Spearman Rank Correlation

You ran the test. You got a number. And now you’re staring at ρ = 0.43 wondering whether that’s actually telling you something useful — or whether you used the wrong test entirely.

That second question matters more than most textbooks admit. Spearman rank correlation is one of the most forgiving tools in statistics, but “forgiving” doesn’t mean “always right.” Run it on the wrong kind of data and it gives you a clean, confident answer that’s completely misleading. This guide covers what the test actually measures, how to calculate it step by step, how to run it in Python, R, and Excel, and — the part every other guide skips — when it’ll quietly lead you astray.

What Is Spearman Rank Correlation?

Spearman rank correlation is a statistical measure of how strongly two variables move together in the same direction — ranked from smallest to largest, rather than using their raw values.

Here’s the key insight that makes everything else click: Spearman’s ρ is mathematically identical to Pearson’s r, except it’s applied to the ranks of your data instead of the original numbers. You convert both variables to ranks — 1st, 2nd, 3rd — and then run a standard Pearson correlation on those ranks. That’s it. The formula exists as a computational shortcut; the underlying concept is just “Pearson on ranks.”

Charles Spearman published this method in 1904 in the American Journal of Psychology, in a paper titled “The Proof and Measurement of Association between Two Things.” He needed to correlate measurements that were ordinal (rankings of mental ability) and couldn’t assume his data was normally distributed. The rank-based approach solved both problems at once. 

The result, ρ (rho), always lands between -1 and +1:

• ρ = +1 means a perfect positive monotonic relationship — every time variable X goes up, variable Y goes up too, without exception
• ρ = -1 means a perfect negative monotonic relationship — every time X goes up, Y goes down
• ρ = 0 means no detectable monotonic pattern between the two

Monotonic vs Linear: Why This Distinction Actually Matters

Most people learn Pearson correlation first and think of correlation as measuring “how linear is this relationship?” Spearman measures something different — it measures whether the relationship is monotonic.

A monotonic relationship is one where the variables consistently move in the same direction, but not necessarily at a constant rate. Prices going up as quality improves is monotonic. Temperature dropping as altitude increases is monotonic. Neither has to be a straight line.

Here’s where it gets practical. Three scenarios:

Scenario 1 — Linear relationship. As study hours increase, exam scores increase at a roughly constant rate. Pearson and Spearman both work here; Pearson is slightly more precise because it captures the actual numerical relationship.

Scenario 2 — Monotonic but non-linear. As income increases, life satisfaction increases, but the gains flatten out at higher incomes. The relationship is real and consistent, but it curves. Pearson underestimates the strength here. Spearman catches it because it only cares about direction, not the rate of change.

Scenario 3 — Non-monotonic (U-shaped). This is the trap nobody warns you about.

Take the relationship between stress and performance. A little stress improves focus. A lot of stress collapses performance. The relationship is strong, but it curves up then back down. Run Spearman on this data and you’ll get ρ near zero — which looks like “no relationship.” But there IS a relationship. A strong one. Spearman just can’t see it because the variable increases in one range and decreases in another. This is what I’ll call the non-monotonic trap throughout this guide, and it’s the single biggest misuse of the test in practice.

The fix: always look at a scatter plot before running any correlation. If your data fans out in a U or an arch, neither Pearson nor Spearman is the right tool. You’d need something like polynomial regression or a non-linear model.

When to Use Spearman Rank Correlation

This section doesn’t exist in most guides. Knowing how to calculate Spearman is easy. Knowing when to reach for it takes more thought.

Use Spearman when:

Your data is ordinal — rankings, Likert scale responses (1–5 satisfaction scores, agree/disagree scales), any data where the categories have a clear order but the gaps between them aren’t necessarily equal. Spearman was designed for this. Pearson technically assumes those gaps are equal, which Likert data never guarantees.

Your data doesn’t follow a normal distribution. If a Shapiro-Wilk test or histogram shows your data is skewed, lumpy, or heavy-tailed, Spearman is the safer choice.

Your data has significant outliers. Here’s why Spearman is outlier-resistant in a concrete way: when you convert raw values to ranks, an extreme outlier gets assigned rank 1 or rank n — it can’t pull the correlation wildly off course the way it would in a Pearson calculation that uses actual values. A student who scored 1,000 points on an exam vs. 800 for second place doesn’t matter to Spearman; they’re both just ranked 1st and 2nd.

Your relationship looks monotonic but curved — the mid-scenario above.

Don’t use Spearman when:

Your relationship is non-monotonic (U-shaped, arch-shaped, or anything that reverses direction). This is the trap. If you’re not sure, plot first.

Your sample is very small — below about 10 pairs, the significance tests become unreliable. Below 5, the results are essentially meaningless. You’re comparing ranks, and with only 5 data points you have almost no statistical power.

You need precise interval-level interpretation. Spearman tells you about direction and rough strength; it discards the actual numerical values by converting to ranks. If the specific magnitude of the relationship matters for your analysis, you’ve lost some information.

The Spearman Rank Correlation Formula

The formula you’ll see most often:

Where:

• d = the difference between the rank of each pair (rank of X minus rank of Y)
• d² = that difference, squared
• Σd² = the sum of all squared differences
• n = the number of data pairs

Why does this work? When there are no tied ranks, the squared rank differences carry all the information about how differently the two variables ordered the same observations. If both variables ranked everything identically, every d = 0, so Σd² = 0, and ρ = 1. If one variable ranked everything in perfect reverse order, the differences are maximized and ρ = -1. Multiplying by 6 and dividing by n(n² − 1) normalizes this to the -1 to +1 range — that’s the bookkeeping, not the insight.

When your data has tied ranks, this simplified formula becomes inaccurate. Two observations with the same value get the average of the ranks they would have occupied. Two people who both scored 78 on an exam, occupying positions 3 and 4 in the ranking, each get rank 3.5. Once you have ties, you should use the full Pearson correlation formula applied to the ranks — which is what Python’s scipy.stats.spearmanr() and R’s cor() do automatically. Don’t try to manually apply the simplified formula when ties exist; you’ll get a wrong answer.

How to Calculate Spearman Rank Correlation by Hand

Let’s use something more interesting than exam scores. Imagine eight film critics and eight regular moviegoers each ranked the same eight films — 1 for their favorite, 8 for their least favorite. We want to know: do critics and audiences agree?

The raw rankings:

Film	Critic rank	Audience rank
Film A	1	3
Film B	2	1
Film C	3	2
Film D	4	6
Film E	5	4
Film F	6	8
Film G	7	5
Film H	8	7

Data is already in rank form here, so we skip the ranking step. That won’t always be the case — see below for how to rank raw data.

Step 1: Calculate d (rank difference) for each film.

Subtract audience rank from critic rank. Order doesn’t matter, since we’re squaring next.

Film	Critic rank	Audience rank	d
Film A	1	3	−2
Film B	2	1	+1
Film C	3	2	+1
Film D	4	6	−2
Film E	5	4	+1
Film F	6	8	−2
Film G	7	5	+2
Film H	8	7	+1

Step 2: Square each d value.

Film	d	d²
Film A	−2	4
Film B	+1	1
Film C	+1	1
Film D	−2	4
Film E	+1	1
Film F	−2	4
Film G	+2	4
Film H	+1	1

Step 3: Sum all d² values.

Step 4: Apply the formula.

That’s a strong positive correlation. Critics and audiences broadly agree on the ranking of these films, though not perfectly. Film F is the biggest disagreement — critics ranked it 6th, audiences ranked it last.

How to Handle Tied Ranks

What if two critics gave the same score to two different films and you need to rank them? Say films C and D both scored 72 points. They would have occupied ranks 3 and 4. So both get the average: (3 + 4) / 2 = 3.5. Rank 3 and rank 4 don’t exist in your table — they’ve been averaged into 3.5 for both tied observations.

Do this for every tie, in both variables separately. Then use the full Pearson formula on the ranked data, or just let the software handle it (all major packages do).

Spearman Rank Correlation in Python, R, and Excel

Python

Two approaches, both correct. The better one for most purposes is scipy.stats.spearmanr() because it gives you both the coefficient and the p-value in one call.

python
from scipy.stats import spearmanr
import numpy as np

critic_ranks = [1, 2, 3, 4, 5, 6, 7, 8]
audience_ranks = [3, 1, 2, 6, 4, 8, 5, 7]

rho, p_value = spearmanr(critic_ranks, audience_ranks)
print(f"Spearman rho: {rho:.3f}")
print(f"p-value: {p_value:.4f}")

Output:

Spearman rho: 0.762
p-value: 0.0279

The p-value of 0.028 means this result is statistically significant at the conventional α = 0.05 threshold — though more on what that actually means in the next section.

If you’re working with a pandas DataFrame and want to compute Spearman across multiple columns at once:

python
import pandas as pd

df = pd.DataFrame({
    'critic': [1, 2, 3, 4, 5, 6, 7, 8],
    'audience': [3, 1, 2, 6, 4, 8, 5, 7]
})

correlation_matrix = df.corr(method='spearman')
print(correlation_matrix)

This returns a correlation matrix, which is useful when you’re comparing more than two variables. The downside vs spearmanr(): no p-values included.

R

r
critic_ranks <- c(1, 2, 3, 4, 5, 6, 7, 8)
audience_ranks <- c(3, 1, 2, 6, 4, 8, 5, 7)

# Coefficient only
cor(critic_ranks, audience_ranks, method = "spearman")

# Coefficient + significance test
cor.test(critic_ranks, audience_ranks, method = "spearman")

Use cor.test() rather than cor() when you need a p-value. The output gives you ρ, the test statistic, degrees of freedom, and p-value. Note that confidence intervals are not returned for Spearman by default in R — unlike Pearson — because no standard closed-form CI exists for rank-based correlation. 

Excel

Nobody covers this, but a lot of researchers live in Excel. Here’s how to do it with a formula, without any plugins.

Your data is in columns A (critic scores) and B (audience scores), rows 2–9.

Step 1: In column C, rank the critic scores:

=RANK.AVG(A2, $A$2:$A$9, 1)

Drag down through C9. RANK.AVG handles ties automatically by averaging ranks — this is the correct method for Spearman.

Step 2: In column D, rank the audience scores the same way:

=RANK.AVG(B2, $B$2:$B$9, 1)

Step 3: Calculate Spearman’s ρ using CORREL on the two rank columns:

=CORREL(C2:C9, D2:D9)

That’s it. Excel doesn’t have a dedicated Spearman function, but Spearman is just Pearson applied to ranks — and CORREL gives you Pearson on whatever columns you feed it. Feed it the rank columns.

For p-values in Excel: Excel doesn’t calculate Spearman significance natively. You’ll need to convert ρ to a t-statistic and use the T.DIST function: t = ρ × √((n−2)/(1−ρ²)), then =T.DIST.2T(t_value, n-2). Or just use Python/R for significance testing and Excel for the coefficient.

Interpreting Your Spearman Result

You have a ρ value. Now what?

Strength Classification

The scale below reflects conventions widely used in behavioral and social science research, consistent with guidance in Cohen, J., Statistical Power Analysis for the Behavioral Sciences (2nd ed., 1988), which remains the most-cited reference for effect size and correlation thresholds in empirical research. 

ρ value	Interpretation
0.0 to < 0.1	No meaningful correlation
0.1 to < 0.3	Weak correlation
0.3 to < 0.5	Moderate correlation
0.5 to < 0.7	Strong correlation
0.7 to 1.0	Very strong correlation

Apply the same thresholds for negative values (−0.7 to −1.0 = very strong negative correlation, etc.).

These thresholds are not universal laws. They’re conventions from social science research. In psychology or education, ρ = 0.4 is often considered a respectable finding. In engineering or clinical diagnostics, you’d want ρ > 0.9 before you’d trust a measurement. Context matters — a lot.

Significance Testing: Is Your Result Meaningful?

Here’s a distinction that trips up a lot of people: statistical significance and practical significance are not the same thing.

Statistical significance tells you whether your result is unlikely to have occurred by chance if there were truly zero correlation in the population. It doesn’t tell you how strong or useful the correlation is.

With a large enough sample, you can get a statistically significant result for ρ = 0.08. That’s essentially a non-existent relationship — but n = 500 gave it enough power to reach p < 0.05. Conversely, ρ = 0.65 with n = 7 might not clear the significance threshold, even though 0.65 is a genuinely strong correlation.

The null hypothesis for Spearman is: H₀: ρ = 0 in the population (there is no monotonic association between the two variables).

When p < 0.05, you can reject that null hypothesis. When p ≥ 0.05, you can’t — but that doesn’t prove there’s no relationship; it just means your data doesn’t give you enough evidence to conclude one exists. And if your sample is small, a non-significant p-value really tells you very little…

Report both numbers. Always.

How to Report Spearman Correlation in APA Format

APA 7th edition format for Spearman:

A Spearman rank-order correlation was conducted to assess the relationship between [variable 1] and [variable 2]. There was a [positive/negative] [strength] monotonic association between [variable 1] and [variable 2], rs(n−2) = [ρ], p = [p-value].

Using our film critic example:

A Spearman rank-order correlation was conducted to assess the relationship between critic rankings and audience rankings across eight films. There was a statistically significant positive monotonic association between the two, rs(6) = .76, p = .028.

Note: the degrees of freedom in parentheses is n − 2, so 8 − 2 = 6. The coefficient is reported with a leading zero dropped (.76, not 0.76) per APA convention.

Spearman vs Pearson vs Kendall’s Tau

Three tests, similar purpose, different contexts. Here’s when to use each:

	Pearson	Spearman	Kendall’s Tau
Data type	Continuous, interval/ratio	Ordinal or continuous	Ordinal or continuous
Distribution assumption	Normal distribution	No assumption	No assumption
What it measures	Linear relationship	Monotonic relationship	Concordant/discordant pairs
Outlier sensitivity	High	Low	Low
Tied ranks	N/A	Handles with correction	Handles naturally
Best for small n	No	Acceptable	Yes — preferred
Gives p-value	Yes	Yes	Yes

The choice between Spearman and Kendall’s tau comes up often enough to deserve a direct answer. Prefer Kendall’s tau when your sample is small (under about 30 pairs) and you have many tied ranks. In those conditions, Kendall’s tau gives more stable estimates. For larger samples with few ties, Spearman and Kendall’s tau usually give similar results — either works.

Real-World Applications

Education research. A teacher ranks 20 students by class participation and by exam performance. Both variables are ordinal rankings. Spearman directly measures whether students who participate more also tend to score higher — no assumption about the distribution of either measure.

UX research. You survey 50 users on a Likert scale (1–7) about ease of use, and separately measure their task completion time. Likert data is ordinal; task completion time is often right-skewed. Spearman handles both cleanly without needing to transform either variable.

Finance. Analysts rank 30 companies by credit rating and separately rank them by bond yield. Ordinal ranks, monotonic expected relationship. Spearman gives a clean measure of how well credit ratings predict yield ranking.

Ecology. A field researcher ranks river sampling sites by a pollution index and separately by species diversity count. Species diversity data is rarely normally distributed in small ecological samples. Spearman is the standard choice here, and has been for decades.

Common Mistakes

1. Using the simplified formula when you have tied ranks. The ρ = 1 − (6Σd²/n(n²−1)) formula only works when all ranks are distinct. Any ties, and you need the full Pearson formula on ranks. All major software handles this automatically; doing it by hand with ties is where errors creep in.

2. Interpreting ρ ≈ 0 as “no relationship.” It means no monotonic relationship. A ρ near zero doesn’t rule out a U-shaped, arch-shaped, or otherwise non-monotonic pattern. Plot your data first. This is the non-monotonic trap again — worth repeating because it’s genuinely how results get misreported.

3. Confusing significance with strength. A very large sample produces significant results for tiny correlations. A very small sample can produce non-significant results for strong correlations. Look at ρ and p together, not just one of them.

4. Running Spearman on n < 10. With fewer than 10 pairs, the test has almost no statistical power. The significance values are unreliable, and a random arrangement of ranks can produce apparently interesting ρ values by chance. If you only have 6 or 7 data pairs, describe your results as preliminary and don’t draw firm conclusions from a p-value.

5. Treating correlation as causation. This mistake isn’t unique to Spearman, but it’s worth saying plainly: ρ = 0.85 doesn’t mean X causes Y. It means they tend to move in the same rank direction. The cause could be reversed, it could be a third variable driving both, or it could be coincidence in your sample — a confounder you haven’t accounted for.

Frequently Asked Questions

The Part Most Analyses Get Wrong

I think the most useful thing this guide can offer isn’t the formula or the Python code — both are easy to find. It’s this: Spearman rank correlation tells you about order, not magnitude. When you get ρ = 0.62, it tells you that the two variables tend to rank observations similarly. It doesn’t tell you that a one-unit increase in X produces any particular change in Y. That information was discarded when you converted to ranks.

This is the right tradeoff for ordinal data, non-normal distributions, and outlier-heavy samples. But for continuous, well-distributed data where you care about the actual numerical relationship, you’ve chosen convenience over precision. Spearman is the practical tool; Pearson is the precise one. Know which you need before you run the test.

And always plot your data first. A scatter plot takes 10 seconds and will stop you from falling into the non-monotonic trap — the one that makes Spearman report “no relationship” when a strong, real relationship exists.