IA modelling, step by step · Power
Power model, step by step: y = a·xⁿ with a log–log plot
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. A straight line means a power law: its gradient is the power n and its intercept is ln a.
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
- The curve passes through (or heads towards) the origin, with no asymptote other than the axes.
- Multiplying x by a fixed factor multiplies y by a fixed factor.
- ln y against ln x is close to straight.
The context
- Physics laws with a power: pendulum period (n = ½), gravity (n = −2), falling distance (n = 2).
- Scaling in biology and geometry: surface area, volume, mass.
- Anything where theory suggests y ∝ xⁿ and you want to test n.
Course fit: AI SL (models of the form axⁿ), AI HL (log–log linearisation); AA through the laws of logarithms. Comparing the fitted power with a theoretical one is strong at any level.
The example data
A student times 10 swings of a pendulum for several string lengths and divides by 10 to get the period.
| i | x: Length of pendulum (m) | y: Period (s) |
|---|---|---|
| 1 | 0.2 | 0.9 |
| 2 | 0.35 | 1.18 |
| 3 | 0.5 | 1.43 |
| 4 | 0.65 | 1.61 |
| 5 | 0.8 | 1.8 |
| 6 | 1 | 2 |
| 7 | 1.2 | 2.21 |
The method: log–log line, then back-substitute
Take natural logs of both variables, fit the least-squares line to (ln x, ln y), and read off n (gradient) and a = e^(intercept). Then refine by least squares in the original units.
Step 1 · Linearise with a log–log plot
If y = axn, taking natural logs gives ln y = n ln x + ln a: a straight line in (ln x, ln y) with gradient n and intercept ln a.
Step 2 · Tabulate the sums
| i | ln x | ln y | (ln x)² | (ln x)(ln y) | (ln y)² |
|---|---|---|---|---|---|
| 1 | −1.6094 | −0.10536 | 2.5903 | 0.16957 | 0.011101 |
| 2 | −1.0498 | 0.16551 | 1.1021 | −0.17376 | 0.027395 |
| 3 | −0.69315 | 0.35767 | 0.48045 | −0.24792 | 0.12793 |
| 4 | −0.43078 | 0.47623 | 0.18557 | −0.20515 | 0.22680 |
| 5 | −0.22314 | 0.58779 | 0.049793 | −0.13116 | 0.34549 |
| 6 | 0 | 0.69315 | 0 | 0.0000 | 0.48045 |
| 7 | 0.18232 | 0.79299 | 0.033241 | 0.14458 | 0.62884 |
| Σ | −3.82401 | 2.96799 | 4.44148 | −0.443845 | 1.84801 |
n = 7 points. The transformed values are rounded here; keep them unrounded in your calculator or spreadsheet.
In the full worked example
- The rest of the method: steps 3 to 9 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.
- Does a theory predict the power n? How close is your value, given the uncertainty in the data?
- Is the log–log plot straight across the whole range, or does it bend at one end?
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 power model
Explorations from our idea list whose mathematics includes this model. Each research question is a starting point to make your own.
- Does a pendulum's period really depend on the square root of its length?: Is the relationship between a pendulum's length and its period a power law with exponent ½, and at what swing angle does the simple model stop working?
- Does light really follow an inverse-square law?: Does the illuminance measured by my phone fall as 1/d² as I move away from a lamp, and at what distances does the model fail?
- Can you predict a tree's height from its trunk?: Is the height of trees in my local park related to trunk diameter by a power law, and does the exponent agree with the value predicted by the theory of elastic similarity (about ⅔)?
- Linear or quadratic drag? Testing air resistance with falling coffee filters: Is the air resistance on a falling paper cone proportional to its speed or to its speed squared, judged by how terminal velocity changes as the mass is doubled and tripled?
- Is BMI's “height squared” right? Evidence from Olympians: If body mass is modelled as a power of height, m = khⁿ, what value of n best fits Olympic athletes in different sports, and what does that imply about using BMI for tall and short people?
- How fractal is a fern? Box-counting dimension from photographs: What box-counting dimension do photographs of a fern, a tree's branches and a cauliflower give, how consistent is the estimate across box sizes, and does it differ between species?
How this earns marks
A power 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 can I tell a power model from an exponential model?
Plot both linearisations: ln y against ln x (straight for a power model) and ln y against x (straight for an exponential). Whichever is straighter, with residuals showing no pattern, is the better candidate.
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 · Logarithmic y = a ln x + b · 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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