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How to interpret Pearson r and r squared

A plain-language guide to reading the correlation coefficient, judging strength and direction, and avoiding the classic misreadings of r.

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Strength and direction in one number

Pearson r packs two facts into a single figure between -1 and 1. The sign tells you the direction: a positive r means the two variables tend to increase together, while a negative r means one tends to fall as the other rises. The distance from zero tells you the strength, so an r of -0.85 describes a stronger relationship than an r of 0.4 even though it is negative. As a rough guide many analysts call values above 0.7 strong, 0.5 to 0.7 moderate, and below 0.3 weak, but sensible thresholds depend on your field.

What r squared adds

Squaring r turns it into r squared, the coefficient of determination. Because squaring removes the sign, r squared only speaks to strength, not direction, and always sits between 0 and 1. Read it as the fraction of the variation in one variable that a straight-line fit to the other can account for. An r of 0.6 looks respectable until you square it to 0.36 and realise the linear relationship explains only about a third of the movement, leaving the rest to other factors and noise.

Common traps to avoid

The famous warning that correlation is not causation is only the start. Pearson r is also fragile to outliers, and a single stray point can push a near-zero r toward the extremes or hide a genuine trend. It measures straight-line association only, so a strong curved relationship can register close to zero. Small samples are unstable too, since two points always produce an r of exactly one. Always plot the data before trusting the number.

Turning a coefficient into a decision

A coefficient on its own rarely settles a question. Pair r with a scatter plot to check the shape, with the sample size to judge reliability, and ideally with a significance test to see whether the pattern could be chance. When you report a result, quote r, r squared and n together so a reader can weigh the effect and the evidence behind it. That habit keeps a tidy single number from being oversold.

Frequently asked questions

Is an r of 0.5 good?

It depends entirely on the context. In a controlled physics experiment 0.5 would be disappointing, while in messy human behaviour research it can be a meaningful signal. Compare it against typical results in your field rather than a universal cutoff.

Can r be exactly zero?

Yes, when the cross-products above and below the means cancel out. A zero r means no linear trend, but the variables can still be strongly related in a nonlinear way, so never read zero as no relationship at all.

Should I report r or r squared?

Report both. r preserves the direction and matches the scatter you see, while r squared gives an intuitive percentage of explained variance. Together they describe the relationship more fully than either alone.