One Number, Four Restrictions, and a Lurking Third Variable

You'll learnwhat r measures and the range it lives in, why the sign is direction and the magnitude is strength, the properties that decide most questions — unitless, symmetric, not the slope, not resistant — the curved patterns it cannot see, and why no value of r turns an association into a cause.

The correlation coefficient compresses a scatterplot into one number, and everything it cannot carry is where the errors live. This page executes the properties instead of reciting them: it converts hours to minutes and shows r not moving while the slope shrinks by a factor of 60; it ranks r = −0.89 above r = +0.42 because strength is the magnitude; it adds one point to a formless cloud and watches r jump from 0.09 to 0.71; it fits a straight line to a cloud built from a curve and still gets 0.925. Then the big one: ice cream sales and beach rescues correlate at 0.91 — and when the page holds temperature fixed, the association vanishes to 0.00, computed.

8 STEPS · 6 QUICK CHECKS · WHAT r IS · PROPERTIES · BLIND SPOTS · CAUSATION · v1

r survives a unit change; the slope does not · one point lifts r from 0.09 to 0.71 · cones and rescues: 0.91, and 0.00 at fixed temperature

Before you start

What you're looking at

The eight students' cloud carrying r = 0.977, and a series of clouds each built to exercise one property of r — or one thing r cannot do.

The question

What exactly does the one number carry — and what gets smuggled in when it is read as more than strength-and-direction of a line?

Watch for

Things that move while r stays still (the slope, under a unit change) and things that move r without any real pattern (one far point). Both are properties, and both are exam questions.

Step 1 / 8