GIFSQ Tools · Module 16 — Statistics, Experimental Design & Research · Tool 310 of 500
Measure the association between two variables: Pearson’s r for linear association and Spearman’s rho for ranked association, with a significance test for r.
Try sample values — fills the form with the worked example below.
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Pearson r
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Spearman ρ
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p-value (r)
Equation: Pearson r = Σ(x − x̄)(y − ȳ) ÷ √(Σ(x − x̄)² Σ(y − ȳ)²). Spearman ρ is Pearson r on the ranks. Test: t = r√(n − 2) ÷ √(1 − r²), df = n − 2.
Method: Pearson product-moment correlation and Spearman rank correlation per ISO 16269-1. The p-value for r comes from the t distribution itself (incomplete beta function). Ranks use average ranks for ties.
Assumptions: the pairs are independent; Pearson assumes an approximately linear relationship and is sensitive to outliers; Spearman only assumes a monotone relationship.
Limitations: correlation is not causation, and a strong r can be produced by a single outlier or by two distinct clusters. Neither coefficient detects a non-monotone (for example U-shaped) relationship.
Rounding: r and ρ to 4 decimal places; p to 4 decimal places; intermediates unrounded.
This is a calculation aid, not a laboratory report — confirm against your accredited method before citing a value.
Method: Pearson product-moment correlation and Spearman rank correlation per ISO 16269-1. The p-value for r comes from the t distribution itself (incomplete beta function). Ranks use average ranks for ties.
Assumptions: the pairs are independent; Pearson assumes an approximately linear relationship and is sensitive to outliers; Spearman only assumes a monotone relationship.
Limitations: correlation is not causation, and a strong r can be produced by a single outlier or by two distinct clusters. Neither coefficient detects a non-monotone (for example U-shaped) relationship.
Rounding: r and ρ to 4 decimal places; p to 4 decimal places; intermediates unrounded.
This is a calculation aid, not a laboratory report — confirm against your accredited method before citing a value.
Worked example: X 1–10 with Y 2, 4, 5, 4, 5, 7, 8, 9, 10, 12: r = 0.9719, ρ = 0.9695, p < 0.0001 — a strong positive linear association.
Enter values above to see results.
If the scatter is curved rather than straight, Spearman’s ρ is the safer summary. Tool 310 of 500 — GIFSQ
