Test Pearson Correlation Significance From r and n

Pearson's r measures linear association between two variables, bounded by [−1, +1]. The significance test asks whether the observed |r| is large enough to rule out ρ = 0, the null of zero population correlation. Test statistic: t = r·√(n − 2) / √(1 − r²), distributed under H₀ as t with n − 2 degrees of freedom.

For a confidence interval on ρ, the t alone isn't enough. r has a skewed sampling distribution near ±1 that the t doesn't correct for. Apply Fisher's z-transform z = ½·ln((1 + r) / (1 − r)), approximately normal with SE = 1/√(n − 3); build the interval in z-space, then transform back. The page returns t, df, two-sided and one-sided p, and the Fisher-based 95% CI on ρ.

A significant correlation isn't automatically meaningful. Cohen's benchmarks for |r|: below 0.1 negligible, 0.1-0.3 small, 0.3-0.5 medium, 0.5-0.7 large, ≥ 0.7 very large. They come from behavioral science and aren't universal. In physics, r = 0.5 between two measured quantities is unimpressive. In epidemiology, r = 0.3 can be substantial. Use the benchmarks as a sanity check, not a target.

The variance explained by a linear relationship is r², not r. So r = 0.7 explains 49% of the variance, not 70%. r = 0.3 explains 9%. r = 0.1 explains 1%. Practical importance comes from r², not the headline correlation.

Sample size determines what's detectable; effect size determines what's meaningful. With n = 1000, r = 0.07 hits two-tailed p ≈ 0.026 but explains 0.5% of the variance. Statistically significant, practically irrelevant. The opposite case: r = 0.45 with n = 15 misses significance (p ≈ 0.09) despite a medium effect, because power is too low. Always report r alongside p, and ideally the 95% CI on ρ.

The CI on ρ is what tells you precision. Width tells you uncertainty. r = 0.42 with CI [0.14, 0.64] is medium but uncertain. r = 0.42 with CI [0.40, 0.44] would be a different story. CI excluding 0 is equivalent to two-tailed rejection at the same α. CI containing 0 means the data don't rule out ρ = 0, which isn't the same as proving ρ = 0.

APA 7 reporting expects r, df, p, and CI together. Format: "r(43) = 0.42, p = .004, 95% CI [0.14, 0.64]." Don't strip out the df or the CI to save space. Both carry information not in the bare correlation.

Small n. With n below ~30, the t-based test assumes bivariate normality of x and y, and the assumption matters. Heavy tails inflate Type I error. Severe skew distorts the sampling distribution of r. For small n with non-normal marginals, Spearman's rank correlation is the safer choice. R's cor.test(x, y, method = "spearman") and scipy.stats.spearmanr handle it. Below n = 5, no correlation test is reliable; the data don't support any conclusion about ρ.

Non-linear relationships. r only catches linear association. A clean parabola y = x² centered at zero gives r ≈ 0 over a symmetric range, even though the relationship is deterministic. Anscombe's quartet (1973) and the Datasaurus Dozen (Matejka and Fitzmaurice 2017) show datasets with identical r and radically different scatter shapes. Always plot before reporting. If the scatter is monotonic but curved, Spearman captures the monotonic trend by working on ranks. If it's non-monotonic, no single correlation coefficient is the right summary and you want a regression with the appropriate functional form.

Outliers. A single influential point can flip r from +0.5 to −0.2 at small n. Investigate before deciding what to do. Data-entry error: fix or remove. Genuine extreme observation: report r with and without the point, and prefer Spearman or biweight midcorrelation (R's WRS2 package). Never quietly drop outliers without disclosing it.

Restricted range. Sampling x over a narrow window suppresses r below its true population value. Studying SAT vs college GPA only on Harvard students gives a misleadingly low r because the SAT range is truncated near the top. If your sample range is much narrower than the population range you care about, the correlation you compute won't generalize.

Causation. r between A and B doesn't tell you A causes B, B causes A, or a third variable C causes both. Coffee drinking correlates with heart disease in observational data, but the direction and any common cause (age, smoking, work stress) need experimental design or causal inference machinery (instrumental variables, DAGs with sufficient assumptions, regression discontinuity) to disentangle. Pearl's "The Book of Why" is the accessible entry point. Correlations are useful for hypothesis generation and for predictive models that don't require causal interpretation, but they don't license causal claims on their own.

Multiple comparisons inflate false positives. Computing dozens of correlations across a wide dataset and reporting only the significant ones is data dredging. With 100 random pairs at α = 0.05, you expect 5 to come out significant by chance alone. Bonferroni or Benjamini-Hochberg corrections handle the family-wise error rate. Pre-register which correlations you care about, or use a two-stage exploratory/confirmatory split.

Selection effects produce spurious correlations and suppress real ones. Berkson's bias arises when sampling conditions on the outcome (hospital-based studies finding negative correlations between unrelated diseases). Range restriction deflates r when sampling truncates x. Both are easy to overlook because they're features of the sampling design, not the data.

Treating Pearson's r as the universal correlation. r assumes linearity. If your scatter is monotonic but bent, r underestimates the strength of the relationship. Spearman's ρ captures the monotonic trend by working on ranks. For non-monotonic relationships, neither correlation coefficient is appropriate; you want a regression model that matches the shape, or distance correlation (Székely, Rizzo & Bakirov 2007) which detects general dependence without assuming monotonicity.