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Spearman's ρ

Spearman's ρ measures rank correlation between two variables, capturing any monotonic association rather than only a linear one — the standard non-parametric alternative to Pearson.

What is Spearman's ρ?

Spearman's ρ is just Pearson's r computed on the *ranks* of the data. By replacing values with their ranks before correlating, it strips out scale and shape information and asks only "does y tend to increase / decrease as x increases?". This makes it robust to outliers and free of any distributional assumptions for inference.

ρ = 1 ⇔ y is a strictly monotonic increasing function of x (perfect concordance of ranks); ρ = −1 ⇔ strictly monotonic decreasing. A non-linear but monotonic relationship like y = x³ has Pearson r < 1 but Spearman ρ = 1.

Less efficient than Pearson on truly bivariate-normal data (~91% efficiency), but the trade-off is essentially always worth it for ordinal data, heavy-tailed data, or data with outliers.

When should I use Spearman's ρ?

One or both variables ordinal (Likert, severity grades, ranks).
Continuous data with skew, heavy tails, or outliers — Pearson misleads here.
Monotonic but non-linear relationships.
Kendall's τ is often preferred over Spearman for small n or many ties (see corr_kendall).

What data does it need?

Two numeric or ordinal columns.

What does it report?

ρ, S statistic, p, 95% CI via Bonett-Wright Fisher-z.

What does it assume?

Independent paired observations.
Outcomes at least ordinal (rank-able).
Monotonic relationship (otherwise ρ still computes but has no clean interpretation).

Formula

ρ = 1 − 6 · Σ dᵢ² / (n · (n² − 1))   when there are no ties,  dᵢ = rank(xᵢ) − rank(yᵢ)

How do I interpret the result?

Same Cohen anchors as Pearson (0.1 / 0.3 / 0.5) but read as ordinal-rank concordance, not linear variance shared.

Big gap between Pearson r and Spearman ρ on the same data ⇒ the relationship is curved (or outliers are driving the Pearson estimate). Plot to confirm.

References

Spearman (1904). The proof and measurement of association between two things. American Journal of Psychology 15(1).