Spearman Rank Correlation Calculator
Paste two lists and get Spearman's rho, the rank based correlation that shrugs off outliers.
What it does
Spearman's rank correlation, often written rS, measures how well two lists move together in order rather than in exact amounts. Rank the values in each column and the coefficient becomes the Pearson correlation of those ranks.
That rank step buys robustness. One huge outlier drags Pearson's r around; Spearman only sees that the value is the highest, not that it is ten times the next one. Curved but consistently rising pairs also score near 1.
The tool accepts 3 to 200 paired values per list, separated by commas or spaces, handles ties by averaging ranks, and prints rS with the pair count. A plain language note reads out strength and direction.
Use it for ordinal data like survey scales, ratings, or rankings, and for numeric data that bends or carries outliers. Avoid it when the relationship rises then falls, because monotonic is the price of admission.
How to use
- Paste the X values, separated by commas, spaces, or semicolons.
- Paste the Y values in the same order; the first X pairs with the first Y.
- Keep both lists the same length, from 3 to 200 entries.
- Read rS, which runs from -1 through 0 to 1.
- Read n, the number of pairs used.
- Read the note for strength words: 0.9 and up very strong, down to little below 0.3.
Formula
- R_i
- Rank of the i-th X value, ties averaged.
- S_i
- Rank of the i-th Y value.
- d_i
- Rank difference R_i minus S_i.
- n
- Number of pairs.
Ties take the average of the ranks they span, and the shortcut formula only holds without ties, so the tool always computes the full rank correlation. Both lists need spread: a constant list makes rS undefined and the tool says so. Rank correlation measures agreement in order, never cause.
Worked example
Example: X 1 to 5 against Y 2, 1, 4, 3, 5
X: 1, 2, 3, 4, 5. Y: 2, 1, 4, 3, 5.
- Ranks of Y: 2, 1, 4, 3, 5, since no values tie.
- Rank differences d: -1, 1, -1, 1, 0.
- Squares: 1, 1, 1, 1, 0, which sum to 4.
- Shortcut: 1 - 6 x 4 / (5 x 24) = 1 - 24/120 = 0.8.
- Five pairs, positive and strong. Tool shows rS 0.8 with n 5.
Reference table
| |rS| | Word | Example run |
|---|---|---|
| 0.9 and up | Very strong | 0.949 with ties, -1 for a perfect reversal |
| 0.7 to 0.9 | Strong | 0.8 on the default lists |
| 0.3 to 0.7 | Moderate to weak | Between the two extremes |
| Under 0.3 | Little | -0.1 on a shuffled pairing |
Limits
- Measures monotonic order, not straight line fit; a U shaped relationship can score near zero.
- At least 3 pairs and at most 200 per run here.
- A list with no spread makes rS undefined.
- No p value is reported; significance needs its own test.
- Ties are averaged, which softens the coefficient slightly.
Common mistakes
- Pairing the lists out of order, which destroys the relationship the tool should measure.
- Reading rS as a percent of agreement.
- Using it on data that rise then fall, where monotonic fails.
- Feeding unequal list lengths and expecting partial results.
- Calling a high rS proof of cause.
Try it yourself
Quick checks. Answers sit tucked under each question.
X 1 to 5 against Y 5, 4, 3, 2, 1?
-1. A perfect reversal, the strongest negative reading possible.
X 10, 20, 20, 30 against Y 1, 2, 3, 4?
0.949. The doubled 20s share rank 2.5, and the tool still reports a very strong positive link.
X 1 to 5 against Y 5, 1, 2, 4, 3?
-0.1, a little bit negative: the two orders barely relate.
Questions
What is Spearman's rho?
The correlation of the ranks of two variables. It reads how consistently one list rises when the other does, ignoring the size of the jumps.
How is it different from Pearson?
Pearson uses raw values and fits a straight line; Spearman uses ranks for monotonic relationships and resists outliers.
How are ties handled?
Tied values share the average of the ranks they span, and the full correlation is computed on those ranks.
How many pairs do I need?
Three minimum and more is better; this tool accepts 3 to 200. With very few pairs, one swapped value moves rS a lot.
Where is the p value?
Not shown. rS here is the effect size; significance testing needs a t approximation or a permutation test.
Does a high rS prove causation?
No. It shows the orders move together. Cause needs an experiment or a careful design.
Sources
- Wolfram MathWorld: Spearman Rank Correlation Coefficient
- The University of Texas at Austin: Spearman's Rank Correlation