---
title: "Absolute versus relative error"
subject: "Numerical Method"
chapter: 1
key: "semester_4/numerical_method:absolute-relative-error-difference"
source: https://www.bctengineer.com/q/semester_4/numerical_method/absolute-relative-error-difference
site: BCT Engineering
---

# Absolute versus relative error

Numerical Method · Chapter 1: Introduction, Approximation and errors of computation

- Asked in: 2073 Bhadra, 4 marks; 2072 Asoj, 4 marks (asked 2 times)
- Video: [1.06 Absolute and Relative error](https://www.youtube.com/watch?v=9jDsHQExlkc) — Phat Mama Physics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/absolute-relative-error-difference
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-1

## Question

Discuss the difference between absolute error and relative error with appropriate examples.

## Short answer

Absolute error is the magnitude of difference between true and approximate value in the same unit. Relative error is that divided by true value, a dimensionless ratio. Example: 1000 m bridge measured 999 m, absolute 1 m, relative 0.001; 10 cm pin measured 9 cm, absolute 1 cm, relative 0.1, showing relative error reveals poor accuracy.

## Answer

If *X* is the true value and *X′* is the approximate value, the difference between them is the error. It can be stated in absolute or relative terms.

- **Absolute error:** Ea = |X − X′|. It gives the size of the error in the same unit as the quantity.
- **Relative error:** Er = Ea / |X|. It compares the error with the size of the true value and is unit-free. Percentage error = Er × 100 %.

| Basis | Absolute error | Relative error |
|---|---|---|
| Formula | \|X − X′\| | \|X − X′\| / \|X\| |
| Unit | Same unit as the quantity | No unit (ratio or %) |
| Shows | Actual size of the mistake | How serious the mistake is compared with the value |
| Use for comparison | Poor, depends on scale | Good, quantities of different size can be compared |
| Value | Can be large even for good accuracy | Small when accuracy is good |

#### Example
Bridge length: true 1000 m, measured 999 m. Pin length: true 10 cm, measured 9 cm.

- Bridge: Ea = 1 m, Er = 1/1000 = 0.001 (0.1 %).
- Pin: Ea = 1 cm, Er = 1/10 = 0.1 (10 %).

Both have the same absolute error (1 unit) but the pin measurement is far worse. The relative error shows this; the absolute error does not. Hence relative error is the better measure of accuracy.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Absolute error Ea is:
   A. |X - X'|, in the same unit as the quantity
   B. |X - X'|/|X|, unit-free
   C. |X'|/|X|
   D. (X - X')*100
   Answer: A. Absolute error is the magnitude of the difference and keeps the unit. (Key point: Absolute error)

2. Relative error is defined as:
   A. Ea * |X|
   B. Ea / |X| (true value)
   C. Ea / |X'| squared
   D. Ea + |X|
   Answer: B. It compares the error to the true value and has no unit. (Key point: Relative error)

3. True 1000 m measured 999 m; true 10 cm measured 9 cm. Which is the worse measurement?
   A. The bridge, because it is longer
   B. Both equal, since absolute errors are 1 unit
   C. The pin, because relative error is 10% vs 0.1%
   D. The bridge, because 1 m > 1 cm
   Answer: C. Equal absolute errors hide the difference; relative error shows the pin is far worse. (Key point: Comparison)

4. Why is relative error better for judging accuracy?
   A. It is always larger
   B. It has the same unit as the value
   C. It ignores the true value
   D. It lets quantities of different size be compared
   Answer: D. Being scale-independent, it shows how serious the error is compared to the value. (Key point: Use)

## Related in Numerical Method

- [Round off and compute errors](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Around-off-significant-figures) — Numerical Method, chapter 1
- [N-R: 4 sin x - e^x](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Anewton-raphson-4sinx-ex) — Numerical Method, chapter 2

## Related in other subjects

- [Relative and absolute stability; R-H of s⁵+s⁴+2s³+2s²+3s+5](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-relative-absolute-s5) — Control Systems, chapter 4, asked 2×
- [Relative and absolute stability; R-H of s⁶+2s⁵+8s⁴+12s³+20s²+16s+16](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-relative-absolute-s6) — Control Systems, chapter 4, asked 2×
- [Absolute vs relative dispersion; horses](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Ahorses-consistency-dispersion) — Probability and Statistics, chapter 1
- [Type I error; difference of two means procedure](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Atype-1-error-difference-two-means) — Probability and Statistics, chapter 6