Pearson's product-moment correlation coefficient, denoted by , measures the direction and strength of the linear relationship between two numerical variables. Its value always satisfies .
In IB Mathematics: Analysis and Approaches, this is part of SL 4.4: Correlation and regression. The coefficient is calculated using technology, so numerical calculation is normally assessed in a technology-required Paper 2 setting.
How to interpret
The sign of gives the direction of the association, while the magnitude (|r|) indicates its strength. Values closer to represent a stronger linear relationship.
| Value of | Interpretation |
|---|---|
| close to | Strong positive linear correlation: as one variable increases, the other tends to increase. |
| close to | Strong negative linear correlation: as one variable increases, the other tends to decrease. |
| close to |
For example, if a GDC gives , the data show a strong negative linear correlation. A suitable interpretation is: “As increases, tends to decrease.”
The word linear is essential. A data set can have a strong curved relationship but an -value close to zero because Pearson's coefficient only measures linear association. Always inspect the scatter diagram before interpreting .
Also, is unit-free and is not the gradient of the regression line. Changing measurement units does not change , although outliers can alter it substantially; therefore, interpret the coefficient together with the scatter diagram and context.
A common misconception is that a strong correlation proves causation. It does not: even when (|r|) is close to , a third variable or coincidence may explain the association.
Exam technique
When asked to describe correlation, state its direction, strength, and linear nature, such as “strong positive linear correlation.” Use a GDC to calculate , but write the value using correct mathematical notation, usually to three significant figures. Never claim that correlation implies causation.