Regression Pitfalls
A correlation coefficient and a fitted line can both look convincing and still mislead you. Here are the traps — and the one habit that catches them all.
A number that hides as much as it shows
Correlation and regression are summaries. You feed in dozens or thousands of points and get back a single coefficient r, or the slope and intercept of one line. That is enormously useful — but a summary throws information away on purpose, and what it throws away is exactly where the traps hide.
This lesson collects four ways a perfectly real, correctly computed number can still lead you somewhere false: mistaking correlation for causation, trusting a line past the edge of the data, letting a single point steer the fit, and forgetting that r alone never tells you what the cloud actually looks like. Every one of them is caught by the same reflex — plot the data first.
Watch for all four:
- Correlation is not causation. A lurking variable can drive both.
- Extrapolation. A line is only trustworthy inside the range of x you actually saw.
- Outliers and leverage. One extreme point can move r and the whole line.
- The same r, different pictures. A coefficient never describes the shape — only a plot does.
The habit that defends against all four: always plot the data before you trust a number.
Trap 1: correlation is not causation
When two variables rise and fall together, the tempting story is that one makes the other happen. But a correlation is perfectly consistent with three different worlds: x causes y, y causes x, or some third variable — a lurking variable (also called a confounder) — drives both while x and y do nothing to each other.
The classic case: across a summer, ice-cream sales and drowning deaths are strongly correlated. Ice cream does not cause drowning. Hot weather independently pushes both up — more cones sold, more people swimming. The correlation is real; the causal story is wrong. Correlation can be a clue that a causal link exists, but on its own it never proves one.
Trap 2 and 3: reaching past the data, and points that pull
Extrapolation is using a fitted line to predict outside the range of x values you actually collected. Inside that range you have evidence the linear pattern holds; outside it you are guessing that the same straight line keeps going — and real relationships bend, level off, or reverse. A line fit to the heights of children aged 2 to 12 would, extended blindly, predict absurd heights at age 40.
Individual points matter too. An outlier sits far from the overall pattern. A high-leverage point sits far from the other x-values — out at the left or right edge — and because the least-squares line pivots to reduce its residual, such a point can tilt the entire line and inflate or deflate r. One well-placed stray point can manufacture a correlation that the rest of the data does not support, or hide one that does.
A famous demonstration (Anscombe's quartet) shows four datasets with identical means, identical standard deviations, and the same correlation r near 0.82 — yet one is a clean straight line, one is a smooth curve, one is a perfect line dragged off by a single outlier, and one is a vertical stack fooled by one far-off point. The summary numbers are indistinguishable; the pictures are nothing alike.
The lesson is blunt: r and the regression line describe a linear pattern only, and never reveal shape. A curve, a cluster, or a lone leverage point can all hide behind a respectable-looking coefficient. Plotting takes seconds and catches what the numbers cannot.
Trap 4 (briefly): the ecological fallacy
One more trap worth naming. A correlation measured on groups — states, schools, countries — need not hold for the individuals inside them. Concluding an individual-level relationship from group averages is the ecological fallacy. Regions with higher average income might show higher average spending on some good, yet within any single region richer people might spend less on it. Group-level and individual-level associations can even point in opposite directions, so be careful which one your data actually measures.
A regression line carries no warning label at the edge of the data. It will happily return a prediction for any x you plug in, including values far beyond anything you observed. Those predictions look just as precise as the trustworthy ones — same equation, same decimals — but they rest on the untested assumption that the straight line continues. Treat any prediction outside your observed range of x with deep suspicion.
- The correlation is almost certainly real: bigger fires do coincide with more property damage. The error is the causal leap from that correlation.
- Look for a lurking variable that drives both. The size of the fire is the obvious one: large fires cause more damage AND prompt dispatchers to send more firefighters. Firefighters are a response to fire size, not the cause of the damage.
- Reversing the logic exposes the absurdity: sending one firefighter to a warehouse blaze would not make the building burn less.
- How to check honestly: hold fire size roughly constant (compare fires of similar size) and see whether more firefighters still track with more damage. Controlling for the confounder typically makes the apparent effect shrink or flip.
Check your understanding
- Correlation is not causation: two variables can move together because a lurking (confounding) variable drives both. Correlation is a clue, never a proof, of cause.
- r measures the strength and direction of a LINEAR relationship only, on a -1 to +1 scale — it says nothing about curves, clusters, or cause.
- Extrapolation — trusting a fitted line outside the observed range of x — turns evidence into guesswork; real relationships bend and reverse beyond the data.
- Outliers and high-leverage points can single-handedly tilt a regression line and distort r.
- The same r can come from completely different-looking data (Anscombe's quartet), so ALWAYS plot the data before trusting a number.
- The ecological fallacy: an association seen across groups need not hold for the individuals within them.