Correlation Coefficient: How to Read r Correctly (2026)

Quick answer: A correlation coefficient, written r, measures how strongly two numeric variables move together on a straight-line scale. It runs from −1 to +1: zero means no linear relationship, +1 means a perfect positive line, and −1 a perfect negative one. The coefficient is unitless and symmetric, so the correlation between height and weight equals that between weight and height. Crucially, r says nothing about causation, and it only captures straight-line relationships.

Part of our statistics series. If you want to see how a correlation is turned into a prediction, read effect size and correlation and regression in jamovi next.

Reading the size of r

The sign tells you the direction; the distance from zero tells you the strength. These bands are conventions, not laws, and they shift by discipline — what counts as “strong” in psychology differs from physics.

Value of r	Strength	Everyday reading
0.00 to ±0.19	Very weak	Essentially no usable linear pattern
±0.20 to ±0.39	Weak	A hint of a trend, much scatter
±0.40 to ±0.59	Moderate	A real relationship with plenty of noise
±0.60 to ±0.79	Strong	Tight enough to predict roughly
±0.80 to ±1.00	Very strong	Points hug the line closely

A common error is to read r as a percentage. It is not: r = 0.5 is not “50% related”. Square it to get r², the coefficient of determination, which is a proportion — an r of 0.5 means the line explains about 25% of the variation in the data. That single squaring step is what turns a flattering number into an honest one.

Pearson, Spearman and when to switch

The default coefficient is Pearson’s r, which measures a straight-line relationship. When the data curve, or when the values are ranks rather than measurements, Spearman’s ρ (rho) or Kendall’s τ is the right tool.

Coefficient	Measures	Use when
Pearson r	Linear association between two numeric variables	Both variables are continuous and roughly linear
Spearman ρ	Monotonic association using ranks	Data are ordinal, skewed or have outliers
Kendall τ	Concordance of pairs	Small samples, many tied ranks
Point-biserial	One numeric, one binary variable	Comparing a score against a yes/no group

# R: Pearson and Spearman on the same pair
cor.test(df$height, df$weight, method = "pearson")
cor.test(df$height, df$weight, method = "spearman")

# R-squared from a fitted line
fit <- lm(weight ~ height, data = df)
summary(fit)$r.squared

Always plot before you trust a number. The classic warnings are Anscombe’s quartet and the Datasaurus: four datasets with identical r can look completely different, and a single outlier can drag a coefficient from 0.2 to 0.9. The visual inspection steps mirror those in the R Commander graphs guide.

Correlation is not causation — and not even linearity

Two variables can correlate strongly because a third drives both. Shoe size and reading ability correlate in children only because age drives both. Two more traps: a strong correlation may be curved rather than straight (Pearson r understates it), and restricting the range of your data — say, sampling only tall people — collapses a relationship that exists in the full population. Before reporting an r, ask what else could move both numbers and whether your sample covers enough of the range.

Reporting r properly

A complete report gives the direction, the size, the sample and a confidence interval: “height and weight were positively correlated, r(48) = 0.71, 95% CI [0.54, 0.82], p < .001″. The degrees of freedom are n − 2 for a Pearson correlation. Pair the coefficient with a scatter plot and, when the relationship is used for prediction, the explained variance. For how intervals are read, see confidence intervals explained, and for the test behind the p-value, hypothesis testing in statistics.

Common mistakes

• Calling r a percentage. Square it for the proportion of variance explained.
• Ignoring curvature. A strong curved relationship can give a low Pearson r; switch to Spearman or fit a curve.
• Letting one outlier decide. Always plot, and consider a rank coefficient.
• Restricting the range. Sampling a narrow slice of the data shrinks r artificially.
• Jumping to cause. Correlation alone never establishes that one variable makes the other change.

Learning to read your own results critically? Ampersand Academy teaches statistics one-to-one, using your own dataset as the curriculum.

Frequently asked questions