Spearman Rank Correlation Calculator
This Spearman rank correlation calculator takes two paired lists of equal length. Example: IQ and weekly TV hours of 10 people give Σd² = 194 and ρ = −0.175758.
Source: Wikipedia – Spearman’s rank correlation coefficient. Updated: .
How it is calculated
Spearman’s correlation measures how well a steadily rising or falling curve, not just a straight line, describes the link between two variables.
How it is calculated
Each list is sorted on its own and given ranks 1 to n. Tied values receive the average of the ranks they would occupy. Without ties, ρ = 1 − 6 · Σd² / (n · (n² − 1)), where d is the difference between the two ranks of a pair. With ties you take the Pearson correlation of the ranks; that is what the calculator does in both cases.
Two worked examples
For x = 86, 97, 99, 100, 101, 103, 106, 110, 112, 113 and y = 2, 20, 28, 27, 50, 29, 7, 17, 6, 12, the x ranks are 1 to 10 and the y ranks are 1, 6, 8, 7, 10, 9, 3, 5, 2, 4. The squared differences add up to 194, so ρ = 1 − 6 · 194 / (10 · 99) = −0.175758. With ties, for example x = 1, 2, 2, 4, 5 and y = 3, 1, 4, 5, 5, the ranks are 1; 2.5; 2.5; 4; 5 and 2; 1; 3; 4.5; 4.5, and ρ = 0.789474.
- ρ always lies between −1 and +1; at ±1 the relationship is strictly monotonic.
- Unlike Pearson’s r, ρ barely reacts to outliers and also captures curved but steadily rising or falling patterns.
- For ordinal data such as grades or league tables, ρ is the usual measure.
Frequently asked questions
How do you calculate Spearman’s rank correlation?
Turn both lists into ranks, then take the Pearson correlation of the ranks. Without ties, ρ = 1 − 6Σd² / (n(n² − 1)) is enough.
What happens with tied values?
Tied values get the average rank: two values in places 2 and 3 both receive 2.5. The simple Σd² formula is then only approximate, so the calculator works on the ranks directly.
When should I use Spearman instead of Pearson?
For ranked data, with outliers, or when the relationship is monotonic rather than linear. For a linear link in normally distributed data, Pearson is more informative.
Sources and legal basis
- Wikipedia – Spearman’s rank correlation coefficient
- Wikipedia – Spearmans Rangkorrelationskoeffizient
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