What r summarises

The Pearson correlation coefficient measures how tightly two variables move together along a straight line. It is computed from the centred values directly: r = Σ(x − x̄)(y − ȳ) / √( Σ(x − x̄)² × Σ(y − ȳ)² ). The result sits between −1 and +1 and is scale-free: multiply every X by a thousand, or add fifty to every Y, and r does not move, so it describes the shape of the association rather than its units. It also computes Spearman's ρ, the same calculation on ranks, and fits the least-squares line of Y on X.

The p-value and the plotted line

Testing whether a correlation differs from zero uses a t statistic built from r and the sample size: t = |r| × √( df / (1 − r²) ) with df = n − 2. This is always a two-sided test here, so the p answers one question only: how compatible are these data with a population correlation of exactly zero? The dashed line on the scatter plot is the regression of Y on X, with slope = r × SD(y)/SD(x) and intercept = ȳ − slope × x̄. Because the fit is directional, the slope is in units of Y per unit of X — a different statement from r, so a steep slope with a weak r and a shallow slope with a strong r are both perfectly possible.

An example using the loaded data

X is 2, 4, 5, 4, 6, 7, 8, 9, 10, 12 and Y is 38, 41, 44, 47, 50, 52, 55, 58, 61, 65, so n = 10 and df = 8. The calculator returns r = 0.981 with r² = 0.962, a t of 14.30 and p displayed as < 0.001. The regression line is ŷ = 2.791·x + 32.401 — each extra unit of X goes with about 2.8 more of Y — and Spearman's ρ is 0.979.

Two details repay attention. First, r² of 0.962 does not mean "96% of the way to a perfect correlation" — it means X accounts for 96% of the variance in Y, while r = 0.6 gives 36%, not 60%. Second, the X column contains the value 4 twice, and tied values receive the average of the ranks they span, which is why ρ lands at 0.979 rather than a rounder number. If every pair lay exactly on a line, r would reach 1 and the p-value would be undefined, so the tool shows a dash instead of a fabrication.

Reading a correlation safely

• It measures linear association only. A symmetrical U-shaped relationship can give an r near zero while X and Y are perfectly dependent, which is why the scatter plot exists — look at it before interpreting the number.
• Outliers and range restriction distort it badly. One extreme point can move r a long way, and restricting the range of X attenuates the correlation toward zero.
• The p-value tests only ρ = 0. The t approximation assumes independent pairs and roughly bivariate normal variables, and it cannot tell you whether two correlations differ from each other. With a small n the interval around r is wide, so a real moderate association can easily miss significance; with a large n, r = 0.1 can be significant while explaining 1% of the variance.
• Nothing here is causal. X may drive Y, Y may drive X, or a third variable may drive both, and no statistic distinguishes those. The strength descriptors — below 0.3 weak, up to 0.7 moderate, above that strong — are conventions rather than natural categories: r = 0.4 can be remarkable in one field and disappointing in another.

Questions that come up

Should I report r or Spearman's ρ?

Use r when the relationship looks linear and no few points dominate it, and ρ when it is monotonic but curved, or when outliers distort it.

Why is r² shown separately?

It answers the question people think r answers: 0.7 sounds like "70% related" but is 49%.

My r is very high but the scatter looks messy. Why?

Look for one or two distant points, which can pull r toward 1 or −1 while the bulk shows little pattern.

Does sample size affect r itself?

Not the estimate, but it controls how much r wobbles between samples: with n = 10 a correlation near 0.63 is needed for p < 0.05, while n = 100 needs only about 0.20.