

Two examples showing why you should look at all three

Example 1: monotonic but nonlinear. x = 1…8, y = 2ˣ: Pearson gives 0.850, while Spearman and Kendall both give 1.000. The relationship is perfectly monotonic, but Pearson underestimates it because the curve is not a straight line.

Example 2: a single outlier can reverse the sign. x = 1,2,…,10 and y = 10,9,8,7,6,5,4,3,2,50: outside the last point, y decreases strictly with x, but that one value of 50 causes Pearson to give +0.358 and Spearman −0.455—opposite signs. Looking only at Pearson leads to the completely wrong conclusion.

So: look at all three together—a large discrepancy means the data have a story to tell. Pearson much smaller than Spearman usually means the relationship is monotonic but curved; opposite signs almost always point to an outlier.

Interpreting r² correctly

r² is the proportion of variance explained by the linear relationship, not an accuracy rate. r = 0.7 looks high, but r² = 0.49, meaning half the variation is still unexplained. At r = 0.5, r² is only 0.25. The squaring relationship makes it easy to overestimate how strong an association really is.

Correlation is not causation—and that is not the whole story

Beyond “correlation does not imply causation,” two more points matter equally:

A correlation of 0 does not mean the variables are unrelated. It only means there is no linear (or monotonic) relationship. A U-shaped relationship can have Pearson r = 0 while the two variables are clearly associated.

At large sample sizes, trivially small correlations can be statistically significant. With n = 1000, r = 0.07 yields p < 0.05, yet explains only 0.5% of the variance. Check r and r² before reading the p-value.

Confidence interval

The Pearson interval is computed using the Fisher z transformation: r is mapped to an approximately normal scale, a symmetric interval is constructed there, then the endpoints are transformed back. The resulting interval is therefore asymmetric around r, and the asymmetry becomes more pronounced as r approaches ±1. With n < 10 the interval will be wide—an honest reflection of limited data, not a defect.