Research Methods · Simplified
What r Cannot See
A correlation coefficient answers one narrow question. Always look at the scatter plot before you trust the number.
The scatter
Twenty-four points, and one you control
The ringed point is yours to move. Everything else stays exactly where it is. Watch what one observation out of twenty-five can do to the number underneath.
Key terms
- r
- Pearson's correlation coefficient. It runs from minus one to plus one and measures how well the points sit along a straight line. That word is the whole difficulty.
- r squared
- The share of the variation in one variable that the straight-line fit accounts for. An r of 0.5 accounts for a quarter of it, not half.
- Influential point
- An observation that changes the answer noticeably when it is removed. Being unusual is not enough; where it sits relative to the rest is what matters.
The ringed point is the one you are moving. The solid line is the best-fitting straight line through all twenty-five points; the dashed line is the best fit through the other twenty-four, so the gap between them is what your one point is doing.
What this shows
The number is a summary, and summaries lose things
r asks how closely the points hug a straight line, and nothing else. It cannot tell a strong curved relationship from no relationship at all, and it can be dragged a long way by one observation.
Key idea: On the arch, the relationship is almost perfect and r sits near zero, because no straight line fits an arch. On the shapeless cloud, one point placed far out in the corner produces a respectable-looking correlation out of nothing at all. In both cases the number is not lying: it is answering the narrow question it was asked. The mistake is to treat that answer as a summary of whether two variables are related. This is why a scatter plot is not an optional illustration of a correlation, it is the thing you have to look at before the correlation means anything.
Two further points. A point that looks extreme is not automatically influential: an unusual observation sitting near the middle of the horizontal range barely moves the line at all, which you can check by sliding the movable point straight up while leaving it at 50 across. Influence comes from being unusual on the predictor as well as far from the fit. And none of this says an outlier should be deleted. Whether an observation is a recording error, a genuine rare case or a sign that the model is wrong is a question about the study, not about the arithmetic, and the honest response is usually to report the analysis both ways.
The longer version adds negative and zero relationships, a change-of-units demonstration, and a guess-the-correlation challenge across many scatter plots. It is at Correlation: Outliers and Non-Linearity in the main collection.