IA modelling, step by step

The modelling workflow, step by step

1. Plot the data first

   Draw a labelled scatter graph before any calculation. Look at the shape: straight, curved, one turning point, repeating, levelling off? Look for outliers and ask whether they are errors or real. Say what you see in words.

2. Choose candidate models from the shape and the context

   Use both. The shape suggests a family (line, quadratic, exponential…); the context says what must be true — an asymptote at room temperature, a ceiling, a period of a year, a line through the origin. Pick two or three candidates and justify each before fitting.

3. Fit each model

   Show at least one fit in full by hand (points and simultaneous equations, completing the square, the least-squares sums, or a linearisation with logs). Then use technology for the rest and say exactly what it did.

4. Check the fit

   Tabulate the residuals (data − model) and plot them against x. A good model leaves residuals that look random; a pattern (a curve, a fan) means the model misses something. Report SSR, and R² or r where it applies.

5. Compare the models

   Put the candidates side by side: SSR or R², the number of parameters, the residual plots, and whether the parameters make sense. The best model is the simplest one that captures the shape and makes sense in context — not automatically the one with the highest R².

6. Test on points you did not use

   Keep one or two points back (or collect new ones), fit on the rest and predict the held-out points. A model that predicts well on new data is far more convincing than one that fits its own data closely.

7. State the domain and the limitations

   Give the domain where the model is valid, say what happens outside it, and discuss extrapolation. Name the assumptions (constant conditions, measurement error) and how they limit your conclusions.

8. Reflect

   What do the parameters mean in the context? Do they agree with theory or other data? How sensitive is the result to your choices (which points, which asymptote, where you split)? What would you do next?

Each step feeds a criterion: choosing and justifying models (C and E), fitting and checking them (E), comparing, testing and discussing limitations (D), and explaining it all clearly (A and B). See Criterion E: use of mathematics and Criterion D: reflection.

Which model? Start from the shape and the context

Not sure between two? That is normal: fit both and compare them properly.

Worked examples, every step shown

Each page takes one small, clearly labelled example data set and fits the model by hand and with technology — every sum, every substitution, the residual table, R² and a graph — plus the key sequences for the TI-84 Plus CE, TI-Nspire and Casio fx-CG50, the Desmos syntax, how to write it up and a Criterion E checklist.

• Straight line y = mx + cA line is the simplest model and the one to beat.
• Quadratic: three points, completing the square, regressionThree ways to fit y = ax² + bx + c to the same data: solve three simultaneous equations through three points, build the vertex form y = a(x − h)² + k from the turning point, and use least-squares regression.
• Sinusoidal y = a sin(b(x − c)) + dFor anything that repeats — daylight, tides, temperatures, a Ferris wheel — find a, b, c and d by hand from the maximum, minimum and period, convert to the calculator's form a sin(bx + c) + d, then refine the fit by minimising the sum of squared residuals.
• Logarithmic y = a ln x + bWhen something keeps growing but more and more slowly — skills, learning, diminishing returns — try y = a ln x + b.
• Reciprocal y = a/x + b and a/(x − h) + kInverse relationships — time against speed, cost per person against group size — curve towards two asymptotes.
• Exponential y = a e^(kx) + c and y = a·bˣGrowth or decay by the same factor in each step — cooling, depreciation, radioactive or drug decay — is exponential.
• Power y = a·xⁿ (log–log)When y is proportional to a power of x — area and length, period and length, metabolic rate and mass — plot ln y against ln x.
• Logistic y = L/(1 + C e^(−kx))S-shaped growth — plants, epidemics, app downloads, the spread of news — starts almost exponential and levels off at a ceiling L.
• Cubic and polynomials (and overfitting)Adding terms to a polynomial always lowers the sum of squared residuals on the data you fitted — until a curve of degree n − 1 passes through every point.
• Piecewise modelsSome data follow one rule and then another: water heats steadily and then boils, a tariff changes at a threshold, a runner speeds up for the finish.
• Choosing and comparing modelsA good modelling IA does not just fit a curve — it argues for one.

How much maths to show at SL and HL

• At SL, fitting with a GDC or Desmos is expected; what earns Criterion E marks is showing one fit by hand, explaining what the technology did, and using the results correctly.
• At HL, examiners look for more sophistication and rigour: derive what you use (least squares by calculus, linearisation with the laws of logarithms), refine iteratively and explain it, test on unused data, or build the model from a differential equation.
• At both levels, the comparison, the interpretation of the parameters, the domain and the limitations are Criterion D reflection — often where modelling IAs lose most marks.

Our summary of the current criteria (exams up to 2028), not the IB's wording. How the IA differs between AA and AI, SL and HL.

Accuracy, rounding and uncertainty

How precise can your results be, and how do rounding and measurement error change them? Apply these to your own data; the reasoning and the wording in your IA must be your own.

Keep full precision, round at the end

• Use the fitted parameters at full precision for every later calculation (predictions, residuals, SSR, R²) and round only the final answers you quote.
• Rounding a parameter early and then using the rounded value changes the model: its predictions, residuals, SSR and R² all change, sometimes a lot (an exponent or a power is very sensitive).
• When you quote a rounded value, write ≈ rather than =, and use the same precision throughout.

Significant figures and decimal places

• Significant figures (s.f.) count from the first non-zero digit; decimal places (d.p.) count the digits after the decimal point. 0.04520 has 4 s.f. and 5 d.p.
• A result calculated from measurements can't be more precise than the measurements: a common rule is to quote it to about the same number of significant figures as the least precise data, or one more.
• Eight or nine figures copied from a calculator suggest a precision your data do not have. Say which precision you chose and why.

Measurement uncertainty and data precision

• Every measurement has an uncertainty. For a scale or a ruler it is often taken as half the smallest division; for a stopwatch, reaction time is usually far larger than the display's resolution.
• State how each variable was measured, its uncertainty and which variable is the least precise.
• Repeating and averaging reduces random error, but not systematic error (a scale that isn't zeroed, a ruler read from the wrong end).
• Compare the residuals with the uncertainty: residuals about the size of the uncertainty are as small as the data allow; much larger residuals mean the model, or the data, misses something.

Reading coordinates off a photo

• The scale comes from something of known size in the photo, so any error in that length changes every coordinate you read.
• Perspective: unless the camera points straight at the object from far enough away, nearer parts look larger and the curve is distorted. Lenses also bend straight lines near the edges of a picture.
• A line in a photo has thickness: decide whether you read its inner edge, outer edge or centre, and do the same every time.
• Each point you read is uncertain by at least the width of the line: estimate that in the units of your axes and say how it could affect the parameters.

Percentage and relative error

• Percentage error = |model value − measured value| ÷ |measured value| × 100%. It compares an error with the size of the quantity, so an error of 2 matters more when the value is 10 than when it is 1000.
• Use it to judge predictions, especially for a value you measured but did not use in the fit.
• Percentage error is not useful when the measured value is zero or close to zero.

How rounding and errors affect R² and the residuals

• Least squares makes SSR as small as possible, so rounding the parameters can only increase SSR and lower R² on the same data (unless the fit was made on transformed values, such as log mode).
• Random error in the measured values adds scatter to the residuals and lowers R², even for the right model: a modest R² with patternless residuals may be the best the data allow.
• Ordinary least squares treats the x values as exact. If x is uncertain too, say so and consider how that affects the fit.

Accuracy and the criteria

• Criterion B (mathematical communication): appropriate notation and precision, ≈ for rounded values, variables defined with units.
• Criterion D (reflection): discuss the accuracy of the data and how errors and rounding affect your results and conclusions.
• Criterion E (use of mathematics): the mathematics should be correct and at an appropriate level of accuracy throughout, with no premature rounding.

The criterion lines are our summary of the current criteria, not the IB's wording. In the free Grapher, “What this model means” under a fit works out the effect of rounding and the percentage errors for your own data, and warns you if rounded parameters are typed back into a model.

Evaluating your model: questions to answer

Questions to answer in your own words, about your own data. They are prompts, not sentences to copy. Answering them well is Criterion D (Reflection).

Strengths and limitations in context

• What does the model capture about your situation, and where does it struggle? Point to particular values or residuals.
• Which limitation matters most for the question you set out to answer?

Extrapolation

• Over what range was the model fitted, and how far beyond it would you trust a prediction? Why that far?
• What does the model give at the edges of a sensible domain (zero, very large values)? Is that possible in reality?

Do the parameters make physical sense?

• What does each parameter mean, and are its sign, size and units plausible?
• Can you check a parameter against an independent value: a known constant, a theory or another measurement?

Residual behaviour

• Is there a pattern in the residual plot (a curve, a trend, a funnel)? What in the situation could cause it?
• Are the residuals about the size of your measurement uncertainty, or much larger?

Comparing models

• Which other models did you try, and why is your choice better for this context, not only by R²?
• Does each extra parameter earn its place (a clearly smaller SSR and better residuals), or is the simpler model enough? Simpler is better when the fit is similar (parsimony).
• Which model best predicts points you did not use to fit it?

Sample size and outliers

• Are there enough points, spread across the whole range, to pin down the shape?
• Is any point far from the rest or from the model? Is there a reason to keep it, check it or leave it out, and does leaving it out change your conclusion?

Assumptions

• What does the model assume (conditions held constant, no other variable changing)? Were those assumptions true while the data were collected?
• Least squares assumes the x values are exact and the errors in y are random. Is that reasonable for your measurements?

Improvements

• What more or better data would help most: more points, a wider range, repeats, a more precise instrument?
• Would a different model, or a piecewise model with a reason for the split, describe the situation better?
• What would you change about your method if you did it again?

Real-world meaning

• What does the model let you say or predict about the real situation, and how sure can you be?
• Who could use the result, and what would they need to be careful about?

Each worked example ends with questions for its own model, and testing on points you did not use shows how to check predictions.

Get feedback on your write-up

Technology

Desmos, GeoGebra and spreadsheets all fit these models; the worked pages give the exact syntax. For graphs, tables and how to describe technology so it earns marks, see using Desmos, GeoGebra and Excel in your IA. For a complete modelling exploration marked criterion by criterion, see the annotated exemplars. Testing a question with data rather than fitting a curve? See IA statistics, step by step.