| # | X | Y | Rank X | Rank Y | d | d² |
|---|---|---|---|---|---|---|
| 1 | 10 | 15 | 1 | 1 | 0 | 0 |
| 2 | 20 | 18 | 2 | 2 | 0 | 0 |
| 3 | 30 | 30 | 3 | 4 | -1 | 1 |
| 4 | 40 | 28 | 4 | 3 | 1 | 1 |
| 5 | 50 | 50 | 5 | 5 | 0 | 0 |
| 6 | 60 | 55 | 6 | 6 | 0 | 0 |
| 7 | 70 | 65 | 7 | 8 | -1 | 1 |
| 8 | 80 | 60 | 8 | 7 | 1 | 1 |
Formula used
Spearman’s rho is the Pearson correlation of the ranked variables: ρ = corr(Rx, Ry). Ranks are computed with your selected tie method.
When there are no ties, the classic shortcut applies: ρ = 1 − (6 Σ d²) / (n(n²−1)), where d = Rx − Ry for each pair.
With ties, the shortcut can be biased; using correlation on ranks remains tie-aware.
How to use this calculator
- Choose an input mode: two lists, two-column paste, or CSV upload.
- Enter paired values so each X matches the corresponding Y.
- Pick a rank method and set epsilon if your data is rounded.
- Select how to handle missing entries, then click Compute.
- Review ρ, p-value, CI, and the rank table, then export.
FAQs
1) What does Spearman’s rho measure?
It measures the strength and direction of a monotonic relationship using ranks. It’s robust to nonlinearity because it compares orderings rather than raw distances.
2) When should I use Spearman instead of Pearson?
Use it for ordinal data, skewed distributions, outliers, or relationships that are monotonic but not linear. Pearson targets linear association and assumes interval scaling.
3) How are ties handled?
Tied values receive ranks based on your chosen rule (average, min, max, or dense). The main rho uses correlation on ranks, which naturally incorporates ties.
4) What if some values are missing or non-numeric?
Choose “Drop incomplete pairs” to remove any pair with an invalid entry. Choose “Stop with an error” when you want strict validation and clean inputs.
5) How do I interpret the rho value?
Values near +1 indicate strong increasing order agreement; near −1 indicate strong decreasing agreement. Around 0 suggests little monotonic association. Use the interpretation label as a quick guide, not a rule.
6) What does the p-value mean here?
It’s an approximate two‑tailed significance test using a t distribution with n−2 degrees of freedom. Small p-values suggest the observed rho is unlikely under zero association, given the approximation.
7) Is there a minimum sample size?
You can compute rho with n ≥ 2, but inference becomes meaningful with larger samples. For very small n, exact methods are preferred; treat p-values and intervals as rough guidance.
8) Can I use ordinal categories?
Yes, if you convert categories to ordered numeric scores first. Spearman relies on ordering, so your coding must reflect the true order of the categories.