An r of 0.85 That Proves Nothing

"Seventy-three per cent explained" is a statement about a line

Squaring r gives 0.727, and the standard phrasing is that the line explains 72.7 % of the variance in y. As a technical description of a least-squares fit that is exact; as ordinary language it invites a causal reading nobody computed. The arithmetic knows only that the two columns move together, and it would produce the same 0.727 if y caused x, if a third variable drove both, or if the pairing were an accident of a small sample. Keep the word "explained" inside its technical meaning, or replace it with "accounted for by the fitted line", which is longer and says exactly what happened.

One point can create the correlation, or destroy it

Pearson's r is built from squared deviations, so a value far from the cloud counts many times more than one near it. With five points, removing a single observation can move r by more than the entire effect being claimed, which is why any correlation on a small sample deserves the same calculation run five times with each point left out in turn. If the answer swings, the correlation belongs to that one observation and not to the data. This is the same weakness that lets four wildly different scatter plots share one r, and it is the reason a plot is not an optional illustration but part of the check.

Frequently asked questions

What counts as a strong correlation?

It depends entirely on the field, and the tables of thresholds handed out in courses are conventions rather than facts. In physics an r of 0.95 can indicate an instrument problem; in psychology or economics 0.4 can be a substantial finding, because the outcome has dozens of causes and one variable is never expected to carry most of it. What travels between fields is not a threshold but a habit: report r with the sample size, the scatter plot and the confidence interval around r, and let the reader judge.

What should I use when the relationship is not a straight line?

Spearman's rank correlation is the usual first move: it replaces each value by its rank, so it detects any relationship that consistently goes up or down, curved or not, and it is far less disturbed by an extreme value. It still misses a shape that rises then falls, like the parabola above, since the ranks go up and then down as well. For those, transform the variable so the relationship straightens — squaring, taking a logarithm — or fit a model that has the shape in it, and use the plot to decide which.