Absolute, relative and percentage error | Year 9

01 / LEARN

The idea, step by step

Absolute and relative error

Absolute error is the distance between the measured value and an accepted reference value. The distance is never negative. To compare errors fairly across different sizes, divide the absolute error by the accepted value. Multiplying this relative error by 100 expresses it as a percentage. These formulas require a non-zero reference value.

e = ∣ m - t ∣, e_{r e l a t i v e} = \frac{∣ m - t ∣}{∣ t ∣}, e_{p e r c e n t} = \frac{∣ m - t ∣}{∣ t ∣} \times 100 %

Rounded measurements have a range

A measurement rounded to the nearest whole centimetre can differ from its unrounded value by at most half a centimetre. If it is rounded to the nearest 0.1 cm, the half-step is 0.05 cm. A rounding interval describes possible original values; it is not a claim that a true reference is known. Keep that idea separate from calculating a percentage error using a supplied accepted value.

A common trap

Dividing the error by the measured value when the accepted value is supplied.

Instead: Use the accepted reference as the denominator. State the unit for absolute error; relative error has no unit.

02 / WORKED EXAMPLES

See the thinking

Example 1

Find three forms of error

A scale reads 102 g for a certified 100 g mass. Find the absolute, relative and percentage error.

Absolute error $= ∣ 102 - 100 ∣ = 2$ g.
Relative error $= 2 / 100 = 0.02$ .
Percentage error $= 0.02 \times 100 % = 2 %$ .

2 g; 0.02; 2%

Example 2

A smaller absolute error is not always better

A 40 cm reference length is measured as 39 cm. A 100 cm reference is measured as 98 cm. Compare their percentage errors.

The first error is $1 / 40 \times 100 % = 2.5 %$ .
The second is $2 / 100 \times 100 % = 2 %$ .
The 100 cm measurement has the larger absolute error but the smaller percentage error.

The 98 cm measurement is closer in relative terms.

Check your understanding

A reading is 18 cm and the accepted value is 20 cm. What is the percentage error?

Show answer

10%.

The absolute error is 2 cm, so $2 / 20 \times 100 % = 10 %$ .

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A measurement is 78 m and the accepted length is 80 m. Find its absolute, relative and percentage error.
3 marks
A 20 g reference is measured as 20.4 g. Find the percentage error.
2 marks
A 250 L tank capacity is estimated as 240 L. Find the absolute and percentage errors.
2 marks
A ruler reading is rounded to the nearest 0.1 cm. State the greatest possible rounding error.
2 marks
A length is recorded as 6.2 m to the nearest 0.1 m. Give the interval of values that round to this reading.
3 marks
A 100 cm reference is measured as 98 cm. A 50 cm reference is measured as 49 cm. Which has the smaller percentage error?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A measurement is 78 m and the accepted length is 80 m. Find its absolute, relative and percentage error.
3 marks
$∣ 78 - 80 ∣ = 2$ m; $2 / 80 = 0.025$ ; $0.025 \times 100 % = 2.5 %$ .
2 m; 0.025; 2.5%
A 20 g reference is measured as 20.4 g. Find the percentage error.
2 marks
Error $= 0.4$ g; $0.4 / 20 \times 100 % = 2 %$ .
2%
A 250 L tank capacity is estimated as 240 L. Find the absolute and percentage errors.
2 marks
Absolute error $= 10$ L; $10 / 250 \times 100 % = 4 %$ .
10 L and 4%
A ruler reading is rounded to the nearest 0.1 cm. State the greatest possible rounding error.
2 marks
Half of a 0.1 cm step is 0.05 cm.
0.05 cm
A length is recorded as 6.2 m to the nearest 0.1 m. Give the interval of values that round to this reading.
3 marks
The half-step is 0.05 m. Subtract and add it to 6.2 m. The upper endpoint rounds up to 6.3 m.
$6.15 \leq L < 6.25$ m.
A 100 cm reference is measured as 98 cm. A 50 cm reference is measured as 49 cm. Which has the smaller percentage error?
3 marks
$2 / 100 \times 100 % = 2 %$ and $1 / 50 \times 100 % = 2 %$ .
Neither; both have 2% error.

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Curriculum reference & lesson scope

Australian Curriculum v9 / South Australian Curriculum

Year 9 surface area and volume of right prisms and cylinders, capacity, absolute/relative/percentage measurement error, precision and practical measurement problems.

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