IA modelling, step by step · Logistic
Logistic model, step by step: 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. Estimate L, linearise to find C and k, then let a least-squares refinement adjust all three.
Example data, invented for this guide. The context is realistic, but the numbers were made up to show the method. Use your own measured or sourced data in your IA.
When to use it
The shape of the data
- An S-shape: slow start, steep middle, then levelling off.
- The fastest growth is roughly halfway to the ceiling.
- Once L is known, ln(L/y − 1) against x is close to straight.
The context
- Growth with a limit: plant height, a population with limited food, adoption of a product.
- Spread through a fixed group: a rumour or illness in a school.
- Anything that must stop at 100% or a known capacity.
Course fit: AI HL (logistic models). It is outside the AA syllabus, but AA and AI SL students can use it if they explain it clearly as new mathematics.
The example data
A sunflower's height is measured every two weeks after it is planted out. The seed packet says this variety grows to about 2.7 m.
| i | x: Weeks after planting out (weeks) | y: Height (cm) |
|---|---|---|
| 1 | 0 | 10 |
| 2 | 2 | 29 |
| 3 | 4 | 67 |
| 4 | 6 | 137 |
| 5 | 8 | 197 |
| 6 | 10 | 237 |
| 7 | 12 | 250 |
| 8 | 14 | 258 |
The method: estimate L, linearise, refine
The seed packet suggests a maximum height of about 270 cm, so start with L = 270. Linearise to find C and k, then refine L, C and k together by least squares.
Step 1 · Estimate the carrying capacity L
From the context, the values level off at about L = 270.
Step 2 · Linearise
Rearranging y = L/(1 + Ce−kx) gives L/y − 1 = Ce−kx, so ln(L/y − 1) = −kx + ln C: a straight line with gradient −k and intercept ln C.
In the full worked example
- The rest of the method: steps 3 to 10 and the residual plot
- What the model tells you, in context
- GDC key sequences: TI-84 Plus CE, TI-Nspire CX and Casio fx-CG50
- Desmos regression syntax
- How to write it up in your IA
- Common mistakes
- Is this good enough for Criterion E? The checklist
Evaluate this model
Questions to answer in your own words, about your own data. They are prompts, not sentences to copy.
- Is there a reason in the context for the ceiling L, and do the data reach far enough to estimate it?
- Is the point of inflection in a believable place?
Then work through the general questions on evaluating a model (extrapolation, the parameters, residuals, comparing models, outliers, assumptions and improvements) and the notes on accuracy and rounding. In the free Grapher, fit the model with ~ and open “What this model means” for the same prompts with your own numbers.
IA ideas that use a logistic model
Explorations from our idea list whose mathematics includes this model. Each research question is a starting point to make your own.
- Exponential or logistic? Modelling my country's population: Which of an exponential, logistic or polynomial model best fits [country]'s population from 1950 to 2000, and which best predicts the actual population from 2000 to today?
- How does bread dough rise? A logistic model in the kitchen: Does the volume of rising bread dough follow a logistic curve, and how do the growth rate and maximum volume change with temperature?
- How fast does news spread through a year group?: Does the number of students who have heard a (harmless, invented) piece of news over a day follow a logistic model, and does a model with “stiflers” who stop spreading fit better?
- Is there a limit to how fast humans can run 100 m?: Which model — linear, exponential decay to an asymptote, or logistic — best fits the progression of the men's (or women's) 100 m world record, and what limiting time does it predict?
- Is solar power growing logistically?: Is the share of electricity from solar in [a country] better described by an exponential or a logistic model, and what saturation level does the logistic model suggest?
- Modelling bacterial growth in the biology lab: Does the growth of a harmless bacterial culture (measured by optical density or colony counts) follow a logistic model, and what doubling time does the exponential phase give?
How this earns marks
A logistic model can earn marks under every criterion: the fit and its explanation under Criterion E, your judgement of it (residuals, the domain, what the parameters mean) under Criterion D, and the reasons you chose it under Criterion C. What examiners look for, criterion by criterion and the mistakes that cost marks are on the modelling guide.
Get feedback on your write-up Self-check your draft Model my data
Frequently asked questions
How do I estimate L for a logistic model?
From the context if you can (a capacity, a population size, 100%), otherwise a little above the largest value once the data are clearly levelling off. Then let the regression refine it and report both.
Next steps
- Criterion E: use of mathematicsWhat “maths at the level of the course” means for a model, at SL and HL.
- Criterion D: reflectionJudging a model: residuals, comparisons, limitations and implications.
- Self-check your draftTick through the criteria for your own exploration.
- Get feedback on your write-upCriterion-by-criterion feedback on your draft, with evidence from your own text.
Related models: Exponential y = a e^(kx) + c and y = a·bˣ · Cubic and polynomials (and overfitting) · Choosing and comparing models. Or model your own data, and see what the IA package adds.
Free: the IA checklist, criterion by criterion
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