Pearson vs Spearman vs Kendall: Choosing a Correlation

Correlation is one of the first numbers a researcher reaches for, and one of the easiest to misuse. The problem is rarely the formula — it is choosing the wrong coefficient for the data. Pearson's r, Spearman's rho (ρ), and Kendall's tau (τ) all return a number between −1 and +1, but they measure different things and rest on different assumptions. Picking the right one is the difference between a defensible result and a misleading one.

This guide explains what each coefficient actually measures, when each is appropriate, and how to report the result properly. The short version: always plot your data first, match the coefficient to your measurement scale and the shape of the relationship, and treat the coefficient itself as your effect size.

Pearson's r: linear association between continuous variables

Pearson's r quantifies the strength and direction of the linear relationship between two interval or ratio variables. It is the covariance of the two variables divided by the product of their standard deviations, so it is scale-free. An r of +1 means the points fall exactly on an upward-sloping line; −1 means a perfect downward line; 0 means no linear trend.

Three assumptions matter. First, the relationship should be linear — Pearson is blind to curves. Second, both variables should be roughly bivariate normal for the significance test and confidence interval to behave. Third, Pearson is sensitive to outliers: a single extreme point can pull r dramatically up or down.

The classic warning is Anscombe's quartet — four datasets with identical means, variances, and an identical Pearson r of 0.816, yet wildly different shapes. One is a clean line, one is a perfect parabola, one is a line dragged by a single outlier, and one is a vertical cluster with one leverage point. The coefficient is the same in every case; only the scatter plot tells them apart. This is exactly why you plot before you compute.

A useful bonus of Pearson is that r² (the coefficient of determination) is the proportion of variance in one variable linearly explained by the other. An r of 0.50 means only 25% of the variance is shared — a reminder that moderate correlations explain less than they appear to.

Spearman's rho: monotonic association via ranks

Spearman's ρ is simply Pearson's r computed on the ranks of the data rather than the raw values. By ranking, it captures any monotonic relationship — one that consistently increases or decreases, even if it bends. If y rises with x but levels off (a logarithmic curve, say), Pearson will understate the link while Spearman captures it fully.

Because it works on ranks, Spearman is the right choice for ordinal data (Likert items, rankings, ordered categories) and for continuous data with outliers or skew, since ranking compresses extreme values. It does not require bivariate normality. The trade-off: Spearman answers a slightly different question — "do the two variables move together in order?" rather than "do they fall on a straight line?"

Kendall's tau: concordant vs discordant pairs

Kendall's τ also uses ranks, but builds on a different idea: it looks at every pair of observations and asks whether they are concordant (both variables agree on direction) or discordant (they disagree). τ is essentially the difference between the number of concordant and discordant pairs, scaled to the −1 to +1 range.

This gives τ a cleaner probabilistic interpretation: it is directly related to the probability that a randomly chosen pair is concordant minus the probability it is discordant. Kendall's τ is generally preferred for small samples, where its sampling distribution is better behaved, and for data with many tied ranks, where the τ-b variant handles ties more gracefully than Spearman. Its values tend to be numerically smaller than Spearman's on the same data — that is expected, not a contradiction, because the two are on different scales.

A worked example

Suppose you measure hours studied and exam score for 12 students. The scatter plot shows a clear upward trend that curves slightly — gains flatten at high study hours. Pearson returns r = 0.81 (95% CI [0.44, 0.94], p = .001), but the curvature means Pearson understates the association. Spearman returns ρ = 0.93 and Kendall returns τ = 0.82, both higher because the relationship is strongly monotonic even though it is not a straight line.

The honest report acknowledges all three: "Study hours and exam scores were strongly, monotonically associated, ρ = .93, p < .001, n = 12. Because the relationship was non-linear, Spearman's ρ is reported in preference to Pearson's r (= .81)." The scatter plot justifies the choice; the coefficients without the plot would have hidden the curve.

Decision guide

Reporting a correlation correctly

The coefficient is the effect size — there is no separate Cohen's d to compute. Rough benchmarks (Cohen) are |.10| small, |.30| medium, |.50| large, but interpret these against your field rather than mechanically. A complete report includes four things:

• The coefficient and its sign (e.g., r = .42).
• A confidence interval, which conveys precision far better than a p-value alone.
• The sample size n, since correlations from small samples are unstable.
• The p-value, reported as "< .001" rather than as a string of zeros.

For Pearson, also report r² as variance explained. And state which coefficient you used and why — reviewers should not have to guess. If you are checking whether your variables meet Pearson's normality assumption, see our guide on checking normality the right way. For framing the coefficient as a standardized effect, our note on reporting effect size in APA style is a useful companion. And if you are comparing groups rather than relating two continuous variables, the same parametric-versus-rank logic appears in ANOVA versus Kruskal-Wallis.

In MindStat you can compute all three coefficients with their confidence intervals from the same panel and overlay the scatter plot, so the visual check and the number arrive together.

Key takeaways

• Plot first. Anscombe's quartet proves a single coefficient can hide a curve, an outlier, or a cluster.
• Pearson for linear, continuous, roughly normal data — and it gives you r² as variance explained.
• Spearman for ordinal data and monotonic-but-curved relationships.
• Kendall for small samples, many ties, and a clean probabilistic reading.
• The coefficient is your effect size — report it with a CI, n, and p, and name the coefficient you chose.