Spearman's Rank Correlation Calculator
Enter two paired data lists and get Spearman's rₛ — a non-parametric correlation coefficient that measures how well a monotonic function describes the relationship, even for non-normal or ordinal data.
Strong positive monotonic correlation
- 1
Σd² — sum of squared rank differences
Σ (rank(xᵢ) − rank(yᵢ))² = 16 - 2
Denominator n(n²−1)
8 × (8² − 1) = 504 - 3
Spearman's rₛ
1 − 6 × 16 ÷ 504 = 0.8095Ranges from −1 (perfect negative) to +1 (perfect positive monotonic agreement).
How does this calculator work?
Spearman's rₛ = 1 − 6Σd²/(n(n²−1)) measures monotonic association between two ranked variables. It ranges from −1 (perfect negative) through 0 (no association) to +1 (perfect positive). Unlike Pearson, it is robust to outliers and works on ordinal data, making it the default for non-normal distributions.
Formula
How this is calculated
Spearman's rank correlation (rₛ) measures the strength and direction of the monotonic relationship between two variables — whether higher values of X consistently correspond to higher (or lower) values of Y, regardless of whether the relationship is linear. It works by converting raw values to ranks and then computing the product-moment correlation of those ranks.
The shortcut formula rₛ = 1 − 6Σd²/(n(n²−1)) applies exactly when there are no ties. When tied values exist, the calculator assigns each tied group the average of the ranks they would occupy (e.g. the two values tied for ranks 3 and 4 each get rank 3.5), which ensures the formula remains accurate. A result of +1 means perfect positive monotonic agreement, −1 means perfect negative monotonic agreement, and 0 means no monotonic relationship.
The t-statistic shown is t = rₛ√(n−2)/√(1−rₛ²) with df = n − 2, which can be compared to a t-table to test the null hypothesis H₀: ρₛ = 0 (no association). This approximation is reasonable for n ≥ 10 and moderate |rₛ|, but for small n or very high |rₛ| a permutation or exact test is more appropriate.
Frequently asked questions
Use Spearman's when your data are ordinal (ranked categories), when the relationship is monotonic but not linear, or when the data contain outliers or clear departures from normality. Pearson's assumes a linear relationship and normally distributed variables — if those assumptions hold, Pearson gives more statistical power.
Tied values receive the average of the ranks they would have occupied. For example, if two observations are tied at positions 3 and 4, both receive rank 3.5. This average-rank method keeps the formula exact and is the standard approach in most statistics software.
There are no universal thresholds, but a common guideline (Cohen 1988) treats |rₛ| < 0.1 as negligible, 0.1–0.3 as weak, 0.3–0.5 as moderate, 0.5–0.7 as strong, and > 0.7 as very strong. Context matters: in social-science surveys 0.6 is quite strong, while in physical measurements it may indicate substantial unexplained variation.
Also known as
TG we-Calculate Editorial Team. (2026). Spearman's Rank Correlation Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/spearmans-rank-correlation-calculator
TG we-Calculate Editorial Team. "Spearman's Rank Correlation Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/spearmans-rank-correlation-calculator.
TG we-Calculate Editorial Team, "Spearman's Rank Correlation Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/spearmans-rank-correlation-calculator
@misc{wecalculate_spearmans_rank_correlation_calculator, title = {Spearman's Rank Correlation Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/spearmans-rank-correlation-calculator}}, year = {2026}, note = {TG we-Calculate} }
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