What Is the Coefficient of Determination (R²) and What Does It Tell You?

The coefficient of determination, $R^2$, is the proportion of variability in the response variable that is accounted for by a chosen regression model. A value closer to $1$ generally indicates that the model fits the observed data more closely.

This is HL-only content in AHL 4.13 Non-linear regression. It applies to models such as linear, quadratic, cubic, exponential, power and sinusoidal regression models fitted using technology.

How to Interpret $R^2$

For example, suppose an exponential model relating time to population has

$$R^2=0.84.$$

This means that $84\%$ of the variability in the observed population values is accounted for by the exponential model. The remaining $16\%$ is not accounted for by that model and may result from other variables, random variation or measurement error.

A common misconception is that a high $R^2$ proves that the model is correct. It does not. A high value does not establish causation, guarantee accurate extrapolation, or show that the model is appropriate across the whole domain. You should also inspect the data, residual behaviour and contextual suitability of the model.

For a linear regression model, $R^2=r^2$, where $r$ is Pearson's product-moment correlation coefficient. This relationship should not be automatically applied to non-linear models.

Exam Technique

When asked to interpret $R^2$, state the percentage and identify both variables in context: “$84\%$ of the variability in population is accounted for by its exponential relationship with time.” Avoid saying simply that the model is “$84\%$ accurate,” because $R^2$ does not measure percentage accuracy.