# Approximation, absolute and percentage error

> IB Mathematics Applications and Interpretation SL · Unit 1: Number and Algebra
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u1-approximation-absolute-and-percentage-error/

This sub-topic teaches you to quantify error when using approximate values for measurements or calculated results. You will learn to calculate absolute error, percentage error, and interpret error bounds for IB AI SL exam questions.

**Prerequisites:** [Rounding and significant figures](https://www.owlsprep.com/study/ib-math-ai-sl-u1-rounding-significant-figures/)

## Learning objectives

- Calculate absolute error for approximate measurements and values
- Calculate percentage error and interpret its significance
- Find maximum error bounds for rounded measurements

## Approximation and Absolute Error

**Absolute Error** — The absolute difference between the true (exact) value of a quantity and its approximate value, measured in the same units as the original quantity.

*Notation:* \Delta x = |x_{\text{true}} - x_{\text{approx}}|

*Example:* If true length = 10 cm, approximate length = 9.8 cm, absolute error = 0.2 cm

Absolute error only tells you the raw size of the error, not how significant it is relative to the size of the quantity you are measuring. An error of 1 cm is much more significant for a 5 cm object than for a 100 cm object.

**Worked example:** A child's true mass is 3.42 kg. It is measured as 3.3 kg on a analog scale. Calculate the absolute error of the measurement.

1. Recall the formula for absolute error:
2. $$\Delta x = |x_{\text{true}} - x_{\text{approx}}|$$
3. Substitute the given values into the formula:
4. $$\Delta x = |3.42 - 3.3| = |0.12| = 0.12$$
5. State the final answer with matching units:
6. Absolute error = 0.12 kg

> **Exam tip:** Always include units for absolute error, as it is a dimensional quantity matching the original measurement.

## Percentage Error

**Percentage Error** — A unitless, relative measure of error that expresses absolute error as a percentage of the true value. It allows comparison of error significance across different sized quantities.

*Notation:* \text{Percentage Error} = \left(\frac{\Delta x}{|x_{\text{true}}|}\right) \times 100\%

*Example:* A 0.12 kg error for a 3.42 kg mass is 3.5% percentage error

Percentage error is the most commonly requested error type in IB AI SL exams, because it gives context to how large the error is. If the true value is unknown, you may use the approximate value in place of the true value for calculation.

**Worked example:** The area of a circle is calculated as $28.27 \text{ cm}^2$ using $\pi = 3.14$. The exact area (using the full value of $\pi$) is $28.2858 \text{ cm}^2$. Calculate the percentage error of the approximation.

1. First calculate the absolute error of the area:
2. $$\Delta A = |28.2858 - 28.27| = 0.0158$$
3. Substitute into the percentage error formula, using the exact area as the true value:
4. $$\text{Percentage Error} = \left(\frac{0.0158}{28.2858}\right) \times 100\% \approx 0.056\%$$
5. Round to an appropriate number of significant figures matching the question:
6. Percentage error $\approx 0.06\%$

> **Exam tip:** Always match the number of significant figures in your error answer to the values given in the question.

## Error Bounds for Rounded Measurements

When a measurement is rounded to a given precision, the true value lies within a range of possible values called error bounds. The maximum possible absolute error (the error bound) is half the precision of the rounded measurement.

**Worked example:** A distance is given as 140 m, rounded to the nearest 10 m. Find the absolute error bound and maximum possible percentage error.

1. The precision of the measurement is 10 m, so the maximum absolute error is half the precision:
2. $$\text{Absolute error bound} = \frac{10}{2} = 5 \text{ m}$$
3. Calculate maximum percentage error using the approximate measurement (true value is unknown):
4. $$\text{Max Percentage Error} = \left(\frac{5}{140}\right) \times 100\% \approx 3.6\%$$
5. State the full range of possible true values:
6. True distance lies between 135 m (lower bound) and 145 m (upper bound)

## Common pitfalls

- **Wrong:** Dividing absolute error by the approximate value when the true value is given
  - Why it fails: The percentage error formula requires dividing by the true value by definition, this mistake costs marks
  - Correct: Always use the given true value for percentage error, only use the approximate value if the true value is unknown
- **Wrong:** Forgetting to multiply by 100% and add the % symbol for percentage error
  - Why it fails: Leaving the answer as a decimal instead of a percentage is marked incorrect in IB exams
  - Correct: Always multiply relative error by 100 and add the % symbol to your final answer
- **Wrong:** Forgetting to add units to absolute error
  - Why it fails: IB markers require units for dimensional quantities like absolute error
  - Correct: Always write the matching units for absolute error, same as the original measurement
- **Wrong:** Using the full precision as the error bound instead of half
  - Why it fails: For a measurement rounded to nearest $n$, the maximum error is $n/2$, not $n$
  - Correct: Always divide the measurement precision by 2 to get the maximum absolute error bound

## Cheatsheet

| Quantity | Formula | Key Exam Note |
| --- | --- | --- |
| Absolute Error | $\Delta x = \|x_{\text{true}} - x_{\text{approx}}\|$ | Has same units as $x$ |
| Percentage Error | $\left(\frac{\Delta x}{\|x_{\text{true}}\|}\right) \times 100\%$ | Most commonly requested error type |
| Maximum Error Bound | $\frac{\text{Measurement precision}}{2}$ | Used when true value is unknown |
| Lower/Upper Bounds | $x_{\text{approx}} \pm \text{error bound}$ | True value lies between these values |

## What's next

Approximation and error are foundational concepts for all measurement-based problems in IB AI SL, from statistics to real-world mathematical modelling. You will reuse error calculation skills when working with experimental data, rounded results from calculators, and real-world measurement problems across the entire course. Understanding error bounds also helps you interpret the accuracy of results from linear regression and other statistical models you will encounter later. Mastering these basic error concepts helps you avoid common mark losses in both Paper 1 and Paper 2 exams.

- [Functions](https://www.owlsprep.com/study/ib-math-ai-sl-u2-overview/)
- [Concept of function, domain, range, and graph](https://www.owlsprep.com/study/ib-math-ai-sl-u2-concept-of-function-domain-range/)

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