Start with sign and scale
Pearson’s r normally ranges from −1 to 1:
- r > 0: larger values tend to accompany larger values in a linear pattern;
- r < 0: larger values tend to accompany smaller values;
- r near 0: little measured linear association;
- |r| near 1: a strong measured linear association.
The sign gives direction, not quality. An r of −.70 is stronger in magnitude than r = .30. Squaring r removes the sign and, in simple linear regression with an intercept, yields R². r versus R-squared explains when that connection applies.
Cohen’s attributed convention
Cohen described conventional effect-size definitions as arbitrary and preferred substantive evidence where available.1 So r = .42 can be described as between Cohen’s medium and large reference values, not as universally “moderate.” Where statistical thresholds come from explains that provenance distinction.
Other published guides use different bands, but their evidentiary status matters. Schober and colleagues call the cutoffs in their own example arbitrary and inconsistent; the table is a rule of thumb, not an anesthesiology standard.2 Replacing Cohen’s table with another unlabeled table would preserve the same problem.
Convention, distribution, and guideline are different claims
| Evidence type | Verified example | What it can support |
|---|---|---|
| Proposed convention | Cohen’s .10/.30/.50 values1 | A named fallback scale, not a natural law |
| Empirical distribution | Schäfer and Schwarz’s distributions from 993 psychology articles5 | What appeared in the sampled literature; no automatic recommendation |
| Distribution-derived guideline | Hemphill’s lower/middle/upper thirds from 380 meta-analytic studies4 | A scoped psychology guideline tied to its source distribution |
| Reasoned guideline | Funder and Ozer’s proposed psychology values7 | An argued convention, not an empirical percentile table |
| Disputed approach | Panzarella and colleagues’ objection8 | Evidence that field-distribution benchmarking itself is contested |
The empirical results also change with what enters the literature. Schäfer and Schwarz reported a median absolute correlation of .36 for non-preregistered studies and .16 for preregistered studies in their sample.5 Lovakov and Agadullina’s publisher abstract reports social-psychology quartiles of .12, .24, and .41 from 134 meta-analyses and frames them as guidelines when better information is unavailable.6 These results are context evidence, not a new universal scale. A literature distribution can reflect designs, measurement, selection, and publication processes as well as the phenomena being studied.
Inspect the scatterplot
A coefficient compresses a cloud of points into one number. Before interpreting strength, inspect:
- whether the pattern is roughly linear;
- whether one or two observations dominate the slope;
- whether groups form separate clusters;
- whether spread changes across the range;
- whether the variable range is restricted;
- whether repeated or clustered observations violate independence.
A U-shaped relationship can have r near 0 despite a strong pattern. A narrow sampled range can attenuate r. A single influential point can create an apparently strong value. correlation matrices and diagnostics and Pearson versus Spearman handle coefficient selection and supporting evidence.
Read uncertainty and sample size
The same r can be estimated precisely in a large sample and imprecisely in a small one. A confidence interval shows which population correlations remain compatible with the data and method. Do not infer importance from the p-value: sample size affects statistical significance even when the observed magnitude is unchanged.
Cohen’s d and effect-size conventions shows the parallel problem for standardized mean differences: a named convention can aid communication but cannot decide practical meaning.
Which coefficient is this?
| Coefficient | Typical use | Key caution |
|---|---|---|
| Pearson r | Linear association between quantitative variables | Sensitive to nonlinearity and influential observations |
| Spearman rho | Monotonic rank association | Ties and rank structure matter |
| Point-biserial r | Quantitative variable with a true binary grouping | Coding determines sign |
| Phi | Two binary variables | Equivalent to Pearson r on binary codes |
| Cramér’s V | Association in a larger contingency table | Nonnegative; table dimensions affect interpretation |
| Partial correlation | Association after linear adjustment for named variables | Depends on the adjustment model |
Do not apply Pearson r bands mechanically to Cramér’s V or another coefficient. Their sampling behavior and context differ.
Correlation is not agreement or causation
Two measurement methods can correlate highly while one is systematically higher, so correlation does not establish agreement. Nor does it establish that changing one variable changes the other. Confounding, selection, shared trends, and reverse direction can all produce association.2
For rater agreement use a coefficient tied to the categorical or quantitative agreement design rather than substituting correlation.
A defensible reporting sentence
“The variables had r = −.42 (95% CI [−.58, −.22]), a negative linear association. Its magnitude falls between Cohen’s medium and large behavioral-science conventions; the scatterplot, sampling context, and domain consequences informed interpretation.”