Errors and Limits of Accuracy
Every measurement is an approximation. The absolute error is always half the smallest unit of the instrument — and when measurements are combined, errors compound. Knowing this prevents catastrophic calculation mistakes.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
A ruler is marked in millimetres. You measure a piece of timber as 45 cm.
Without calculating — what is the largest it could actually be? What is the smallest? How does this matter if you are cutting 20 of these pieces from a single plank?
Error calculations start with three core formulas. Every bounds and percentage-error question is just a rearrangement of these.
Absolute error is always half the smallest unit — it captures the maximum possible difference between the measured value and the true value. Upper and lower bounds put a bracket around the true value. Percentage error tells you how significant the error is relative to the measurement.
Key facts
- Absolute error = ½ × smallest unit of measurement
- Upper bound = value + absolute error; lower bound = value − absolute error
- Percentage error = (absolute error ÷ measurement) × 100%
- Errors compound when measurements are added or multiplied
Concepts
- Why every measurement has an inherent uncertainty
- How measurement precision affects reliability of calculations
- Why a small measurement has a larger percentage error than a large one
Skills
- State the absolute error for any given instrument precision
- Calculate upper and lower bounds for a measurement
- Calculate percentage error
- Find bounds for calculated quantities (area, perimeter)
When you read a measurement from any instrument, you round to the nearest marked graduation. This introduces an uncertainty of up to half that graduation on either side.
- Ruler in mm: precision = 1 mm, absolute error = 0.5 mm
- Scale in 0.1 kg: precision = 0.1 kg, absolute error = 0.05 kg
- Thermometer in 1°C: precision = 1°C, absolute error = 0.5°C
- Odometer in 0.1 km: precision = 0.1 km, absolute error = 0.05 km
A measured value of $x$ with absolute error $e$ means the true value lies in the interval $[x-e, \; x+e]$.
What to write in your book
- Absolute error $= \frac{1}{2} \times \text{smallest unit}$ — this is always the starting step.
- Upper bound $= \text{measurement} + \text{AE}$; Lower bound $= \text{measurement} - \text{AE}$.
- True value lies in the interval $[\text{lower bound},\; \text{upper bound}]$.
- Example: 34 cm measured on a 1 cm ruler → AE = 0.5 cm → bounds: [33.5, 34.5] cm.
Did you get this? True or false: a ruler graduated in millimetres has an absolute error of 1 mm.
Worked examples · 3 in a row, reveal as you go
A length is measured as 34 cm using a ruler marked in centimetres (precision = 1 cm). (a) State the absolute error. (b) Find the upper and lower bounds of the true length.
A mass is recorded as 45 kg using scales with precision 0.5 kg. Find the percentage error, correct to 2 decimal places.
A rectangle is measured as 8 m × 5 m using a tape measure with precision 0.1 m. (a) State the bounds for each dimension. (b) Find the maximum and minimum possible area.
What to write in your book
- % error formula: $\% \text{ error} = \dfrac{\text{AE}}{\text{measurement}} \times 100\%$ — percentage error depends on both AE and the size of the measurement.
- A 0.5 cm error on a 5 cm measurement = 10% error; on a 500 cm measurement = 0.1% error.
- Compounding in area: Max area = upper bound × upper bound; Min area = lower bound × lower bound.
- Compounding in perimeter: Max perimeter = sum of all upper bounds; Min perimeter = sum of all lower bounds.
Quick check: A distance is measured as 6.0 m with precision 0.1 m. What is the percentage error?
Common errors · the 3 traps that cost marks
What to write in your book
- AE = ½ × unit — never the full unit.
- Always check AE and measurement share the same unit before dividing for % error.
- Max area = upper × upper; Min area = lower × lower.
- Max perimeter = sum of uppers; Min perimeter = sum of lowers.
Fill the gap: A rectangle is measured as 12 cm × 9 cm with precision 1 mm. The absolute error is mm. The upper bound of the 12 cm side is 12. cm.
Quick-fire practice · 5 calculations
A length is measured as 72 mm using a ruler with 1 mm graduations. State the absolute error and the upper and lower bounds.
A container holds 2.4 L, measured using a jug marked in 0.1 L divisions. Find the absolute error and bounds.
A plank is measured as 85 cm using a ruler with 1 mm precision. Find the percentage error (to 3 significant figures).
A measurement of 0.6 kg is taken with precision 0.1 kg. Find the percentage error.
Three lengths of rope each measure 2.5 m with precision 0.01 m. They are joined end to end. Find the maximum and minimum total length.
Odd one out: Three of these statements are correct. Which one is wrong?
Earlier you estimated the largest and smallest possible length of the 45 cm timber piece. Let's check:
Ruler in mm → precision = 1 mm = 0.1 cm → AE = 0.5 mm = 0.05 cm
Upper bound: $45 + 0.05 = 45.05$ cm. Lower bound: $45 - 0.05 = 44.95$ cm.
Over 20 pieces: maximum total = $20 \times 45.05 = 901.0$ cm; minimum total = $20 \times 44.95 = 899.0$ cm. That is a possible variation of 2 cm across the full plank — which could be critical when fitting timber.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. A length is measured as 8.5 cm using a ruler with 1 mm graduations. (a) State the absolute error of this measurement. (b) State the upper and lower bounds for the true length. (c) Calculate the percentage error, correct to 2 significant figures. (3 marks)
Q2. The dimensions of a swimming pool are measured as 25 m × 12 m using a tape measure with precision 0.5 m. (a) State the absolute error and the bounds for the 25 m measurement. (b) Find the maximum and minimum possible area of the pool. (3 marks)
Q3. A surveyor measures three sides of a triangular paddock as 120 m, 85 m, and 95 m using a distance wheel with precision 1 m. (a) State the absolute error for each measurement. (b) Find the maximum possible perimeter. (c) Find the minimum possible perimeter. (d) What is the range of possible perimeters? (4 marks)
📖 Comprehensive answers (click to reveal)
Drill 1: AE = 0.5 mm; bounds: $[71.5,\; 72.5]$ mm · 2: AE = 0.05 L; bounds: $[2.35,\; 2.45]$ L · 3: AE = 0.5 mm = 0.05 cm; % error $= 0.05/85 \times 100 \approx 0.0588\%$ · 4: AE = 0.05 kg; % error $= 0.05/0.6 \times 100 = 8.\overline{3}\%$ · 5: Each: $[2.49,\; 2.51]$ m; Max = $7.53$ m; Min = $7.47$ m
Q1 (3 marks): (a) $\frac{1}{2} \times 1 \text{ mm} = 0.5 \text{ mm}$ [1]. (b) $[8.45,\; 8.55]$ cm [1]. (c) $\frac{0.05}{8.5} \times 100 \approx 0.59\%$ [1].
Q2 (3 marks): (a) AE = 0.25 m; bounds for 25 m: $[24.75,\; 25.25]$ m; bounds for 12 m: $[11.75,\; 12.25]$ m [1]. (b) Max: $25.25 \times 12.25 = 309.3125 \text{ m}^2$ [1]. Min: $24.75 \times 11.75 = 290.8125 \text{ m}^2$ [1].
Q3 (4 marks): (a) AE $= 0.5$ m for each [1]. (b) Max: $(120.5 + 85.5 + 95.5) = 301.5$ m [1]. (c) Min: $(119.5 + 84.5 + 94.5) = 298.5$ m [1]. (d) Range $= 301.5 - 298.5 = 3$ m [1].
Five timed questions on absolute error, percentage error and bounds. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering errors and limits of accuracy questions. Pool: lesson 13.
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