Spearman's Rho and Pearson's r
Tests of Correlation: Spearman's Rho and Pearson's r
Both Spearman's rho (rs) and Pearson's r are correlation coefficients that measure the strength and direction of the relationship between two variables — ranging from −1.00 (perfect negative) to +1.00 (perfect positive). They differ in their assumptions, the level of data they require, and whether they detect linear or monotonic relationships.
Pearson's r (Parametric)
Pearson's product-moment correlation coefficient measures the strength and direction of the linear relationship between two variables. It requires:
- Data at interval or ratio level
- An approximately normal distribution for both variables
- A linear (straight-line) relationship between the variables
Pearson's r uses the actual numerical values of all data points. It is more powerful than Spearman's rs when its assumptions are met — i.e. more sensitive to detecting genuine correlations.
Significance: significant if |r| ≥ critical value from the Pearson's r table for the given N and α.
Spearman's Rho rs (Non-Parametric)
Spearman's rank correlation coefficient measures the strength and direction of the monotonic relationship between two variables (a relationship that consistently increases or decreases, even if not in a straight line). It works by converting raw scores to ranks and computing the correlation on the ranks. It requires:
- Data at ordinal level or above
- No distributional assumptions
Because it ranks data first, Spearman's rs is robust to outliers (an outlier can only be the highest or lowest rank, limiting its influence) and is appropriate when the distribution is substantially non-normal.
Significance: significant if |rs| ≥ critical value from the Spearman's rho table for the given N and α.
Spearman's Rho: Calculation Procedure
- Rank each participant's score on Variable 1 (1 = lowest; tied ranks receive the mean of the ranks they would occupy).
- Rank each participant's score on Variable 2.
- Calculate d = Rank₁ − Rank₂ for each participant.
- Square each d to get d².
- Sum all d² values: Σd².
- Apply the formula: rs = 1 − (6Σd²) / (N(N²−1))
The obtained rs is then compared to the critical value table for the given N and α.
Comparison
| Pearson's r | Spearman's rs | |
|---|---|---|
| Type | Parametric | Non-parametric |
| Data level | Interval/ratio | Ordinal or above |
| Distribution | Normal required | No assumption |
| Detects | Linear relationships | Monotonic relationships |
| Outlier sensitivity | High | Low (uses ranks) |
| Power | Higher (when valid) | Lower |
Key Takeaways
- Both coefficients range −1.00 to +1.00: sign = direction, absolute value = strength.
- Pearson's r: parametric — interval/ratio data, normal distribution, detects linear relationships. Significant if |r| ≥ critical value.
- Spearman's rs: non-parametric — ordinal data, no distribution assumption, detects monotonic relationships. Robust to outliers (ranks limit their influence).
- Spearman formula: rs = 1 − (6Σd²) / (N(N²−1)). Rank both variables, calculate d = Rank₁ − Rank₂, square, sum, apply formula.
- Both significant if |obtained| ≥ critical value for given N and α from their respective tables.
- Prefer Pearson when assumptions are met (more powerful). Use Spearman for ordinal data, outliers, non-normal distributions, or monotonic-but-non-linear relationships.