Spearman's Correlation

Measure rank-based monotonic association between two variables.

Last updated: March 2026

Calculator

X Values

Comma or space separated

Y Values

Must match X count

Spearman's rₛ

-0.1758

Very weak / none negative

Σd² (rank diff²)194.00

t-statistic-0.5050

Pairs (n)10

What is Spearman's Correlation?

Spearman's rank correlation coefficient (rₛ or ρ) measures the monotonic relationship between two variables. Unlike Pearson's correlation which measures linear relationships, Spearman's works with ranks (ordinal positions) of data values. This makes it robust to outliers and suitable for non-linear, monotonic trends.

The key difference: convert raw data to ranks (1, 2, 3, ..., n), then calculate correlation on the ranks. Tied values get the average rank. Spearman's correlation ranges from −1 (perfect negative rank correlation) to +1 (perfect positive rank correlation), with 0 indicating no monotonic relationship.

Applications include rating scales (movie ratings vs user engagement), ordinal data (economic ranking vs health outcomes), and detecting non-linear but monotonic patterns. It's distribution-free and doesn't assume normality, making it popular in non-parametric statistics.

How to Calculate

Step-by-Step Process

Step 1: Pair X and Y values (must have equal count, n ≥ 3)

Step 2: Rank X values (1 to n), handling ties by averaging ranks

Step 3: Rank Y values same way

Step 4: Calculate dᵢ = rankₓᵢ − rankᵧᵢ for each pair

Step 5: Sum squared differences and apply formula

Key Formulas

Spearman's rₛ:

rₛ = 1 − (6 × Σd² / (n(n² − 1)))

Where:

d = difference between ranks, n = number of pairs

t-statistic (for hypothesis test):

t = rₛ × √((n − 2) / (1 − rₛ²))

Interpretation:

Spearman vs Pearson

Spearman's

• Rank-based, non-linear

• Robust to outliers

• For ordinal data

Pearson's

• Value-based, linear

• Sensitive to outliers

• For continuous data

Real-World Example

Movie Ratings vs Box Office Success

Context:

Critic ratings (1-10) vs box office revenue (millions) for 8 films: higher rating may correlate with higher revenue monotonically, but not necessarily linearly.

Data:

Critic Ratings: [6.5, 7.2, 8.1, 5.9, 8.5, 7.8, 6.3, 8.9]
Box Office ($M): [45, 62, 105, 28, 180, 95, 52, 220]

Analysis:

Step 1: Rank critic ratings

5.9 (1), 6.3 (2), 6.5 (3), 7.2 (4), 7.8 (5), 8.1 (6), 8.5 (7), 8.9 (8)

Step 2: Rank box office revenue

28 (1), 45 (2), 52 (3), 62 (4), 95 (5), 105 (6), 180 (7), 220 (8)

Step 3: Calculate Σd²

d values show close rank alignment. rₛ ≈ 0.88 (strong positive)

Result: Strong monotonic relationship between critic ratings and box office success

Frequently Asked Questions

When should I use Spearman instead of Pearson?

Use Spearman when: data has outliers, relationship is monotonic but not linear, data is ordinal (ranks/ratings), or normality assumption is violated. Pearson assumes linear relationship; Spearman doesn't.

How do ties affect Spearman's correlation?

When values are tied, assign average rank. Example: if two scores are tied for 3rd place, both get rank 3.5. The calculator handles ties automatically by averaging ranks in the tied group.

Can Spearman's be > 1 or < −1?

No, rₛ ranges from −1 to +1. Perfect −1: strictly decreasing ranks. Perfect +1: strictly increasing ranks. 0: no monotonic relationship.

What does t-statistic tell me?

t-statistic tests whether correlation is statistically significantly different from 0. Compare t to critical value (depends on sample size and significance level α). Larger |t| suggests stronger evidence of correlation.

Can I use Spearman with very small samples?

Yes, minimum n = 3, but power is very low. With n < 10, large changes yield big rₛ swing. Larger samples (n > 30) give more reliable estimates and better statistical power.

How does Spearman differ from other non-parametric tests?

Spearman measures association strength between two ranked variables. Kendall's τ is similar but uses different formula. Spearman is more widely used and easier to interpret for most applications.

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