Historical Context & Motivation
Humans have always looked for patterns in numbers. Farmers in ancient civilisations tracked seasonal rainfall and crop yields, hoping to predict the future from past observations. The mathematical leap — turning scattered data points into a smooth, predictive curve — took centuries of development. Regression modelling is the modern toolset that lets us do exactly that: find the equation that best fits a set of data so we can interpolate (estimate values within the range) or extrapolate (predict values outside the range).
The central question regression addresses is straightforward: Given a set of data points, what type of function — linear, quadratic, or exponential — best describes the underlying relationship, and how well does that function actually fit? In SL 2.7 you will learn to answer this question using technology rather than manual calculation.
Core Principles & Definitions
Before you press any buttons on your calculator, it helps to understand the ideas that make regression work. Every regression analysis rests on a handful of foundational concepts that guide your choice of model and your interpretation of results.
Scatter Plot
Regression Line / Curve
Coefficient of Determination (r² or R²)
Correlation Coefficient (r)
Residuals
Visual Explanation — Choosing the Right Model
The diagram below shows the same set of data fitted with three different regression models — linear, quadratic, and exponential. Notice how the shape of each curve changes and how closely each one follows the data points. This visual comparison is exactly what you do on a GDC when deciding which model to report.
When you look at a scatter plot, ask yourself three questions. First, does the overall trend go steadily upward or downward in a roughly straight band? If so, start with a linear model. Second, does the data curve like a hill or valley, rising then falling (or vice versa)? That suggests a quadratic model. Third, does the data grow slowly at first and then shoot upward (or decay rapidly toward zero)? That pattern points to an exponential model. After your visual guess, you confirm with technology by comparing R² values.
Mathematical Framework
Each of the three regression types produces an equation in a specific form. You do not need to derive these equations by hand — your GDC or software handles the heavy algebra. However, you do need to recognise the general form, understand what each parameter means, and use the equation to make predictions.
Detailed Breakdown — Choosing & Interpreting Models
Selecting the right model is a two-stage process. First you make a visual judgement from the scatter plot, then you confirm quantitatively by comparing R² values. The diagram below walks through the decision-making flowchart that experienced IB students use.
After you have selected a model, you should also consider whether the equation makes sense in context. For example, an exponential growth model that predicts a population of ten billion people in a small town within two years is probably being extrapolated beyond its useful range. Always check whether your predictions are reasonable given the real-world situation.
| Feature | Linear | Quadratic | Exponential |
|---|---|---|---|
| General form | y = ax + b | y = ax² + bx + c | y = a × bˣ |
| Number of parameters | 2 (a, b) | 3 (a, b, c) | 2 (a, b) |
| Shape of graph | Straight line | Parabola (U or ∩) | J-curve or decay curve |
| Typical real-world use | Constant rate of change, e.g. distance vs. time at constant speed | Projectile height, profit vs. price | Population growth, radioactive decay, compound interest |
| Fit statistic given by GDC | r and r² | R² only | R² (or r for log-transformed data) |
Worked Example — Modelling Plant Growth
A biology student measures the height (cm) of a sunflower seedling every week for six weeks. The data are shown in the table below. We will determine the best regression model, write the equation, and use it to predict the height at week 8.
| Week (x) | Height in cm (y) |
|---|---|
| 1 | 3.2 |
| 2 | 5.1 |
| 3 | 8.7 |
| 4 | 14.0 |
| 5 | 22.8 |
| 6 | 36.5 |
STAT → Edit and type each value.ZOOM → 9:ZoomStat to see the points. The data rises slowly at first then accelerates — this suggests exponential growth rather than a straight line.STAT → CALC select LinReg, QuadReg, and ExpReg in turn. Record each equation and R² value.Strengths, Limitations & Comparisons
Every regression model is a simplification of reality. Understanding when each model works well — and when it breaks down — helps you make responsible predictions and earn full marks on the IB exam.
| Model | Strengths | Limitations |
|---|---|---|
| Linear | Simple to interpret; gradient gives constant rate of change; reliable for interpolation within the data range. | Cannot capture curvature; may over-simplify trends that accelerate or decelerate. |
| Quadratic | Captures a single turning point (max or min); good for projectile motion and optimisation scenarios. | Extrapolation is unreliable — parabolas eventually go to ±∞. Not suitable if data has no turning point. |
| Exponential | Excellent for growth/decay processes; naturally models percentage change per unit of x. | Grows without bound — unrealistic over long time frames. Cannot model data that levels off (use logistic instead). |
Connection to Advanced Topics
The regression skills you develop in SL 2.7 form a springboard to more sophisticated modelling in IB Higher Level, university statistics, and data science. Here is a glimpse of how the ideas extend.
| SL 2.7 Concept | Advanced Extension |
|---|---|
| Linear regression (y = ax + b) | Multiple linear regression with several independent variables: y = a₁x₁ + a₂x₂ + … + b |
| R² as a measure of fit | Adjusted R², AIC, BIC — metrics that penalise over-fitting when many parameters are used |
| Choosing between three model types | Logistic, sinusoidal, power, and piecewise regression — plus machine-learning algorithms like random forests and neural networks |
| Visual residual checks | Formal residual analysis: normality tests, heteroscedasticity detection, and diagnostic plots |
Even if you never take another statistics course, the core idea — fitting a mathematical model to data and measuring how well it fits — is a skill you will use in economics, biology, engineering, and everyday decision-making. The SL 2.7 toolkit gives you a solid, technology-supported foundation for all of these applications.
Practice Problems
Lesson Summary
Regression modelling is the process of fitting a mathematical function to a set of data points using technology. In SL 2.7 you work with three model types: linear (y = ax + b) for constant-rate relationships, quadratic (y = ax² + bx + c) for data with a turning point, and exponential (y = a × bˣ) for growth or decay patterns. You select the best model by first inspecting the scatter plot shape, then comparing R² values from your GDC — the model with the highest R² generally fits best.
Always remember that a high R² does not guarantee the model is appropriate for extrapolation. Check that your predictions make sense in context, and state your regression equation with coefficients rounded to three significant figures. The correlation coefficient r (for linear models) tells you the direction and strength of the linear association. Mastering these skills prepares you for real-world data analysis and for the IB examination, where you are expected to justify your model choice with technology-generated evidence.