A correlation coefficient is a single number that summarizes two things about the relationship between two variables: its direction (does one variable tend to increase as the other increases, or decrease?) and its strength (how closely do the two variables track each other?). Every correlation coefficient in common use is bounded between −1 and +1. A value of +1 means a perfect positive relationship, −1 means a perfect negative relationship, and 0 means no relationship of the specific type that coefficient is built to detect — for Pearson’s r, that means no linear relationship specifically, not necessarily no relationship at all.
That last distinction is the source of more misinterpretation than any other single fact about correlation coefficients, and it is why this guide gives real weight to plotting your data before you trust any r value, however clean it looks in a table.
Because a correlation coefficient summarizes the relationship between two variables at once, it is typically computed only after each variable has been examined individually — see CASRAI’s guide to descriptive statistics for the central tendency, dispersion, and shape statistics that describe one variable at a time.
The three coefficients you’ll actually use
Pearson’s r: the default for continuous, linear data
Pearson’s product-moment correlation coefficient (Pearson’s r) is what people usually mean when they say “correlation coefficient” without qualification. It measures the strength of a linear relationship between two continuous variables. Its assumptions matter because violating them doesn’t just make the test technically invalid — each violation distorts r in a specific, predictable direction:
- Continuous variables. Pearson’s r is designed for interval/ratio data. Applying it to ordinal data (e.g., Likert-scale responses treated as if evenly spaced) can misrepresent the relationship; use Spearman’s rho instead.
- Linearity. Pearson’s r only captures straight-line association. A strong curvilinear (e.g., U-shaped or exponential) relationship can produce an r near 0 even though the variables are tightly related — this is exactly the failure mode Anscombe’s quartet was built to demonstrate.
- Approximate normality. Significance testing on r assumes the sampling distribution is approximately normal, which is more robust with larger samples but shakier with small, skewed samples.
- Homoscedasticity. The spread of one variable should be roughly constant across the range of the other. Fan-shaped (heteroscedastic) data can still produce a misleadingly moderate r.
- No extreme outliers. Because Pearson’s r is calculated from squared deviations, a single extreme outlier can swing r substantially — toward a spurious correlation where none really exists, or away from a real one.
Spearman’s rho and Kendall’s tau: rank-based alternatives
Spearman’s rank correlation coefficient (rho, ρ or rs) and Kendall’s tau (τ) both work by converting your data to ranks before calculating the association, which is what makes them the standard choice when:
- Your data is ordinal rather than continuous (rankings, Likert items, severity grades).
- The relationship is monotonic but not linear — consistently increasing or decreasing, but not at a constant rate.
- Your data has outliers you can’t justify removing, since rank-based methods are far less sensitive to extreme values than Pearson’s r.
- Sample sizes are small and you can’t safely assume normality.
Spearman’s rho is essentially Pearson’s r calculated on ranks and is the more commonly reported of the two, especially where results need to be compared against a body of literature that already uses it. Kendall’s tau is often preferred in smaller samples or when many tied ranks are present, and it has a more direct probabilistic interpretation (it reflects the difference between the probability that two randomly chosen observations are ranked in the same order versus the opposite order across both variables). Neither coefficient assumes linearity — only that the relationship is monotonic (consistently one direction) — and both will understate association for a relationship that reverses direction partway through the data (e.g., an inverted-U).
Point-biserial and phi: coefficients for binary variables
Two further variants matter often enough to name: the point-biserial correlation is mathematically equivalent to Pearson’s r calculated where one variable is genuinely continuous and the other is naturally binary (e.g., correlating a continuous test score with pass/fail group membership). The phi coefficient is the corresponding measure when both variables are binary (e.g., two yes/no items), and is closely related to the chi-square test of independence for a 2×2 table. Both are interpreted on the same −1 to +1 scale as Pearson’s r.
Interpreting magnitude: use verbal bands cautiously
Many textbooks give fixed verbal bands for interpreting |r| — commonly something like 0.1 = weak, 0.3 = moderate, 0.5 = strong, following benchmarks popularized by Jacob Cohen’s work on statistical power. These bands are a convenient starting heuristic, not a universal standard, and treating them as one invites two real problems:
- They are field-dependent. An r of 0.3 between two psychological self-report measures might be considered a solid, well-established effect, while the same r of 0.3 relating a physical measurement to a clinical outcome might be considered far too weak to be useful. What counts as a “strong” correlation in a given field is a matter of that field’s typical effect sizes and its measurement precision, not a fixed statistical fact.
- Magnitude and importance are not the same thing. A small correlation can still be practically important (e.g., a modest correlation with a rare but serious outcome, aggregated across a large population), and a large correlation between two variables that are conceptually close to the same thing can be trivial.
r² as variance explained
Squaring the correlation coefficient (r²) gives the proportion of variance in one variable that is statistically associated with (“explained by,” in the purely statistical sense) the other, under a linear model. An r of 0.5 corresponds to an r² of 0.25 — 25% of variance accounted for, not 50%. This squared relationship is worth stating explicitly in any interpretation: it’s why moving from r = 0.3 to r = 0.5 represents a much bigger practical jump (roughly triple the explained variance, from about 9% to 25%) than the raw r values suggest at first glance.
Why you must plot the data
The single most practically important fact about correlation coefficients is that identical values of r can come from data that look nothing alike, and a table of coefficients alone cannot show you this — only a scatterplot can. A quick check of each variable’s own histogram is a natural first step before that scatterplot — it can already reveal the skew, bimodality, or outliers that Anscombe’s quartet shows a bare r value hides.
Anscombe’s quartet, constructed by the statistician Francis Anscombe in 1973, is the classic demonstration: four small datasets that share nearly identical summary statistics — including the same Pearson correlation coefficient (approximately 0.816) — despite looking completely different when plotted. One dataset shows a genuine linear relationship; another shows a clear curved (nonlinear) relationship that a straight-line fit misrepresents; a third shows a near-perfect linear relationship distorted by a single outlier; and a fourth shows almost no real relationship at all, with the entire correlation driven by one extreme point. All four produce essentially the same r.
The Datasaurus Dozen, created by Justin Matejka and George Fitzmaurice (Autodesk Research) and presented in their 2017 paper “Same Stats, Different Graphs,” extends the same lesson with a set of datasets that share the same mean, standard deviation, and Pearson correlation coefficient to two decimal places — yet when plotted, the points trace out entirely different shapes, including one that visibly forms the outline of a dinosaur.
The practical takeaway is not that correlation coefficients are useless, but that they are a summary, and summaries discard information by design. Reporting r without ever having looked at the corresponding scatterplot means you cannot rule out nonlinearity, a bimodal or clustered structure, or a single influential outlier driving the whole result. Plot first; compute second.
Correlation is not causation
A correlation coefficient, however large and however statistically significant, does not by itself establish that one variable causes the other. The relationship could run in the opposite direction (reverse causality), both variables could be driven by a third, unmeasured variable (confounding), or the relationship could be an artifact of how the sample was assembled (selection effects). Any of these can produce what CASRAI’s guide to spurious correlation covers in depth: a statistically real association with no direct causal link between the two variables. CASRAI covers this distinction, including the Bradford Hill criteria researchers use to weigh observational causal claims, in a dedicated comparison: Correlation vs. Causation, and in the fuller causal analysis guide covering what does and doesn’t license a causal claim from observational data. This guide focuses on what the coefficient itself measures and how to compute, interpret, and report it correctly; treat the causation question as a separate, necessary step that a correlation coefficient alone cannot answer.
Sources of distortion worth checking before you interpret r
Attenuation from measurement error
Any unreliability in how a variable is measured — a noisy instrument, an inconsistent rater, a short or poorly validated scale — pulls the observed correlation toward zero relative to the “true” correlation between the underlying constructs. This is called attenuation, and it means a low observed r can reflect measurement noise as much as a genuinely weak relationship. Measurement consistency is exactly what reliability statistics are built to quantify; see CASRAI’s comparison of test-retest vs. inter-rater reliability and the guide to types of validity in research for how to assess and improve it before trusting a correlation that depends on it.
Restriction of range
If your sample only captures a narrow slice of the true range of one or both variables — for example, correlating exam performance with an aptitude test only among students who were already accepted into a selective program — the observed correlation will typically be smaller than the correlation in the full, unrestricted population. Range restriction is a common, often invisible cause of an apparently weak correlation.
Ecological and atomistic fallacies
A correlation computed at the group or aggregate level (e.g., country-level averages) does not necessarily hold at the individual level, and vice versa. Inferring an individual-level relationship from a group-level correlation is the ecological fallacy; inferring a group-level relationship from individual-level data is the less commonly named but equally real atomistic fallacy. Always match the level at which a correlation was calculated to the level at which you’re interpreting it.
Significance vs. magnitude
A correlation’s p-value tells you how likely you’d be to observe a correlation at least this large in your sample if the true population correlation were zero — it does not tell you how large or practically meaningful the correlation actually is. With a large enough sample size, even a trivially small correlation (r = 0.05, for instance) can be statistically significant. Conversely, a moderately large correlation in a small sample may not reach significance at all. See CASRAI’s guide on what a p value actually measures, and the guides on p-values and how to report p-values, for the general version of this problem.
Because of this, best practice is to report a confidence interval for r alongside (or instead of) a bare significance test — a 95% CI communicates the precision of the estimate directly, in the same units as r itself, rather than collapsing that information into a single significant/non-significant judgment. See CASRAI’s guide on the confidence interval for how these are calculated and interpreted more generally, and the related concept of effect size, of which a correlation coefficient is itself one common form.
Multiple comparisons and correlation matrices
Correlating many variables against each other — a full correlation matrix across, say, twenty measured variables — produces a large number of pairwise tests simultaneously. Even if none of the underlying relationships are real, some pairs will show a “significant” correlation by chance alone at the conventional p < .05 threshold, simply because so many comparisons are being run. Reporting the single largest correlation found by scanning a full matrix, without adjusting for the number of comparisons made or without having specified that comparison in advance, substantially overstates the evidence for that specific relationship. Standard responses include correcting the significance threshold for the number of comparisons (e.g., a Bonferroni-style correction), pre-registering which specific correlations are of interest, or treating matrix-wide exploratory findings as hypothesis-generating rather than confirmatory.
Reporting a correlation coefficient in APA style
APA style reports a Pearson correlation with the sample size implied by the degrees of freedom (df = n − 2), the coefficient itself, and the p-value, typically alongside a confidence interval:
There was a moderate positive correlation between the two variables, r(48) = .42, p = .002, 95% CI [.17, .62].
For Spearman’s rho, report it as rs rather than r, along with sample size (Spearman’s degrees of freedom convention differs from Pearson’s, so many style guides recommend reporting N directly rather than df): for example, rs(38) = .35, p = .03. Always specify which coefficient was used (Pearson, Spearman, Kendall, point-biserial) rather than the generic term “correlation,” since the assumptions and interpretation differ across them.
Calculating a correlation coefficient: R, Python, and SPSS
R
cor(x, y, method = "pearson") returns the coefficient itself; method can be switched to "spearman" or "kendall". For the coefficient plus a significance test and confidence interval in one call, use cor.test(x, y, method = "pearson").
Python (SciPy)
from scipy.stats import pearsonr, spearmanr, kendalltau, then pearsonr(x, y) or spearmanr(x, y) returns both the coefficient and its p-value as a tuple. Recent SciPy versions also expose a .confidence_interval() method on the result object for Pearson’s r.
SPSS
Analyze > Correlate > Bivariate lets you select Pearson, Spearman, and/or Kendall’s tau-b simultaneously for a set of variables, producing a correlation matrix with significance values; a scatterplot should always be requested and reviewed alongside it, per the plotting guidance above, rather than reading the matrix in isolation.
Frequently asked questions
What does a correlation coefficient tell you?
It tells you the direction (positive or negative) and strength of association between two variables, summarized as a single number between −1 and +1. It does not, by itself, tell you why that association exists.
What is considered a strong correlation coefficient?
There is no universal cutoff. Commonly cited benchmarks (around 0.5 for “strong”) are field-dependent heuristics, not fixed statistical standards — what counts as strong in one discipline may be considered weak or trivial in another with more precise measurement or larger typical effects.
Can a correlation coefficient be greater than 1?
No. Pearson’s r, Spearman’s rho, and Kendall’s tau are all mathematically bounded between −1 and +1 by construction.
Is r or r² the effect size?
Both are used as effect sizes in different contexts. r describes direction and strength directly; r² describes the proportion of variance explained and discards the direction information (since it’s always positive).
Should I use Pearson or Spearman?
Use Pearson’s r for continuous, roughly linear, roughly normally distributed data without extreme outliers. Use Spearman’s rho (or Kendall’s tau) for ordinal data, non-linear-but-monotonic relationships, or data with outliers you can’t justify excluding.
