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.

50–55 min MS-M2 — MEDIUM 3 MC 3 SA Lesson 13 of 22 Free
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Think First

A ruler is marked in millimetres. You measure a piece of timber as 45 cm. 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?

Type your initial response below — you will revisit this at the end of the lesson.

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Saved

Come back to this at the end of the lesson.

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Error Formulas — This Lesson

$\text{Absolute error} = \dfrac{1}{2} \times \text{smallest unit}$
Absolute error — also called the "limit of reading"; always half the precision of the instrument e.g. ruler graduated in mm: smallest unit = 1 mm, absolute error = 0.5 mm
$\text{Upper bound} = \text{measurement} + \text{absolute error}$
Upper bound — the largest possible true value Lower bound = measurement − absolute error
$\text{\% error} = \dfrac{\text{absolute error}}{\text{measurement}} \times 100\%$
Percentage error — expresses absolute error as a percentage of the measured value A smaller measurement has a larger percentage error for the same absolute error
MEASUREMENT ERROR BOUNDS AE AE lower bound measured value upper bound AE = ½ × smallest unit SAME AE, DIFFERENT % ERROR Measure: 5 cm AE: 0.5 cm % err = 10% Measure: 50 cm AE: 0.5 cm % err = 1% Larger measurement → smaller % error

🧠 Know

  • 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

💡 Understand

  • 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

✅ Can Do

  • 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)
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Key Terms

Absolute error The maximum possible difference between the measured value and the true value; equal to half the smallest graduation of the measuring instrument
Bounds The upper bound is the largest possible true value; the lower bound is the smallest possible true value; true value lies in $[\text{lower}, \text{upper}]$
Percentage error The absolute error expressed as a percentage of the measured value; indicates the relative size of the error
Precision The smallest unit that an instrument can measure; a more precise instrument has a smaller unit and therefore a smaller absolute error

Misconceptions to Fix

✗

Wrong: Capacity and volume are completely different concepts.

✗

Right: Capacity and volume are related; 1 litre = 1000 cm³ = 1 dm³.

Absolute Error and Bounds

Key Point

Always check your units before substituting into formulas. Converting to consistent units is a common source of errors in assessment tasks.

Key Terms
FormulaA rule showing the relationship between variables using symbols.
SubstitutionReplacing variables with their known values in an equation.
Unit ConversionChanging a measurement from one unit to another.
CapacityThe amount of liquid a container can hold, measured in litres or millilitres.
PerimeterThe total distance around the outside of a shape.
AreaThe amount of space inside a two-dimensional shape.

Every Measurement Has an Error

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.

A measured value of $x$ with absolute error $e$ means the true value lies in the interval $[x-e, \; x+e]$.

Worked Example 1 Upper and Lower Bounds

Problem

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.

Solution

1 (a) Absolute error $= \frac{1}{2} \times 1 \text{ cm} = 0.5 \text{ cm}$ Precision is 1 cm; absolute error is always half the smallest unit

Percentage Error

Expressing Error as a Percentage

Absolute error alone doesn't tell you how significant the error is. A 0.5 cm error in a 5 cm measurement is very significant; in a 500 cm measurement it is trivial. Percentage error puts errors in context.

$$\text{\% error} = \frac{\text{absolute error}}{\text{measurement}} \times 100\%$$

Key insight: Smaller measurements always have larger percentage errors (for the same instrument). This is why short lengths measured with a ruler are less reliable than long lengths.
Worked Example 2 Percentage Error

Problem

A mass is recorded as 45 kg using scales with precision 0.5 kg. Find the percentage error, correct to 2 decimal places.

Solution

1 Absolute error $= \frac{1}{2} \times 0.5 = 0.25 \text{ kg}$ Precision = 0.5 kg

Compounding Errors in Calculations

Bounds for Calculated Quantities

When calculated quantities depend on measured values, errors combine. For a sum (like perimeter), errors add. For a product (like area), the bounds are found by using the upper/lower bounds of each measurement.

Worked Example 3 Bounds for Area

Problem

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.

Solution

1 (a) Error $= 0.05$ m; Length: $[7.95, 8.05]$ m; Width: $[4.95, 5.05]$ m Absolute error $= \frac{1}{2}(0.1) = 0.05$ m; subtract and add from each measurement
Practice

Practice Questions

Section A — Absolute Error and Bounds

  1. A length is measured as 72 mm using a ruler with 1 mm graduations. State the absolute error and the upper and lower bounds.
  2. A container holds 2.4 L, measured using a jug marked in 0.1 L divisions. Find the absolute error and bounds.
  3. A temperature is recorded as 37.2°C using a thermometer with 0.2°C graduations. Find the bounds.
  4. A car odometer reads 1256.8 km (precision 0.1 km). State the bounds for the actual distance.

Section B — Percentage Error

  1. A plank is measured as 85 cm using a ruler with 1 mm precision. Find the percentage error (to 3 significant figures).
  2. Two students measure the same length: Student A records 8 m (precision 0.5 m); Student B records 8.0 m (precision 0.1 m). Compare their percentage errors.
  3. A measurement of 0.6 kg is taken with precision 0.1 kg. Find the percentage error.

Section C — Compounding Errors

  1. A rectangle is measured as 12 cm × 9 cm with precision 1 mm. Find the maximum and minimum area.
  2. 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.

Q1

Abs. error = 0.5 mm; bounds: $[71.5, 72.5]$ mm

Q2

Abs. error = 0.05 L; bounds: $[2.35, 2.45]$ L

Q3

Abs. error = 0.1°C; bounds: $[37.1, 37.3]$°C

Q4

Bounds: $[1256.75, 1256.85]$ km

Q5

Abs. error = 0.5 mm = 0.05 cm; % error $= 0.05/85 \times 100 \approx \mathbf{0.0588\%}$

Q6

Student A: $0.25/8 \times 100 = 3.125\%$; Student B: $0.05/8 \times 100 = 0.625\%$; Student B is more precise

Q7

Abs. error = 0.05 kg; % error $= 0.05/0.6 \times 100 = \mathbf{8.\overline{3}\%}$

Q8

Abs. error = 0.5 mm = 0.05 cm; Max: $12.05 \times 9.05 = 109.0525 \approx \mathbf{109.05 \text{ cm}^2}$; Min: $11.95 \times 8.95 = 106.9525 \approx \mathbf{106.95 \text{ cm}^2}$

Q9

Each piece: bounds $[2.49, 2.51]$ m; Max total: $3 \times 2.51 = \mathbf{7.53 \text{ m}}$; Min total: $3 \times 2.49 = \mathbf{7.47 \text{ m}}$

Revisit Your Initial Thinking

Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?

Multiple Choice

1 A balance scale measures mass to the nearest 0.5 g. A sample is recorded as 12 g. The upper bound of the true mass is:

A   12.05 g
B   12.25 g
C   12.5 g
D   13 g

? Regarding this topic, 1 A balance scale measures mass to the nearest 0.5 g. A sample is recorded as 12 g. The upper bound of the true mass is:

A     12.05 g
B     12.25 g
C     12.5 g
D     13 g
B - Correct!
B — Absolute error $= \frac{1}{2}(0.5) = 0.25$ g; upper bound $= 12 + 0.25 = 12.25$ g.

2 A distance is measured as 6.0 m with precision 0.1 m. The percentage error is:

A   0.83%
B   1.67%
C   0.17%
D   8.33%

? Regarding this topic, 2 A distance is measured as 6.0 m with precision 0.1 m. The percentage error is:

A     0.83%
B     1.67%
C     0.17%
D     8.33%
A - Correct!
A — Absolute error $= 0.05$ m; % error $= 0.05/6.0 \times 100 = 0.8\overline{3}\% \approx 0.83\%$.

3 A square has side length measured as 10 cm with absolute error 0.5 cm. The maximum possible area of the square is:

A   100.25 cm²
B   105.25 cm²
C   110.25 cm²
D   110 cm²

? Regarding this topic, 3 A square has side length measured as 10 cm with absolute error 0.5 cm. The maximum possible area of the square is:

A     100.25 cm²
B     105.25 cm²
C     110.25 cm²
D     110 cm²
C - Correct!
C — Upper bound of side $= 10 + 0.5 = 10.5$ cm; max area $= 10.5^2 = 110.25$ cm².

Short Answer

01

SA 4 3 marks A length is measured as 8.5 cm using a ruler with 1 mm graduations.

(a) State the absolute error of this measurement.  (1 mark)

(b) State the upper and lower bounds for the true length.  (1 mark)

(c) Calculate the percentage error, correct to 2 significant figures.  (1 mark)

Work in your book
Saved

(a)

$\frac{1}{2} \times 1 \text{ mm} = \mathbf{0.5 \text{ mm}}$

(b)

$[8.45 \text{ cm}, \; 8.55 \text{ cm}]$

(c)

$\frac{0.05}{8.5} \times 100 \approx \mathbf{0.59\%}$

02

SA 5 3 marks 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.  (1 mark)

(b) Find the maximum and minimum possible area of the pool.  (2 marks)

Work in your book
Saved

(a)

Abs. error $= 0.25$ m; bounds for 25 m: $[24.75, 25.25]$ m; bounds for 12 m: $[11.75, 12.25]$ m

(b)

Max: $25.25 \times 12.25 = \mathbf{309.3125 \text{ m}^2}$; Min: $24.75 \times 11.75 = \mathbf{290.8125 \text{ m}^2}$

03

SA 6 4 marks 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.  (1 mark)

(b) Find the maximum possible perimeter of the paddock.  (1 mark)

(c) Find the minimum possible perimeter.  (1 mark)

(d) What is the range of possible perimeters?  (1 mark)

Work in your book
Saved

(a)

Abs. error $= \frac{1}{2} \times 1 = \mathbf{0.5 \text{ m}}$ for each measurement

(b)

Max: $(120.5 + 85.5 + 95.5) = \mathbf{301.5 \text{ m}}$

(c)

Min: $(119.5 + 84.5 + 94.5) = \mathbf{298.5 \text{ m}}$

(d)

Range $= 301.5 - 298.5 = \mathbf{3 \text{ m}}$

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Errors and Limits of Accuracy