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IB DP Mathematics: Applications and Interpretation | SL & HL | Statistics and Probability — Topic 4.13
📋 Learning Intentions
- Identify when a non-linear model (quadratic, cubic, exponential, power) is more appropriate than a linear model for a given data set.
- Use the GDC to fit non-linear regression models to data and extract the equation of the regression curve.
- Calculate the coefficient of determination \( R^2 \) using the GDC and interpret its value in context.
- Compare competing regression models using \( R^2 \) to justify the best-fit model for a given data set.
- Interpret the parameters of a non-linear regression equation in the context of a real-world problem.
- Evaluate the reliability and limitations of a regression model based on \( R^2 \) and the nature of the data.
Real-World Introduction
A biologist tracking the growth of a bacterial colony records the population every hour. Plotting the data reveals a curve that sweeps upward steeply — not a straight line. Fitting a linear model would give a poor description of the growth pattern and lead to wildly inaccurate predictions.
Similarly, a physicist measuring the distance a ball travels under gravity, or an economist modelling diminishing returns, will encounter data that curves. The question is not just which curve fits best, but how well any model fits. The coefficient of determination \( R^2 \) answers that second question with a single number between 0 and 1.
🌍 Why This Matters Climate scientists use non-linear regression to model the relationship between atmospheric CO₂ concentration and global mean temperature — a relationship that is decidedly not linear over long timescales. The \( R^2 \) value tells researchers how much of the observed temperature variation is explained by the chosen model, directly informing policy decisions.
Core Concepts & Definitions
When a scatter plot shows a clear pattern that is not straight — a curve that bends upward, levels off, or oscillates — a linear regression model \( y = mx + c \) will systematically under- or over-predict values. The residuals (differences between observed and predicted values) will show a pattern rather than random scatter, signalling that a non-linear model is needed.
🔑 Key Definition — Non-linear Regression Non-linear regression is the process of fitting a curve (rather than a straight line) to a set of bivariate data. The GDC determines the parameters of the chosen model by minimising the sum of squared residuals. IB Math AI considers four non-linear models: quadratic \( y = ax^2 + bx + c \), cubic \( y = ax^3 + bx^2 + cx + d \), exponential \( y = ab^x \) (or \( y = ae^{bx} \)), and power \( y = ax^b \).
Each model has a different shape and is suited to different physical or real-world situations. Choosing the correct model type requires inspecting the scatter plot and considering the context — for example, exponential models suit