Absolute, Relative & Percentage Error

Statement

When measuring or approximating values, errors are inevitable. There are three main ways to quantify them:

• Absolute error: the difference between the true value and the approximation.
• Relative error: the absolute error compared to the true value, expressed as a ratio.
• Percentage error: the relative error expressed as a percentage.

\[ \text{Abs. error} = |x - \hat{x}|, \quad \text{Rel. error} = \frac{|x - \hat{x}|}{|x|}, \quad \% \text{ error} = \frac{|x - \hat{x}|}{|x|}\times 100\% \]

Why it’s true (short reason)

• Absolute error measures the raw difference.
• Relative error compares this to the size of the true value, giving a sense of scale.
• Percentage error is simply relative error expressed as a percentage for easier interpretation.

Recipe (how to use it)

1. Find the difference between measured (or approximate) and true values: \(|x-\hat{x}|\).
2. For relative error, divide this difference by the true value.
3. For percentage error, multiply the relative error by 100.

Spotting it

• Questions asking “How accurate?” or “What is the error as a percentage?”
• Problems involving rounding, truncation, or approximate values.
• Data comparisons in physics, chemistry, and engineering often require error measures.

Common pairings

• Bounds of accuracy: linking absolute error with upper and lower bounds.
• Approximations of irrational numbers: e.g., comparing \(\pi\) to 3.14.
• Scientific experiments: reporting measurement errors alongside results.

Mini examples

1. Given: True length 10 cm, measured 9.8 cm. Abs. error: 0.2 cm. Rel. error: 0.02. % error: 2%.
2. Given: True speed 25 m/s, estimate 24 m/s. Abs. error: 1. Rel. error: 0.04. % error: 4%.

Pitfalls

• Dividing by approximate instead of true value: always use the true value for relative error.
• Forgetting absolute value bars: errors are non-negative.
• Mixing units: keep the same units for true and measured values.

Exam strategy

• Underline true and approximate values.
• Always compute absolute error first — it forms the basis for the others.
• State answers with correct units for absolute error, but relative and percentage errors are unitless.
• Percentage error should always include “%”.

Summary

Absolute, relative, and percentage errors are three linked ways of measuring the accuracy of approximations. Absolute error shows the raw difference, relative error gives it in proportion to the true value, and percentage error expresses this ratio in a simple, intuitive percentage. These tools are central in science and mathematics for assessing the reliability of calculations and measurements.