Pearson vs Spearman vs Kendall Correlation Coefficient: Formulas, Differences, and Which to Use

Pearson correlation (r) measures linear association between two continuous variables. Formula: r = Σ(x - x̄)(y - ȳ) / √(Σ(x - x̄)² × Σ(y - ȳ)²). Ranges from -1 to +1. Assumes both variables are normally distributed and the relationship is linear. Sensitive to outliers. Spearman correlation (ρ or r_s) measures monotonic association between two variables using ranked data. Calculate Pearson correlation on the ranks. Ranges -1 to +1. No distributional assumptions; works with ordinal data or non-normal continuous data. Detects monotonic but non-linear relationships that Pearson misses. Kendall correlation (τ) measures monotonic association based on concordant and discordant pairs. τ = (C - D) / (n(n-1)/2), where C is the count of concordant pairs and D is the count of discordant pairs. Ranges -1 to +1. No distributional assumptions; more robust than Spearman for small samples and ties; more intuitive interpretation (τ is literally the probability difference between concordance and discordance). When to use each: Pearson for continuous, normally distributed, linear relationships (and if outliers are controlled). Spearman for non-normal continuous data, ordinal data, or when the relationship may be monotonic but non-linear. Kendall when sample size is small (under 30-40), when there are many tied values, or when you want a more robust and interpretable measure. In large samples with clean data, all three typically agree in sign and approximate magnitude. This content is for educational purposes only and does not constitute statistical advice.

Formula: r = Σ(xᵢ - x̄)(yᵢ - ȳ) / √(Σ(xᵢ - x̄)² × Σ(yᵢ - ȳ)²) Or equivalently: r = Cov(X,Y) / (σ_X × σ_Y) Pearson measures the strength and direction of a LINEAR relationship between two continuous variables. A positive r means both variables tend to increase together; negative means one increases while the other decreases; zero means no linear relationship. Assumptions: 1. Both variables are continuous (interval or ratio scale) 2. Both variables are normally distributed (or approximately so) 3. Relationship between them is linear 4. No extreme outliers (r is highly sensitive to outliers) 5. Homoscedasticity (variance of y is roughly constant across x) Violations of these assumptions can produce misleading r values. Worked example: heights (x) and weights (y) of 6 adults. x: 60, 62, 65, 68, 70, 72 (inches) y: 115, 130, 145, 160, 170, 185 (lbs) x̄ = 66.17, ȳ = 150.83 Deviations and products: (60-66.17)(115-150.83) = -6.17 × -35.83 = 221.07 (62-66.17)(130-150.83) = -4.17 × -20.83 = 86.86 (65-66.17)(145-150.83) = -1.17 × -5.83 = 6.82 (68-66.17)(160-150.83) = 1.83 × 9.17 = 16.78 (70-66.17)(170-150.83) = 3.83 × 19.17 = 73.42 (72-66.17)(185-150.83) = 5.83 × 34.17 = 199.21 Σ(x-x̄)(y-ȳ) = 604.16 Σ(x-x̄)² = 38.07 + 17.39 + 1.37 + 3.35 + 14.67 + 33.99 = 108.84 Σ(y-ȳ)² = 1283.76 + 434.08 + 33.99 + 84.09 + 367.48 + 1167.59 = 3370.99 r = 604.16 / √(108.84 × 3370.99) = 604.16 / √366,891.79 = 604.16 / 605.72 = 0.997 Very strong positive linear correlation, as expected for height and weight in adults. Interpret: as height increases, weight tends to increase linearly. Effect size interpretation (Cohen's conventions): - r = ±0.1: small - r = ±0.3: medium - r = ±0.5: large - r = ±0.7+: very large Coefficient of determination (r²) = 0.994 in this example. 99.4% of variance in weight is explained by variance in height (very unrealistic for real data, but illustrates the principle).

Spearman correlation is simply Pearson correlation computed on the ranks of the data. This has two consequences: 1. No distributional assumptions — ranks don't care whether the original data is normal, skewed, or bimodal. 2. Detects monotonic non-linear relationships — if y is a non-linear monotonic function of x (like y = x² for positive x, or y = log(x)), Pearson will understate the association but Spearman will pick it up. Formula (when no ties): ρ = 1 - (6 × Σd²) / (n × (n² - 1)) Where d = difference between the ranks of each pair, and n = number of pairs. With ties, use the Pearson formula applied to the ranks. Worked example: data showing a non-linear but monotonic relationship. x: 1, 2, 3, 4, 5, 6 y: 1, 8, 27, 64, 125, 216 (y = x³) Ranks: since the order matches exactly, rank of x_i = i, rank of y_i = i. All d values = 0. ρ = 1 - (6 × 0) / (6 × 35) = 1.0 Perfect Spearman correlation. Pearson for this data would be less than 1.0 because the relationship is non-linear (cubic). This demonstrates Spearman's key advantage: it captures monotonic association regardless of linearity. When Spearman outperforms Pearson: - Relationship is monotonic but clearly non-linear (e.g., exponential growth) - Data has outliers that bias Pearson - One or both variables are ordinal (cannot assume interval scale) - Data is non-normal (skewed, bimodal) When Pearson outperforms Spearman: - Data is truly linear and meets Pearson's assumptions - You care about the specific shape (magnitude, not just direction) of the relationship - Coefficient of determination (r²) interpretation is needed For most real-world data analysis, computing both and comparing is informative. If they agree, report Pearson (more interpretable). If they disagree substantially, investigate why (outliers, non-linearity) and typically report Spearman with an explanation. This content is for educational purposes only and does not constitute statistical advice.

Kendall's tau uses a different mathematical foundation — counting concordant and discordant pairs. A concordant pair is two data points where the variables move in the same direction (both higher or both lower). A discordant pair is two data points where the variables move in opposite directions. Formula (no ties): τ = (C - D) / (n(n-1)/2) Where C = number of concordant pairs, D = number of discordant pairs, and n(n-1)/2 is the total number of possible pairs. τ ranges from -1 (all pairs discordant) to +1 (all pairs concordant), with 0 indicating random association. Worked example: 5 data points. (x, y) pairs: (1,2), (2,4), (3,3), (4,6), (5,5) List all pairs and classify: (1,2)-(2,4): x increases (1→2), y increases (2→4). Concordant. (1,2)-(3,3): x increases, y increases (2→3). Concordant. (1,2)-(4,6): x increases, y increases (2→6). Concordant. (1,2)-(5,5): x increases, y increases (2→5). Concordant. (2,4)-(3,3): x increases (2→3), y decreases (4→3). Discordant. (2,4)-(4,6): x increases (2→4), y increases (4→6). Concordant. (2,4)-(5,5): x increases (2→5), y increases (4→5). Concordant. (3,3)-(4,6): x increases, y increases (3→6). Concordant. (3,3)-(5,5): x increases, y increases (3→5). Concordant. (4,6)-(5,5): x increases (4→5), y decreases (6→5). Discordant. Concordant: 8. Discordant: 2. Total pairs: n(n-1)/2 = 5×4/2 = 10. τ = (8 - 2) / 10 = 0.6 Interpretation: τ has an intuitive interpretation as a probability. τ = 0.6 means that for a random pair of observations, the probability of concordance minus the probability of discordance is 0.6. Or equivalently, if you randomly pick two pairs, the ranks are 80% concordant and 20% discordant (because (C + D)/10 = 1, and C - D = 6 means C = 8 = 80%, D = 2 = 20%). Kendall vs Spearman: for most data they produce similar rankings of variable pairs. Kendall is preferred when: - Sample size is small (under 30-40) - There are many tied values - Robustness against distribution assumptions is critical - The interpretation as 'probability of concordance minus discordance' is useful Kendall τ values tend to be smaller in magnitude than Spearman ρ for the same data. This is not a defect — it reflects different measurement scales. Don't directly compare Kendall and Spearman numbers; use the appropriate one for your situation.

Decision tree: Step 1: What's your data type? - Both variables continuous, interval/ratio scale → proceed to step 2 - At least one variable ordinal → Spearman or Kendall (not Pearson) - Nominal data → correlation is not appropriate; use chi-square or other measures Step 2: Is the relationship linear? - Yes, clearly linear → proceed to step 3 - No, but monotonic (always increases or always decreases) → Spearman or Kendall - Non-monotonic → correlation is not appropriate; consider regression with polynomial terms or non-linear models Step 3: Are both variables approximately normally distributed? - Yes → Pearson is the best choice - No, skewed or non-normal → Spearman or Kendall Step 4: Are there outliers? - No outliers → Pearson (with other assumptions met) - Outliers present → Spearman or Kendall (robust to outliers) Step 5: Is sample size large (n ≥ 40)? - Yes → Pearson or Spearman generally work well - No (small sample) → Kendall is preferred (better small-sample properties) Final recommendations: For standard linear analysis with normal, large data: Pearson. For non-normal data, ordinal data, or suspected non-linear monotonic: Spearman. For small samples (n < 30-40) or many ties: Kendall. For complete analyses reporting multiple correlations: compute all three. Divergence between them is informative — it reveals non-linearity, outliers, or distributional issues. This content is for educational purposes only and does not constitute statistical advice.

Common mistakes: 1. Using Pearson when it's inappropriate. Applying Pearson to non-normal data, ordinal variables, or relationships with obvious outliers produces misleading results. Always examine your data visually (scatterplot) before choosing a correlation coefficient. 2. Interpreting correlation as causation. No correlation coefficient establishes causation, regardless of magnitude. This is first-week statistics content but constantly forgotten. Correlation only quantifies association. 3. Over-interpreting small correlations. r = 0.1 or r = 0.15 in a large sample can be statistically significant but explains less than 2% of variance. Statistical significance and practical importance are different things. 4. Ignoring non-linearity. A near-zero Pearson correlation doesn't mean no relationship — it means no LINEAR relationship. A U-shaped relationship can have r = 0 while still being a strong non-linear association. Always plot your data. 5. Sample size effects on significance. The p-value for a correlation depends on both magnitude and sample size. Very large samples produce statistical significance for trivial correlations. Report effect size (the coefficient itself) alongside any p-value, not instead of it. 6. Attenuation from measurement error. Unreliable measurements systematically understate correlation (the attenuation phenomenon). If your variables have substantial measurement error, true correlations are higher than observed. In reliability-focused contexts, use disattenuated correlation formulas. 7. Using correlation when regression would be better. Correlation measures association symmetrically (same result regardless of which variable is X or Y). Regression measures how one variable predicts another — different question, different interpretation. Best practices: 1. Always plot the data first (scatterplot). 2. Choose the coefficient based on data properties, not habit. 3. Report the coefficient, sample size, and confidence interval (not just p-value). 4. State which coefficient you used and why. 5. For non-standard situations, compute and report multiple coefficients. This content is for educational purposes only and does not constitute statistical advice.