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 calculatingwhat 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 $[x-e, \; x+e]$.
Absolute error = half the smallest graduation of the measuring instrument. Percentage error = (absolute error ÷ measurement) × 100%. Measurements are reported as: value ± absolute error. Smaller instruments give smaller percentage errors.
Pause, copy the absolute error formula (half the smallest graduation of the instrument), the percentage error formula (absolute error ÷ measurement × 100%), and the measurement notation (value ± absolute error) into your book.
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.
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
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 confidencethat 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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