StatReason
What this answers
This page answers "when do Pearson and Spearman correlation agree, and when do they tell a genuinely different story about the same data?" Use the scatter plot before choosing either coefficient, since a single number can look identical for very different underlying patterns.
What each coefficient measures
Pearson correlation measures the strength and direction of a linear relationship. Spearman correlation measures the strength and direction of a monotonic relationship after values are converted to ranks. A Pearson r near 1 or minus 1 indicates a strong linear pattern. A Pearson r near zero indicates little linear association, but it does not rule out a curved relationship, exactly the situation the U-shaped pattern above demonstrates.
Reading the four patterns
The linear pattern gives both coefficients a value at or near 1, since a straight-line relationship is also perfectly monotonic. The monotonic-curve pattern (values that consistently rise but not in a straight line) gives Spearman a higher value than Pearson, since Spearman only needs the ranks to move consistently, not the raw values to fall on a line. The U-shaped pattern gives both coefficients a value near zero, correctly reflecting that neither a straight line nor a consistently rising rank pattern describes this relationship, even though the two variables are clearly related. The outlier pattern shows how a single extreme point can pull Pearson's r toward it much more strongly than Spearman's rank-based rho, since one extreme raw value has less leverage once it is converted to a rank.
Worked example
Consider a data set where sleep hours increase from five to nine and alertness ratings also increase, but the relationship curves rather than following a straight line. Spearman may show a strong monotonic association even when Pearson is smaller, because the straight-line model is a poor description of the actual curve. A U-shaped pattern, where alertness first drops and then rises again, could give both correlations a value near zero even though the variables are strongly related; only the scatter plot reveals a relationship a single coefficient cannot capture.
Neither coefficient identifies a cause
A positive association can arise because X affects Y, Y affects X, a third variable affects both, a selected sample creates a pattern, or pure chance produces an unusual sample. Time order, assignment, measurement quality, and confounding require design knowledge beyond what any correlation calculator can supply; see Correlation Does Not Prove Causation for the specific alternative explanations to rule out first.
Source
This comparison follows the correlation guidance in the NIST/SEMATECH e-Handbook of Statistical Methods and the shared statistical reasoning contract every StatReason engine is built against.
Limitations
Report the coefficient, sample size, and a plain description of direction; avoid universal strong-or-weak labels unless you name a field-specific convention. Neither coefficient is immune to every problem: many tied values, a very small sample, or a sharply curved pattern can still make a single correlation misleading even after this comparison.