Correlation

Two things vary together: taller people tend to weigh more, more study hours tend to mean higher grades. Correlation puts a single number on that co-movement: the correlation coefficient r, which runs from −1 to +1 and captures both the direction and the strength of a straight-line relationship.

Reading the number

• Sign = direction. Positive r: as one goes up, so does the other. Negative r: as one goes up, the other goes down.
• Size = strength. Near ±1 the points hug a straight line; near 0 they're a shapeless blob with no linear trend.

Numbers are abstract, though. The fastest way to build intuition is to see what an r of 0.3 versus 0.9 actually looks like, and to break a tidy relationship by dragging a single point.

Three things people get wrong

All three warnings collapse into one habit: plot your data. The classic proof is Anscombe's quartet: four datasets that share almost every summary number yet look nothing alike. (For two numeric variables the scatterplot is nearly always right; when one of them turns out to be categorical, the chart chooser hands you the display that fits instead.)

Every dataset above reports r = 0.82 and the exact same line, yet only the first is the tidy linear relationship that number implies. The second is a smooth curve, the third is a perfect line derailed by one outlier, and the fourth has no trend at all until a single far-flung point invents one. Summary statistics are a compression; the scatter is the truth.

Two reasons r comes out too small

A weak correlation sometimes means the relationship is weak. Two duller explanations come first, and both make an honest relationship look smaller than it is.

Restriction of range. Correlation feeds on spread, so looking at only part of the range starves it. Suppose an admissions test correlates .60 with later grades across all applicants. Compute that correlation inside the students you actually admitted, the top quarter of the test distribution, and .60 shows up as roughly .35; in the top 10% it falls to about .30. Nothing about the underlying relationship changed. You removed the variation that made it visible. This is why selective samples (admitted students, hired employees, patients sick enough to be referred) return correlations that look disappointing, and why a validity study run inside the selected group understates the very test that did the selecting.

Unreliable measures. Every measurement carries noise, and noise correlates with nothing. The shrinkage this causes has a name, attenuation. If your two variables have reliabilities r_xx and r_yy, the correlation you observe is the true one shrunk by a factor of √(r_xx · r_yy). Two questionnaires at a respectable α = .70 apiece turn a true correlation of .60 into an observed .42. The same arithmetic sets a ceiling: with those instruments, even a perfect underlying relationship could not report above .70.

Undoing attenuation, and why it bites

Attenuation has an obvious-looking inverse. If unreliability shrinks r by a factor of √(r_xx · r_yy), dividing that factor back out should recover the correlation between the true scores:

r_corrected = r_observed ⁄ √(r_xx · r_yy)

That is the correction for attenuation, and on the example above it works exactly: .42 divided by .70 returns the .60 the noise was hiding. What the formula does not show is that the divisor is itself an estimate from the same modest sample, and dividing by an uncertain number below 1 magnifies everything, sampling error included.

Simulating the whole procedure puts a size on that. Two 8-item scales at α ≈ .70, 100 respondents, a true correlation of .60, with both alphas estimated from the same sample: the corrected estimate averages .60, so the correction is unbiased enough, but 90% of its values land between .41 and .78, a spread about a third wider than the raw r's. Raise the true correlation to .90 and the same procedure returns a corrected value above 1.0 in roughly one sample in ten.

So report the correction as a supplement, never as a replacement: the observed r, the reliabilities you used, and the corrected value labeled as corrected. A reader can then judge whether "stronger than it looks" rests on the data or on dividing by .70. And if the corrected value comes out above 1, say so. It means the reliability estimates are too low for the correlation you observed, which is evidence about your instruments rather than about the constructs.

The interval around r

An r of .42 from 60 people is one sample's answer, and the fair next question is how far it would move if you ran the study again. The usual add-and-subtract formula cannot answer it. A correlation cannot pass 1, so as the true value climbs, the sampling distribution of r piles up against that ceiling and grows a long tail on the low side only. Symmetric limits would run off the end of the scale.

Fisher's transformation moves the problem somewhere the ceiling does not exist. Writing z = ½ · ln[(1 + r) / (1 − r)] stretches the correlation scale until it runs from minus infinity to plus infinity, and on that scale the sampling distribution is close to normal with a standard error of 1/√(n − 3), which depends on the sample size and nothing else. Add and subtract 1.96 standard errors there, then map both limits back with r = tanh z.

On this lesson's own numbers: r = .42 with n = 60 gives z = .448 and a standard error of 1/√57 = .132, so the interval on the transformed scale is .448 ± .260, or [.188, .707]. Mapped back, that is 95% CI [.19, .61]. The estimate sits .23 above its lower limit and .19 below its upper one, and that lopsidedness is what the transformation exists to handle. It is the number the Correlation & Regression Calculator prints beside r under the label "Fisher z", and the one the APA Results Formatter writes into the sentence.

Two things follow. The interval is wide: .19 to .61 covers everything from below Cohen's "medium" to above his "large", so an r reported bare invites a reader to take .42 far more literally than 60 people can support, and getting the half-width down to .10 around this estimate would take n = 263. And the transformation is more than a reporting device. Correlations are averaged on the z scale rather than the r scale for the same reason, which is one of the conversions behind pooling results across studies.

Want a real cloud to try this on? screen-time.csv holds 60 people, one influential point, and the worked answer for what r does when you drop it.

Problem 20 of the practice problems computes r from six pairs of numbers by hand, then asks what besides causation could have produced it.