Spearman Correlation: Interpretation and APA Reporting

The Basics

When to Use It

Use it when you've measured two ordered variables on the same people (continuous scores, ratings or rankings) and want to know whether higher values on one consistently go with higher, or lower, values on the other. It suits a monotonic relationship that bends, skewed scores and stray extreme points, where Pearson's *r*, which measures how closely the scores follow a straight line, can mislead. For example:

Do pianists who practice more sight-read better?
Is the number of stressful life events in the past year related to depression symptom scores?
Do teachers' rankings of students' reading ability match the students' ranks on a standardized reading test?

Vocabulary You Need

Definition Rank

A score's position when all the scores are lined up from lowest to highest: 1 for the lowest, 2 for the next, and so on. Tied scores share the average of the positions they fill. Ranks keep the order of the scores but throw away the distances between them, so one extreme score counts only as the highest and can't dominate a rank-based test, and taking logs or square roots of the scores leaves the ranks unchanged. Glossary →

Definition Spearman's Rho (

r_s

)

Rank each variable separately, then compute Pearson's r (how closely two variables follow a straight line, from −1 to 1) on the ranks. It runs from −1 (the two variables put people in exactly opposite orders) through 0 (no consistent order) to 1 (exactly the same order). Because it uses only the order of the scores, it measures any monotonic trend, straight or curved, and an extreme score counts only as the highest or lowest rank, however far out it is. It has no units, so it is its own effect size. Report it as rₛ; ρₛ (“rho”) is the population value, and you'll also see ρ used loosely for the sample value. Glossary →

Definition Monotonic Relationship

A relationship in which higher values of one variable go with higher values of the other all the way along (increasing), or with lower values all the way along (decreasing), though not necessarily at a steady rate. A straight line is monotonic, and so is a curve that levels off, like skill that improves quickly with the first hours of practice and slowly after that. A U shape or an inverted U isn't, because the direction changes partway. Rank-based measures of association, such as Spearman's rho, gamma and Somers' d, measure how consistently a relationship keeps one direction, not whether it's straight. Glossary →

Assumptions and Inference

1. Paired Ordered Scores Each person has a score on both variables, both can be put in order, and both vary.

What It Means: r_s ranks each variable and pairs each person's two ranks, so both scores must come from the same person and each must run from less to more of something. Continuous scores, counts, ratings and rankings all qualify; unordered categories don't. If everyone has the same score on either variable, r_s can't be computed.
Applies To: Variables
How to Check: By design. Check that both columns on a row belong to the same person, that a higher score means more on each variable, and that each variable takes a range of values.
How Much It Matters: It decides whether r_s fits. Practice hours and sight-reading scores are both ordered and both spread widely (1.5 to 17 hours, 32 to 98 points).

What to Do

Alternative Test Somers' D. If both variables are a handful of ordered categories, a measure built for an ordered table handles the many ties better.

2. Monotonic Relationship (for What rₛ Means) rₛ measures how consistently the two variables move in one direction; it can be computed for any shape, but a relationship that rises and then falls can give a low rₛ even when the variables are strongly related.

What It Means: The relationship keeps one direction across the whole range of scores, though it doesn't have to be a straight line or change at a steady rate. Rank-based coefficients summarize how consistently it keeps that direction, so a relationship that rises and then falls can show a coefficient near zero even when the two variables are strongly related.
Applies To: Variables
How to Check: Look at a scatterplot, ideally with a smooth trend line through it. For a small table of ordered categories, look instead at the percentages within each category (or a stacked bar chart): the share at the high end should move one way across the categories. A trend that bends but keeps going the same way is fine; one that clearly turns around, like a U or an inverted U, is not. It's a question about the shape, so decide by looking, not by a test.
How Much It Matters: A lot, for what r_s means. It measures a consistent one-direction trend, so a relationship that rises and then falls can give an r_s near zero. In the practice study, sight-reading scores are steeply higher across the first few hours of weekly practice and then level off; the three heaviest practicers dip a little, but the trend runs one way.

What to Do

Proceed with Caveat If the rise-then-fall is the finding, don't summarize it with one coefficient. Show the scatterplot and describe where the turn happens.
Alternative Test Multiple Linear Regression. A regression with the predictor and its square (or a spline) can describe a relationship that changes direction; adding the square is what lets the line bend.
Design Change If theory predicts one direction only over part of the range, restrict the question to that range and say so before collecting data.

3. Independent Observations (for the p-Value and Interval) The people (or other units) in your study don't influence each other.

What It Means: Each person's data comes from a separate, unconnected source, so knowing one person's scores tells you nothing about anyone else's. In designs that measure people more than once, their own scores are meant to be related; it's different people who must be independent. Methods built for clustered data, such as multilevel models and GEE, also allow members of the same group to be related, as long as the analysis names the grouping; then it's the groups that must be independent.
Applies To: Observations
How to Check: By design, not by a test. Did each person contribute their own data, with no clusters such as roommates, couples, therapy groups or classrooms whose members influence each other? If there are clusters, does the analysis model every one of them?
How Much It Matters: A lot. Ranking doesn't fix dependence. Pairs of scores that depend on each other, such as pianists who share a teacher and practice together, make p too small. And if studios differ in both average practice and average reading, r_s mixes the studio-level relationship with the individual one, and can even point the wrong way.

What to Do

Design Change Collect one pair of scores per independent person.
Alternative Test If people come in clusters (studios, classrooms, families), a multilevel model can estimate the relationship while accounting for the clustering.

4. Few Ties, or a Large Enough Sample (for the p-Value and Interval) The usual p-value and interval for a rank statistic are approximations that work best with a moderate sample and few tied scores.

What It Means: Software usually gets a rank statistic's p-value and interval from a formula that treats it as roughly normal or t-distributed. That works well once samples are moderately large, but with a small sample, or many people sharing the same score, the statistic's actual sampling distribution can differ, and the p-value and interval can be off. A permutation test, which shuffles one variable many times and recomputes the statistic, handles ties exactly; exact tables work only without ties.
Applies To: Observations
How to Check: Count the sample and the tied scores on each variable. With fewer than about 20 people, or long runs of identical scores, compare the usual p with a permutation p.
How Much It Matters: Some, for p and the interval only. With 40 pianists the approximations hold up, even though practice hours take only 16 different values (seven pianists reported exactly 3.0): a permutation test that shuffles the sight-reading scores 100,000 times gives p = .011, against .010 from the t approximation.

What to Do

Robust Estimation Shuffle one variable many times, recomputing r_s each time, and count how often a shuffled r_s is at least as far from zero as yours. It handles ties exactly. In R, coin::spearman_test(..., distribution = approximate(nresample = 10000)).
Robust Estimation Resample pianists with replacement and take the middle 95% of the resampled r_s values, when the sample is small or heavily tied.

Limitations

This Is Not Your Test If…

Your trend is a straight line and neither variable is badly skewed Pearson Correlation →
Your variables form a small table of ordered categories Somers' D →
Your relationship rises and then falls, so you need to model the curve itself (a squared term or a spline) Multiple Linear Regression →
You want to predict one variable from the other Simple Linear Regression →
You're comparing correlations between groups Comparing Correlation Coefficients →
Your variables are unordered categories Categorical Data and Proportions →

What It Doesn't Tell You

Why. Practice and sight-reading move together, but nothing here shows that practice built the skill; pianists who already read well may simply enjoy practicing more.
How much. r_s describes how consistently the orders line up, not how many points higher sight-reading tends to be for each extra hour. A model of the curve can describe that, but only as a difference between pianists, not as what an extra hour of practice would do for any one of them.
Who else. The result describes adult amateurs like these 40, over the range of practice they reported. Only three practiced more than 8 hours a week, so the data say little about heavy practice.
That there's no relationship, if p had been above .05. A non-significant r_s would have meant the data were inconclusive about a one-direction trend. A relationship that rises and then falls can even have an r_s near zero.

Common Mistakes

Don'tSwitch to a one-sided p because the result went the way you hoped.

InsteadUse a one-sided test only if you predicted the direction before collecting data, and say so.
Don'tRun Spearman because Pearson's r wasn't significant.

InsteadChoose before seeing the results, from the measures' scale and the shape you expect. If unsure, report both and say why.
Don'tDescribe a correlation as one variable causing, raising or improving the other.

InsteadSay the variables are associated. Causal claims need an experiment or a design that rules out third variables.
Don'tRead r_s² as the share of variation in the raw scores that the variables share.

InsteadSay it's the share of variation in one variable's ranks accounted for by a straight-line prediction from the other's ranks. It says nothing about variance on the original scales.
Don'tUse the 1 − 6Σd² shortcut (d is the gap between a person's two ranks) when many scores are tied.

InsteadCorrelate the midranks directly, as software does. The shortcut is exact only without ties.
Don'tAssume ranking removes every assumption.

InsteadCheck independence, look at the scatterplot for a trend that changes direction, and with a small sample or many ties check p with a permutation test.
Don'tRead a p-value as the chance that the null hypothesis is true.

InsteadRead it as: if the null were true, how often data at least this extreme would turn up.

Result Decoder

Significance and Effect Size Guidelines

Benchmarks Significance (p Value)

p < .001
Very strong evidence against the null hypothesis.
.001 ≤ p < .01
Strong evidence against the null hypothesis.
.01 ≤ p < .05
Statistically significant at α = .05. Moderate evidence against the null hypothesis.
.05 ≤ p < .10
Not significant. Weak evidence; sometimes called a “trend.” Treat it as inconclusive.
p ≥ .10
Not significant. Little evidence against the null, which is not the same as evidence of no effect.

These evidence labels are one common convention for describing p in graded language rather than as significant or not; the boundaries are rough guides, not rules, and other authors draw them differently. Muff et al. (2022) argue for this kind of language of evidence over a single cutoff. α is chosen before the analysis, and p never says how big or important an effect is.

More about statistical significance ↓

Benchmarks Effect Size (Spearman's

r_s

)

|rₛ| < .10 Below Small
Smaller than the conventional small benchmark. Whether an effect this size matters depends on the setting.
.10 ≤ |rₛ| < .30 Small
The two orderings line up a little, often too little to see in a scatterplot without computing it.
.30 ≤ |rₛ| < .50 Medium
A moderate tendency for the two orderings to agree; whether it shows in a scatterplot depends on the data.
|rₛ| ≥ .50 Large
A strong tendency for the two orderings to agree, though ties and a small sample can still blur the plotted pattern.

The sign shows direction; the benchmarks apply to its size. They are Cohen's (1988, 1992) benchmarks for Pearson's r, borrowed by convention, since rₛ is Pearson's r on ranks. They are not measures of practical or clinical importance.

More about effect size ↓

How It's Reported

Equation Form

Example

r_s(38) = .40, 95% CI [.09, .64], t(38) = 2.70, p = .010

Table 1. Spearman and Pearson Correlations between Weekly Practice and Sight-Reading

Example

	Coefficient	95% CI	$t$	$p$
Spearman's $r_s$ (ranks)	.40	[.09, .64]	2.70	.010
Pearson's $r$ (raw scores)	.24	[−.07, .52]	1.55	.130

Note. N = 40, so both tests have 38 degrees of freedom. The Spearman interval uses Fisher's z with Bonett and Wright's (2000) standard error; the Pearson interval uses the usual 1/√(n − 3).

Full Text (APA Style)

Example

Pianists practiced a median of 4.0 hours a week (range 1.5–17.0) and had a median sight-reading score of 73 out of 100 (range 32–98). Spearman's correlation was chosen before the analysis, because practice hours were expected to be right-skewed and sight-reading scores to level off at higher practice levels. Pianists who practiced more tended to sight-read better, r_s(38) = .40, 95% CI [.09, .64], a medium correlation by convention. The Spearman correlation was statistically significant, t(38) = 2.70, p = .010. Pearson's correlation on the raw scores was weaker and not significant, r(38) = .24, 95% CI [−.07, .52], p = .130: scores leveled off after about 6 hours a week, and the three pianists who practiced 12 or more hours were not the best sight-readers.

Name the coefficient: write r_s (APA style) so readers don't read it as Pearson's r, and keep ρ_s for the population value.
Say why you chose Spearman, and that you chose it before seeing the results.
Report medians and ranges rather than means and SDs when a variable is skewed enough to call for ranks.
Say how the interval was built. Several methods exist for r_s, and they give slightly different limits.
If you show Pearson's r too, say why they differ. Here the scatterplot's bend and three heavy practicers explain it.

Reading Each Piece

The equation form above, piece by piece, in this order: how big, how certain, then how surprising.

r_s(38) = .40	Pianists who practiced more tended to sight-read better. A rank correlation of .40 is medium by convention. The 38 in parentheses is the degrees of freedom, n − 2. Benchmarks ↑
95% CI [.09, .64]	The true rank correlation is plausibly anywhere from below small (.09) to large (.64). The whole range is above zero, but 40 pianists can't pin the strength down precisely.
t(38) = 2.70	t ≈ 2.70 on 38 degrees of freedom is an approximate test statistic for the null hypothesis that the population rank correlation is zero, and it's where p comes from. SPSS doesn't print t.
p = .010	If practice and sight-reading were truly unrelated, a rank correlation at least this far from zero, in either direction, would turn up about 1% of the time. That's below .05, so the correlation is statistically significant. Benchmarks ↑

Calculations

The Running Example

Example

The practice study: 40 adult amateur pianists, each with weekly practice hours and a sight-reading score, asking whether more practice goes with better sight-reading.

Null Hypothesis:: $\rho_s = 0$
Alternative (Two-Sided):: $\rho_s \neq 0$

$\rho_s$ (“rho s”) is the population Spearman correlation: how consistently more practice goes with better sight-reading among all pianists like these, which the sample's r_s = .40 estimates.

What Your Data Should Look Like

One row per person and one column for each variable, as raw scores; the software does the ranking. Both scores on a row must come from the same person.

Example

participant	practice_hours	sight_reading_score
1	2.0	73
2	5.0	98
3	2.0	54
4	3.0	57
5	2.5	81
6	4.5	65
… 34 more rows

	n	Mdn	Q₁–Q₃
Practice Hours	40	4	3–6
Sight Reading Score	40	73	65–79.25

Q₁–Q₃ is where the middle half of the scores lies, from the 25th to the 75th percentile (R's default quantile definition).

Pearson correlation between the two: r = .24

Simulated data.

Download all 40 rows (data.csv)

Napkin Math

The whole test, one idea at a time. Under each heading, a line says why the step matters, and the green note reads the equation in plain English with the practice study's numbers. Collapse all to read just the headings: they tell the whole story on their own.

1. Replace Each Score with Its Place in Line

Spearman cares only about order, so line up each variable from lowest to highest and replace every score with its position. People who tie share the average of the positions they fill.

$\text{tied rank} = \dfrac{\text{first position} + \text{last position}}{2}$ = $\dfrac{10 + 16}{2}$

= 13

Seven pianists reported exactly 3.0 hours, filling positions 10 through 16 in the practice line-up, so each gets the average, (10 + 16) ÷ 2 = 13. The heaviest practicer, at 17 hours, becomes rank 40: one step above the next person, however far ahead in hours.
2. Correlate the Ranks

Now run an ordinary Pearson correlation on the two columns of ranks. Pearson's r is the usual measure, from −1 to 1, of how closely two variables follow a straight line; on ranks, a straight line means the two orders match. The result is Spearman's correlation, and it's also the effect size.

$r_s = \dfrac{\text{cov}(R_x, R_y)}{s_{R_x} \, s_{R_y}}$ = $\dfrac{54.57}{11.63 \times 11.68}$

= .40

$1 - \tfrac{6\sum d^2}{n(n^2 - 1)}$ = 0.41

$R_x$ and $R_y$ are each pianist's practice rank and sight-reading rank. Their covariance is 54.57, and their SDs are 11.63 and 11.68, so $r_s$ = 54.57 ÷ (11.63 × 11.68) = .40. The textbook shortcut, $1 - 6\sum d^2 / (n(n^2 - 1))$ , where $d$ is each pianist's practice rank minus their sight-reading rank, gives .41, slightly off because so many pianists tied on hours.
3. See What Pearson's r Makes of the Raw Scores

For contrast, correlate the raw hours and scores instead of their ranks. Pearson's r looks for a straight line, so a bend and a few far-out points count against it.

$r = \dfrac{\text{cov}_{xy}}{s_x \, s_y}$ = $\dfrac{9.62}{3.20 \times 12.37}$

= .24

On the raw scores the covariance is 9.62 hour-points, and the SDs are 3.20 hours and 12.37 points: 9.62 ÷ (3.20 × 12.37) = .24. The three pianists practicing 12 to 17 hours sit far to the right with middling scores, and they pull the straight line flat.
4. Account for How Much Information We Had

Most software judges r_s against a t distribution with n − 2 degrees of freedom, as for Pearson's r. For ranks that's an approximation, not an exact result, but it works well once samples are moderately large.

$df = n - 2$ = $40 - 2$

= 38

$n$ is our 40 pianists. 40 − 2 = 38.
5. Approximate the Rank Correlation's Significance

Even if practice and sight-reading were unrelated, a sample rank correlation would drift away from zero by chance. Software commonly turns r_s into a t statistic with the same formula as for Pearson's r. For ranks, the t distribution is an approximation, especially with small samples or many ties.

$t = \dfrac{r_s\sqrt{n - 2}}{\sqrt{1 - r_s^2}}$ = $\dfrac{0.402\sqrt{38}}{\sqrt{1 - 0.161}}$

= 2.70

$r_s$ is our .402 and $n - 2$ is 38. .402 × √38 (6.16) = 2.48, and dividing by √(1 − .161) (0.916) gives 2.70. A permutation test, which shuffles the sight-reading scores 100,000 times and needs no t distribution, gives p = .011 (in R, with set.seed(2026)), close to the t approximation's .010, so the approximation holds up here despite the ties.
6. Stretch the Correlation onto a Scale Where Uncertainty Is Even

A correlation can't pass 1, so its chance variation gets squashed near the edges. Fisher's transformation stretches the scale so an interval can be built the same way everywhere.

$z_{r_s} = \tfrac{1}{2}\ln\!\left(\dfrac{1 + r_s}{1 - r_s}\right)$ = $\tfrac{1}{2}\ln\!\left(\dfrac{1.402}{0.598}\right)$

= 0.43

$\ln$ is the natural log. 1.402 ÷ 0.598 = 2.34, its log is 0.85, and half of that is 0.43.
7. Give a Range of Plausible Values for the True Rank Correlation

Build the range on the stretched scale, then convert both ends back. This interval doesn't come from step 5's t. Bonett and Wright's standard error for r_s is a bit wider than Pearson's 1/√(n − 3), because r_s varies a little more from sample to sample when scores are roughly normal. With heavy-tailed scores the reverse can hold (de Winter et al., 2016). It's an approximate interval: its coverage can vary with the sample size, the shape of the scores and how many are tied, and a bootstrap interval is an option when those are challenging.

$\tanh\!\left(z_{r_s} \pm 1.96\sqrt{\dfrac{1 + r_s^2/2}{n - 3}}\right)$ = $\tanh\!\left(0.43 \pm 1.96\sqrt{\dfrac{1 + 0.16/2}{37}}\right)$

= [.09, .64]

$SE_{z}$ = 0.171

$n - 3$ is 37 and $1 + r_s^2/2$ is 1.08, so the standard error is √(1.08 ÷ 37) = 0.171. 1.96 × 0.171 = 0.335, so the range is 0.426 ± 0.335, from 0.09 to 0.76. $\tanh$ undoes Fisher's stretch, converting the ends back to correlations: tanh(0.09) = .09 and tanh(0.76) = .64.
8. Ask How Surprising the Result Would Be If There Were No Real Relationship

Finally, ask how often chance alone would produce a rank correlation at least this far from zero if practice and sight-reading were truly unrelated.

$p = 2 \times P(T_{df} \le -|t|)$ = $2 \times P(T_{38} \le -2.70)$

= .010

$P(T_{38} \le -2.70)$ is the chance that a t with 38 df lands at or below −2.70. Doubling it counts both directions: .010, about 1 in 98.

Run It in Software

The code below reads the running example's data from data.csv; save it in the same folder you run the code from.

SPSS (version 30, as documented by IBM)

Analyze → Correlate → Bivariate.
Move practice_hours and sight_reading_score into Variables.
Under Correlation Coefficients, uncheck Pearson and check Spearman. Leave Two-tailed checked.
Click Confidence interval, check the box to estimate it, and under Spearman choose Bonett and Wright. Click Continue, then OK.

Or paste this syntax:

NONPAR CORR
  /VARIABLES=practice_hours sight_reading_score
  /PRINT=SPEARMAN TWOTAIL NOSIG FULL
  /CI METHOD(BW) CILEVEL(95)
  /MISSING=PAIRWISE.

Reading the Output

SPSS prints the correlation as a matrix, with 1.000 down the diagonal. Read the off-diagonal cell in this order: Correlation Coefficient, then the 95% Confidence Interval, then Sig. (2-tailed). SPSS doesn't print t.

Correlations

			practice_hours	sight_reading_score
Spearman's rho	practice_hours	Correlation Coefficient	1.000	.402*
		Sig. (2-tailed)	.	.010
		N	40	40
	sight_reading_score	Correlation Coefficient	.402*	1.000
		Sig. (2-tailed)	.010	.
		N	40	40

* Correlation is significant at the 0.05 level (2-tailed).

Confidence Intervals (Spearman's rho)

	Spearman's rho	Sig. (2-tailed)	95% CI Lower	Upper
practice_hours – sight_reading_score	.402	.010	.090	.641

SPSS's default interval method for Spearman is Fieller, Hartley and Pearson's, which gives [.094, .640] here. METHOD(BW) asks for Bonett and Wright's, to match the Napkin Math. Both are Spearman-based intervals, which Bishara and Hittner (2017) found held their coverage with non-normal data, where the usual Pearson interval often fell short. Unchecking Pearson in step 3 is optional: leave it checked to get Pearson's r = .243 (p = .130) for comparison.

Output laid out from this page's computed results, not captured from a live run; labels and layout may differ slightly in your version.

R

practice <- read.csv("data.csv")

cor.test(practice$practice_hours, practice$sight_reading_score,
         method = "spearman", exact = FALSE)

# 95% CI (Fisher z with Bonett and Wright's standard error)
library(statpsych)
ci.spear(0.05, practice$sight_reading_score, practice$practice_hours)

	Spearman's rank correlation rho

data:  practice$practice_hours and practice$sight_reading_score
S = 6378.6, p-value = 0.01021
alternative hypothesis: true rho is not equal to 0
sample estimates:
      rho
0.4016369

 Estimate      SE     LL     UL
   0.4016 0.14333 0.0904 0.6414

cor.test leads with the test and puts rho last; read rho first, then the interval from ci.spear, then p-value. S is not a t: it's the Σd² that would give this r_s if there were no ties. With tied scores R can't compute an exact p, so exact = FALSE asks for the t approximation used here (without it, R does the same and prints a warning). cor.test gives no interval for Spearman; ci.spear from Bonett's statpsych package does. Its SE (0.143) is on the correlation scale: it's the Napkin Math's z-scale 0.171 times (1 − r_s²). Neither is the 0.149 behind t, which serves the test, not the interval.

Output from a real run: R 4.6.1 (statpsych 2.0.0), run 2026-10-03.

Python

import numpy as np
import pandas as pd
from scipy import stats

practice = pd.read_csv("data.csv")
res = stats.spearmanr(practice["practice_hours"], practice["sight_reading_score"])
res.statistic   # 0.4016
res.pvalue      # 0.01021

# 95% CI: Fisher z with Bonett and Wright's standard error
n = len(practice)
z = np.arctanh(res.statistic)
se = np.sqrt((1 + res.statistic**2 / 2) / (n - 3))
np.tanh([z - 1.96 * se, z + 1.96 * se])   # array([0.0904, 0.6414])

Impact

Effect Size

Spearman's r_s is its own effect size. Like Pearson's r, it has no units and runs from −1 to 1, so it can be compared across studies. It measures how consistently the two variables put people in the same order, whatever the shape of the trend.

r_s = \dfrac{\text{cov}(R_x, R_y)}{s_{R_x} \, s_{R_y}} = \dfrac{54.57}{11.63 \times 11.68} = .40 \qquad r = .24

Pianists who practiced more tended to sight-read better, a rank correlation of .40: medium by convention. Pearson's r on the same scores is only .24, because sight-reading levels off after about 6 hours a week and the three heaviest practicers weren't the best readers.

Squaring it, r_s² = .16, gives the share of variation in one variable's ranks accounted for by its best-fitting straight-line prediction from the other's ranks, not the share of variation in hours or points. It's less useful here than for Pearson's r, so report r_s itself. The benchmarks are Cohen's for Pearson's r, borrowed by convention; for normally distributed scores r_s runs very close to r, so the borrowing is roughly fair, unlike for pair-counting measures such as Kendall's τ-b, which run lower.

Ties matter too. Seven pianists reported exactly 3.0 hours, and many others share a value. With heavy ties or a small sample, Kendall's τ-b, which counts pairs that are ordered the same way against pairs ordered the opposite way, is a common alternative. Choose one before the analysis and say which you used.

Whether practice is worth the hours is a question these data can't answer: they show who practices and who reads well, not what practice does.

Benchmarks Spearman's

r_s

at a Glance

Below Small

|rₛ| < .10

Small

.10 ≤ |rₛ| < .30

Medium

.30 ≤ |rₛ| < .50

Large

|rₛ| ≥ .50

Full benchmarks and caveats ↑

Statistical Significance

p answers one narrow question: if the two variables were unrelated in the population, how often would a sample's rank correlation land at least this far from zero, in either direction? Below the conventional α = .05, the correlation is called statistically significant. Most software gets p from the same t test used for Pearson's r, an approximation that works well once samples are moderately large. With small samples (roughly under 20), use an exact or permutation p instead; R's cor.test computes a more accurate one by default when there are no ties.

If practice and sight-reading were truly unrelated, a sample of 40 pianists would show a rank correlation at least this far from zero about 1% of the time, roughly 1 in 98. That's below .05, so it's significant. Pearson's r on the raw scores gave p = .130: same data, different question.

p is not the probability that there's no relationship, and it says nothing about how strong the relationship is. That's what r_s and its interval are for.

Benchmarks p at a Glance

Very Strong

p < .001

Strong

.001 ≤ p < .01

Significant

.01 ≤ p < .05

Weak

.05 ≤ p < .10

Little

p ≥ .10

Strength of evidence against the null hypothesis.

Full benchmarks and caveats ↑

Power and Sample Size

Power is the chance that a study detects an effect that's really there. A real relationship doesn't guarantee a significant result: with too few people, it can easily come out non-significant. Power depends on four things: how strong the true correlation is, how many people you have, α, and whether the test is one- or two-sided.

Suppose, before collecting any data, we expected practice and sight-reading to have a rank correlation of about .40. The calculator below says 48 pianists would give 80% power. Our 40 gave about 72%: a real .40 would be missed more than a quarter of the time.

Researchers usually aim for at least 80% power. Plan it before collecting data, using the smallest correlation that would matter. The calculator is an approximation for Spearman's test, not an exact calculation. It uses Fisher's z, judging the test against a correlation of zero (where Bonett and Wright's, 2000, standard error is the usual 1/√(n − 3)) and letting r_s vary around the true value by their larger standard error, √((1 + r_s²/2) ÷ (n − 3)). That combination is ours, not a procedure from their paper, so we checked it by simulation. For each setting we generated 20,000 samples of pairs from a bivariate normal distribution, with its correlation chosen so the population Spearman correlation equals the planned value; for some settings we also skewed and rounded the scores to produce ties like ours (hours to the nearest half hour, scores to whole points). We ran the usual t-approximation test on each sample and counted how often it was significant. Across true rank correlations from .2 to .6 and samples from 20 to 87, the simulated power was within 1.2 percentage points of the calculator's (48 pianists gave 80% to 81%). Heavy ties or an unusual shape of relationship can change that, so for a high-stakes study simulate data like yours. Bonett and Wright's paper plans sample size for the width of a confidence interval instead, which is the other well-grounded route. Try it:

Other Info

Follow-Up and Post Hoc Tests

None needed. This test answers a single question, so there are no planned contrasts or post hoc tests to run.

Related Methods

General CasePearson Correlation: Spearman's r_s is Pearson's r computed on ranks instead of raw scores. Many programs, SPSS among them, get its p from the same t formula, but that is an approximation for ranks, not exact, especially in small samples or with many ties; exact and permutation p-values are also used. Pearson asks about a straight line through the scores; Spearman asks about a consistent direction in their order.
Alternative MeasureGoodman and Kruskal's Gamma: a pair-counting measure for a small table of ordered categories, where most people share a category. It leaves out every tied pair, so on the same data it runs well above r_s or Kendall's tau-b.
Alternative MeasureSomers' D: an asymmetric pair-counting measure for ordered categories when one variable is the natural predictor. It leaves out pairs tied on the predictor, so it suits a small table with many ties better than r_s. Kendall's tau-b is the geometric mean of its two directions.
AlternativeMultiple Linear Regression: An alternative if “Monotonic Relationship” doesn't hold. A regression with the predictor and its square (or a spline) can describe a relationship that changes direction; adding the square is what lets the line bend.

References

Spearman, C. (1904). The proof and measurement of association between two things. The American Journal of Psychology, 15(1), 72–101. https://doi.org/10.2307/1412159
Bonett, D. G., & Wright, T. A. (2000). Sample size requirements for estimating Pearson, Kendall and Spearman correlations. Psychometrika, 65(1), 23–28. https://doi.org/10.1007/BF02294183
de Winter, J. C. F., Gosling, S. D., & Potter, J. (2016). Comparing the Pearson and Spearman correlation coefficients across distributions and sample sizes: A tutorial using simulations and empirical data. Psychological Methods, 21(3), 273–290. https://doi.org/10.1037/met0000079
Bishara, A. J., & Hittner, J. B. (2017). Confidence intervals for correlations when data are not normal. Behavior Research Methods, 49(1), 294–309. https://doi.org/10.3758/s13428-016-0702-8
“Only the Spearman rank-order and RIN transformation methods were universally robust to nonnormality.” (Abstract)
Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155–159. https://doi.org/10.1037/0033-2909.112.1.155
“small, medium, and large ESs are respectively .10, .30, and .50” (p. 156)
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
Muff, S., Nilsen, E. B., O'Hara, R. B., & Nater, C. R. (2022). Rewriting results sections in the language of evidence. Trends in Ecology & Evolution, 37(3), 203–210. https://doi.org/10.1016/j.tree.2021.10.009
“we suggest language of evidence that allows for a more nuanced approach to communicate scientific findings as a simple and intuitive alternative to statistical significance testing” (Abstract)