Spearman Correlation Calculator

Calculator

Enter paired observations. Ranks are computed automatically.

Input mode

Choose the fastest way to enter pairs.

Decimals

Controls how results and ranks display.

Significance level (alpha)

Used to label p-value as significant or not.

Tie tolerance (optional)

Treat nearly-equal values as ties (advanced).

Reset

#	X value	Y value	Action
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Tip: leave blanks to ignore a row.

Example Data Table

This example includes a tie in X (two 18 values), showing average ranks.

#	X	Y	Rank X	Rank Y
1	12	10	1	1
2	18	16	2.5	2
3	18	22	2.5	3
4	25	24	4	4
5	30	35	5	5
6	45	40	6	6

Formula Used

Spearman’s rho measures how well a monotonic relationship fits the data.

Tie-aware approach (recommended)

Convert X and Y to ranks (average ranks for ties).
Compute Pearson correlation of the rank vectors.
This works with ties and non-linear monotonic trends.

ρ = corr( rank(X), rank(Y) )

Classic shortcut (no ties only)

If there are no ties in either column:

ρ = 1 − (6 Σ d²) / ( n (n² − 1) )

Here, d is the difference between ranks for each pair.

p-value note: This calculator uses a t-approximation with df = n − 2 for a two-tailed test. For very small samples, exact methods can differ.

How to Use This Calculator

Choose an input mode: table rows or pasted pairs.
Enter at least three valid (X, Y) pairs.
Set decimals and an optional significance level.
Press Calculate to view rho and p-value.
Use Download CSV or Download PDF after results appear.

When Spearman Fits Best

Spearman correlation summarizes whether two variables move in the same order. It is ideal when data are ordinal, skewed, or contain outliers that distort linear metrics. Use it for survey ratings, ranks, response times, or any setting where the relationship is monotonic but not necessarily straight. A rho near +1 indicates higher X tends to pair with higher Y; a rho near −1 indicates the opposite ordering. Unlike Pearson, it focuses on order, not spacing, so unequal gaps between values do not mislead much.

Ranking and Ties

The method converts each column to ranks, then compares those ranks. If values repeat, tied observations share the average of the tied rank positions. Ties are common in rounded measurements and Likert scales. When ties increase, rho typically shrinks toward zero because many pairs become indistinguishable. Reporting whether ties exist matters, especially if more than about 10% of entries repeat in either column.

Computing ρ and d²

Without ties, a shortcut uses the rank differences d for each pair: rho = 1 − 6Σd² / (n(n²−1)). The table in this calculator lists Rank X, Rank Y, d, and d² so you can verify every step. With ties, rho is computed by correlating the rank vectors directly, which remains valid. Check the Σd² row to see how disagreement between orderings accumulates.

Significance and Sample Size

To test whether the association differs from zero, the calculator applies a two‑tailed t approximation with df = n − 2. Larger samples stabilize rho and narrow uncertainty. As a rule of thumb, n ≥ 10 provides a more reliable p‑value, while n ≥ 30 supports clearer inference. Small samples can show large rho by chance, so interpret p‑values alongside context, measurement quality, and tie frequency.

Reporting and Decisions

Report n, rho, the p‑value, and a short practical interpretation. Common strength bands use |rho| < 0.20 very weak, 0.20–0.39 weak, 0.40–0.59 moderate, 0.60–0.79 strong, and ≥ 0.80 very strong. If the goal is prediction, also inspect a scatterplot of ranks to confirm monotonicity. Use the CSV and PDF exports to document inputs, ranks, and results for audits.

FAQs

What does Spearman rho represent?

It measures how consistently two variables move in the same ranked order. Values range from −1 to +1, where the sign shows direction and the magnitude shows monotonic strength.

When is it better than a linear correlation?

Use it for ordinal scores, skewed data, or relationships that rise or fall without a straight line. It is also less sensitive to extreme outliers because it works on ranks.

How does the calculator handle ties?

Repeated values receive the average of their tied rank positions. The final rho is computed from the ranked columns, which remains appropriate when ties exist.

How many data pairs are recommended?

At least three pairs are required, but n ≥ 10 is a safer minimum for inference. Larger samples improve stability and make the p-value more informative.

Can I analyze categories like 'low, medium, high'?

Yes, if you map categories to ordered numbers consistently, such as 1, 2, 3. Spearman depends on ordering, so preserve the intended rank structure.

Why does the table show d and d²?

Those columns explain rank differences for each pair. When there are no ties, Σd² plugs into the classic shortcut formula, offering a transparent cross-check of the main result.