# Approximation, error and estimation

> IB Mathematics: Applications and Interpretation HL · IB DP AI HL (First Assessment 2021)
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u1-approximation-error-and-estimation/

This module covers approximation techniques, error calculation, and estimation for IB AI HL. You will learn to round appropriately, quantify errors in measurements, and quickly estimate results to check your calculations.

**Prerequisites:** [Basic arithmetic and introductory rounding rules](https://www.owlsprep.com/study/ib-math-ai-hl-u1-basic-number-concepts/)

## Learning objectives

- Distinguish between exact and approximate values for measurement and calculation
- Calculate absolute, relative and percentage error for approximate values
- Round values to an appropriate number of significant figures or decimal places
- Estimate results of calculations to check for reasonableness
- Identify common sources of error in applied problems

## Significant Figures and Rounding

**Significant figures** — Digits that carry meaningful information about the precision of a measurement. Leading zeros are never significant; trailing zeros are significant only after a decimal point.

*Notation:* abbreviated s.f.

*Example:* 0.0042 has 2 s.f., 2400 has 2 s.f., 2400.0 has 5 s.f.

Rounding to a specified number of significant figures is required for almost all calculations in IB exams, as it communicates the precision of your result. Follow these consistent rules to avoid mistakes:

1. Start counting from the first non-zero digit on the left
2. Count the required number of significant digits
3. Round the last digit up if the next digit is 5 or greater, otherwise leave it unchanged

**Worked example:** Round 0.024571 to 3 significant figures

1. The first non-zero digit is 2 (1st s.f.), so we count 3 digits: 2 (1st), 4 (2nd), 5 (3rd). The next digit after 5 is 7.
2. Since 7 ≥ 5, round the 3rd digit up from 5 to 6.
3. Final rounded value: 0.0246

## Absolute, Relative and Percentage Error

**Error measures** — Quantify the discrepancy between a true (exact) value and an approximate or measured value. Three standard measures are used in IB AI HL:

Errors arise from rounding, measurement device precision limits, and estimation. Each error type communicates different information about the size of the discrepancy.

$$\text{Absolute error: } |\Delta x| = |x_{\text{true}} - x_{\text{approx}}|$$

$$\text{Relative error: } \frac{|\Delta x|}{|x_{\text{true}}|}, \quad x_{\text{true}} \neq 0$$

$$\text{Percentage error: } \frac{|\Delta x|}{|x_{\text{true}}|} \times 100\%$$

**Worked example:** A runner's race time is measured as 42.3 seconds. The actual official time is 42.7 seconds. Calculate the absolute, relative and percentage error of the measurement.

1. Identify values: true time $x_{true} = 42.7$ s, approximate time $x_{approx} = 42.3$ s
2. Calculate absolute error:
3. $$|\Delta x| = |42.7 - 42.3| = 0.4 \text{ s}$$
4. Calculate relative error:
5. $$\text{Relative error} = \frac{0.4}{42.7} \approx 0.0094$$
6. Calculate percentage error:
7. $$\text{Percentage error} = 0.0094 \times 100\% \approx 0.94\%$$

> **info**
>
> If no true value is given, the maximum absolute error for a measurement is half the precision of the device. For example, a scale that measures to the nearest 0.1 kg has maximum absolute error 0.05 kg.

## Estimation of Calculations

Estimation is used to quickly check if a final calculated answer is reasonable, or to get an approximate result when an exact value is not needed. The standard technique is to round every value in the calculation to 1 or 2 significant figures before computing.

**Worked example:** Estimate the value of $\frac{10.2 \times 58.7}{1.98}$

1. Round each value to 1 significant figure:
2. $$10.2 \approx 10, \quad 58.7 \approx 60, \quad 1.98 \approx 2$$
3. Substitute and simplify:
4. $$\frac{10 \times 60}{2} = 300$$
5. The exact value is ~ 302.6, so the estimate is accurate for a quick check.

**Check your understanding**

Test your understanding of maximum error

1. A mass is given as 1400 g to 2 significant figures. What is the maximum possible absolute error?

   - 0.5 g
   - 5 g
   - 50 g
   - 100 g

   *Why:* 1400 g to 2 s.f. means the true value lies between 1350 g and 1450 g, so the maximum error is $1450 - 1400 = 50$ g.

## Common pitfalls

- **Wrong:** Counting leading zeros as significant figures
  - Why it fails: Leading zeros only mark the position of the decimal point, they do not contribute to precision. Overcounting leads to incorrect rounding
  - Correct: Always start counting significant figures from the first non-zero digit from the left
- **Wrong:** Using the approximate value instead of the true value in the denominator for percentage error
  - Why it fails: The IB syllabus defines percentage error relative to the true value, so this will always give an incorrect result
  - Correct: Always divide the absolute error by the true value when calculating relative or percentage error
- **Wrong:** Writing 1500 to indicate 4 significant figures, with no additional notation
  - Why it fails: Trailing zeros without a decimal point are ambiguous, and examiners will assume they are placeholders
  - Correct: Write it as $1.500 \times 10^3$ to clearly show 4 significant figures
- **Wrong:** Forgetting to include units for absolute error
  - Why it fails: Absolute error is a measure of the difference in the original quantity, so it shares the original units. Examiners often penalize missing units
  - Correct: Add the correct units to your absolute error answer, leave relative error unitless and add a % symbol for percentage error

## Cheatsheet

| Term | Formula | Key Note |
| --- | --- | --- |
| Absolute Error | $\|x_{true} - x_{approx}\|$ | Same units as original quantity |
| Relative Error | $\frac{\|x_{true} - x_{approx}\|}{\|x_{true}\|}$ | Unitless proportional error |
| Percentage Error | $\frac{\|x_{true} - x_{approx}\|}{\|x_{true}\|} \times 100\%$ | Add % symbol to final answer |
| Max Measurement Error (precision $u$) | $\frac{u}{2}$ | Use when true value is unknown |
| Estimation Rule | Round all values to 1 s.f. | Quick check for calculation reasonableness |

## What's next

Approximation and error analysis is a foundational skill that appears across every topic in IB AI HL, from descriptive statistics to differential equations to real-world modelling. Examiners enforce strict significant figure rules, so mastering this topic helps you avoid losing easy marks on every exam question. The focus on error also aligns with the applications focus of AI HL, as all real-world data has inherent uncertainty. Your next topic will build on this foundation to work with large and small numbers in scientific notation.

- [Arithmetic and geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-hl-u1-arithmetic-and-geometric-sequences-and/)
- [Financial applications of geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-hl-u1-financial-applications-of-geometric-sequences/)
- [Laws of Logarithms](https://www.owlsprep.com/study/ib-math-ai-hl-u1-laws-of-logarithms/)

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