Absolute, relative and percentage error
These questions compare a measurement with a stated positive reference treated as exact
\[\begin{aligned}\text{Absolute error}&=|\text{measured}-\text{reference}|\\\text{Relative error}&=\frac{\text{absolute error}}{\text{reference}}\end{aligned}\]
- Use matching units Convert before subtracting. A conversion changes the unit, not the quantity \(\displaystyle 278\text{ mm}=27.8\text{ cm}\)
- Express the ratio per hundred The percentage and decimal ratio represent the same value \(\displaystyle 0.0375=3.75\%\)
- Round at the end Keep the ratio exact until you round the requested percentage
- Interpret carefully Zero error means this reading matches this reference. It does not establish the accuracy of every reading from a device
Common mistakes
Dividing by the reading
✕\(\displaystyle \text{Relative error}=\frac{3}{72}\)✓\(\displaystyle \text{Relative error}=\frac{3}{75}\)
With a reference mass of 75 g and a measured mass of 72 g, the absolute error is 3 g. Divide by the reference, not by the readingKeeping the sign in absolute error
✕\(\displaystyle |72\text{ g}-75\text{ g}|=-3\text{ g}\)✓\(\displaystyle |72\text{ g}-75\text{ g}|=3\text{ g}\)
Absolute error is the non-negative size of the differenceChanging the quantity during conversion
✕\(\displaystyle 278\text{ mm}\div10=27.8\text{ cm}\)✓\(\displaystyle 278\text{ mm}=27.8\text{ cm}\)
The two measurements are equal. Dividing the physical length by ten would change the lengthDifficulty levels
introFind exact whole-number percentage errors with matching units
standardFind exact non-whole percentage errors from decimal readings
challengeConvert units when needed and round only the final percentage
examChoose a same-unit percentage error from diagnostic alternatives
Questions about this topic
Can this percentage error be negative?+
No. It reports an absolute error magnitude. Signed discrepancies record direction separately
Why must the reference be positive?+
These questions use positive measurement sizes as the comparison base. A zero reference would require division by zero
Does this account for uncertainty in the reference?+
No. These questions treat the stated reference as exact
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 1: Number and financial mathematics
In this chapter
Evaluate an integer expressionFactors, multiples, HCF, LCM and primesRound to decimal places or significant figuresEstimate an expression using rounded inputsConvert between fractions, mixed numbers and decimalsClassify real numbers and compare exact valuesAdd, subtract, multiply and divide fractions and mixed numbersWrite a ratio from countsWrite ratios in simplest formComplete equivalent ratiosUse a constant rateDivide a quantity in a given ratioDecide which option is the better valueConvert between fractions, decimals and percentagesFind a percentage of a quantityExpress one quantity as a percentage of another1F Percentages and money: Find the original amount from a percentageIncrease or decrease a quantity by a percentageFind percentage changePercentage error against a reference value1G Percentage change and percentage error: Find the original amount from a percentageProfit, mark-up and discountIncome and taxSimple interestCompound interest (optional extension)