Skip to main content
iBacalao
Try the browser appInstall the app

IB Math AI exploration: why a high R² does not mean your model is good

The coefficient of determination, R², measures how much sample variation a fitted model explains relative to a simple baseline. It does not prove that the model describes a causal relationship, predicts unseen observations accurately, or is the best model for your investigation. Inspect the residuals and the meaning of your variables before deciding that a regression is useful.

Back to all guides
Add as a Google preferred sourceReview us on G2
Math AI regression-check workflow connecting a fitted model, residual plot, and limitations
Regression check: the residual pattern matters as much as the fitted equation.

The coefficient of determination, R², measures how much sample variation a fitted model explains relative to a simple baseline. It does not prove that the model describes a causal relationship, predicts unseen observations accurately, or is the best model for your investigation. Inspect the residuals and the meaning of your variables before deciding that a regression is useful.

The short answer

The coefficient of determination, R², measures how much sample variation a fitted model explains relative to a simple baseline. It does not prove that the model describes a causal relationship, predicts unseen observations accurately, or is the best model for your investigation. Inspect the residuals and the meaning of your variables before deciding that a regression is useful.

The IB's 2026 Mathematics: Applications and Interpretation examiner instructions describe the exploration as a short mathematical report that emphasizes the mathematics, mathematical communication and thoughtful reflection. The worked dataset below is synthetic. It is deliberately small enough that every calculation can be checked by hand; it is not evidence of any real relationship.

A regression result you can reproduce

Imagine that four students record the number of scheduled extracurricular sessions in a week, x, and a numerical fatigue self-rating, y. This is a hypothetical teaching device. It is not an ethical or scientifically adequate survey, and no relationship inferred from four observations should be generalized to students.

Observation x: sessions y: fatigue Fitted ŷ = 1.5+0.8x Residual y-ŷ
A 1 2 2.3 −0.3
B 2 3 3.1 −0.1
C 3 5 3.9 +1.1
D 4 4 4.7 −0.7

The mean response is 3.5. The total sum of squares around that mean is:

SST=(2-3.5)^2+(3-3.5)^2+(5-3.5)^2+(4-3.5)^2=5.00.

The sum of squared residuals from the fitted line is:

SSE=(-0.3)^2+(-0.1)^2+(1.1)^2+(-0.7)^2=1.80.

So R^2=1-SSE/SST=1-1.80/5.00=0.64. The line accounts for 64% of the observed sample variation under this specific definition of R². It does not show that sessions cause fatigue, nor that a fifth student with five sessions would have a fatigue rating of exactly 5.5.

The residual plot tells a different story

Plot the four signed residuals against x, not just the four original points and the regression line. The residuals are −0.3, −0.1, +1.1 and −0.7. The positive residual at x=3 means this observation is above the model's prediction by 1.1 rating units. With only four points you cannot reliably diagnose a pattern, but the plot makes an important question visible: does the error vary systematically with x, or is a single observation dominating the fit?

Do not automatically remove C because it reduces R². Investigate whether it is a recording error, a genuine value or a consequence of an omitted variable. Deleting a valid point to improve the statistic changes the evidence rather than the mathematical model.

The three-model decision table

Candidate Why try it? What must be checked? What not to claim
Linear A transparent starting hypothesis Residuals, units, plausible range "A line is true because calculator output exists"
Quadratic Curvature might be conceptually plausible Whether extra complexity is justified on independent data "Higher training R² proves better prediction"
No useful model Association may be weak or unstable Data quality, alternative variables, scope "Every investigation must produce a strong correlation"

With only four points, a quadratic can fit the same sample more closely simply because it has extra flexibility. That is not compelling evidence that it will predict new data well. If your exploration is genuinely predictive, hold back observations or test another period, and compare prediction errors rather than just fitting statistics.

Reflection that actually changes the interpretation

An effective mathematical reflection here would say: "The linear model gave R^2=0.64 for four constructed observations, but the residual near x=3 was much larger than two of the others. Because a fatigue self-rating is subjective and the sample has only four points, an extrapolation beyond the measured range is not justified. A better investigation would define a reproducible measurement, obtain an ethically sourced larger dataset and compare out-of-sample errors."

That is stronger than merely describing R² as "quite good." It explains how the mathematics constrains your real conclusion. When building your own exploration, obtain genuine ethically collected data, show calculations, justify your choice of model and distinguish explanatory from predictive aims.

Primary sources and transparency

All measurements and interpretations in the worked table are fabricated educational data, explicitly labelled as such. They are not real student health or survey findings. ibacalao.com provides independent draft-feedback tools.

Continue with related guides

Check your assessment session against the linked official IB sources. Read our editorial policy and sourcing method.