Back to Mathematical Modelling

Higher Applications of Mathematics

Units, errors and tolerance

Working with units, accuracy and acceptable ranges.

Modelling referenceModelling mixed quiz

Modelling reference

Check variables, formulae, model types, units, errors and assumptions.

Mixed revision quiz

Review Mathematical Modelling with adaptive feedback.

Before you start

  • Be confident substituting numbers into simple formulae.
  • Check the units in the question before calculating.
  • Be ready to explain what an answer means in the real situation.

Method helper

Which model or method do I use?

Does the output change by a fixed amount each step?

Use a linear model.

Is there a starting fee plus a rate per unit?

Use y = mx + c and interpret the intercept.

Does the situation involve area, a turning point or a squared variable?

Use a quadratic model.

Is the amount increasing or decreasing by a percentage repeatedly?

Use an exponential model.

Have you been given a formula and values?

Substitute into the formula carefully.

Do you need to make one variable the subject?

Rearrange the formula one step at a time.

Are measurements rounded or allowed within a range?

Calculate tolerance or error.

Do the units look inconsistent?

Convert units and check dimensional sense.

Are you testing several assumptions or scenarios?

Use technology to compare model outputs.

Does the prediction seem unrealistic?

Comment on model limitations.

Modelling lesson

Key idea

  • This topic focuses on checking units, rounding, measurement error and acceptable ranges. Mathematical modelling is about using maths to represent a real situation well enough to support a decision.
  • A good model identifies the variables, uses a sensible formula or graph, and states the assumptions being made.
  • For Higher Applications, the interpretation matters. You should explain whether the output is realistic, what it means in context, and what might make the model less reliable.

Key formulae, definitions and methods

  • Tolerance interval: target ± allowed tolerance.
  • Absolute error = |measured value − actual value|
  • Percentage error = absolute error ÷ actual value × 100%

Worked examples

Modelling walkthrough 1

Build the model

A local workshop checks whether cut timber lengths are within a building tolerance.

  1. Identify the target measurement and tolerance.
  2. Calculate the lower and upper acceptable limits.
  3. Check whether the measured value lies inside the interval.

Tolerance intervals help decide whether a measurement is acceptable.

Modelling walkthrough 2

Use the model

A local workshop checks whether cut timber lengths are within a building tolerance.

  1. Find the absolute error by subtracting from the target or actual value.
  2. Convert to a percentage error if comparing different sizes.
  3. Round the percentage suitably.

Percentage error makes errors easier to compare across different scales.

Modelling walkthrough 3

Evaluate the model

A local workshop checks whether cut timber lengths are within a building tolerance.

  1. Check that all measurements use the same units.
  2. Convert mm to cm or metres where needed.
  3. Explain whether the result is acceptable for the job.

Unit checks prevent sensible-looking but wrong model outputs.

Watch out

  • Using a model without defining the variables and units.
  • Choosing a linear model when the rate of change is not constant.
  • Treating a model prediction as an exact fact rather than an estimate.
  • Forgetting to convert units before substituting values.
  • Giving a calculation without commenting on assumptions or limitations.

Technology and data connection

Related Higher Applications topics

Spreadsheet: Checking formula errorsSpreadsheet: Interpreting spreadsheet outputs

Next step

Move into practice

Use the learning notes to identify variables, assumptions and units, then try varied formulas, model types and reasonableness checks.

Modelling mixed quiz