IA idea · Modelling with functions
Exponential or logistic? Modelling my country's population
Research question
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?
Make it yours, then agree it with your teacher.
Next step: plan this IA free — the planner opens with this question filled in. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.
Why it makes a good exploration
Population models are a classic for a reason: the choice between exponential and logistic is a real scientific question. Using your own country — and testing predictions on data the model did not see — makes it yours.
The mathematics you'll need
- Exponential and logistic functions
- Linearising with logarithms
- Least-squares regression and residuals
- Using a model for out-of-sample prediction
- HL: the logistic differential equation dP/dt = kP(1 − P/L)
Modelling the data, step by step
This idea usually needs exponential or logistic models. See it worked step by step, with a criterion tip at every step: Exponential y = a e^(kx) + c and y = a·bˣ · Logistic y = L/(1 + C e^(−kx)) · Choosing and comparing models.
Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.
Where the data comes from
Download the population series for your country from the UN World Population Prospects or Our World in Data.
- UN World Population Prospects 2024 — Official UN population estimates 1950–present and projections to 2100, CSV bulk download.
- World Bank Open Data — Development indicators (GDP, life expectancy, mortality, internet use) for every country, usually from 1960.
Real data for this idea
- Primary source: World Bank Open Data: Population, total (SP.POP.TOTL). Download the CSV and turn one country's row into year and population columns. Licence: CC BY 4.0 for most datasets (the World Bank's licence page lists any exceptions).
- Also useful: UN World Population Prospects: World Population Prospects data portal. Choose total population, or population by five-year age group, for one country and export CSV. Licence: Check the terms on the UN site.
- Readable chart: Our World in Data: Population. Use it to see the pattern; download your numbers from the primary source or the chart's own download. Cite as: Our World in Data, ‘Population’, CC BY 4.0. Original source: HYDE (2023); Gapminder (2022); UN World Population Prospects (2024).
- Readable chart: Our World in Data: Population growth rate. Use it to see the pattern; download your numbers from the primary source or the chart's own download. Cite as: Our World in Data, ‘Population growth rate’, CC BY 4.0. Original source: UN, World Population Prospects (2024).
Links checked October 2026; we link to the data and never copy it. More free data sets.
Ready to use: World population in the IA data bank — cited, licensed for reuse, and one click to open in the grapher.
A possible outline
- Explain why each model is plausible for population.
- Fit each model to 1950–2000 only.
- Compare fits and then compare predictions for 2000–present.
- Interpret the logistic carrying capacity and compare with UN projections.
- Reflect on events (migration, policy, conflict) the models cannot capture.
Pitfalls that cost marks
- Fitting to all the data and calling the fit a prediction.
- Choosing a high-degree polynomial because R² is higher.
- Ignoring that the carrying capacity is very sensitive to the data range.
Showing personal engagement
- Choose a country you have a connection to and explain its history.
- Compare two countries with different demographic histories.
- Compare your prediction with the official UN projection and explain the difference.
See Criterion C: personal engagement for what examiners look for.
Which course is it for?
| Course | Fit | Maths to lean on |
|---|---|---|
| AA SL | Good fit | Exponential and logistic functions; Linearising with logarithms |
| AA HL | Good fit | Exponential and logistic functions; Linearising with logarithms |
| AI SL | Good fit | Exponential and logistic functions; Linearising with logarithms |
| AI HL | Good fit | Exponential and logistic functions; Linearising with logarithms |
Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.
How this idea reaches the top bands
Reflection (D)
For this idea, start with the weak points: fitting to all the data and calling the fit a prediction; and choosing a high-degree polynomial because R² is higher. Say how each one affects your answer.
Use of mathematics (E)
SL: exponential and logistic functions; linearising with logarithms; least-squares regression and residuals; using a model for out-of-sample prediction, each explained in your own words and carried out correctly.
HL adds: the logistic differential equation dP/dt = kP(1 − P/L), each explained in your own words and carried out correctly.
How modelling explorations reach the top bands in general: Modelling with functions IA ideas. Guides: A · B · C · D · E · self-grade sheet.
Taking it further
HL: fit the logistic DE directly with Euler's method, or explore how the fitted carrying capacity changes as you add each decade of data.
Extending it for HL
This idea already has HL mathematics in it: the logistic differential equation dP/dt = kP(1 − P/L). Make that the centre of the exploration and use it with full rigour.
See a complete IA, marked
When can we launch? Modelling the tide at our sailing club's slipway (AA SL) is a complete modelling exploration, marked on every criterion.
Frequently asked questions
Which IB Maths course is this IA idea for?
It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.
What mathematics does this IA use?
Exponential and logistic functions, linearising with logarithms, least-squares regression and residuals, using a model for out-of-sample prediction and hL: the logistic differential equation dP/dt = kP(1 − P/L).
Where can I get data for this IA?
Download the population series for your country from the UN World Population Prospects or Our World in Data. Checked sources and tools: World Bank Open Data, UN World Population Prospects, Our World in Data and UN World Population Prospects 2024.
Turn this idea into your IA
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