Year 9 Measurement Error Worksheets & Practice

Worked examples

Year 9 measurement error: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

1. Example 1 · Finding absolute and percentage error

   A student measures the length of a book as 28 cm. The true length is 27.5 cm. Find the absolute error, the relative error and the percentage error.

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   Key idea: Absolute error is how far the measurement is from the true value. Relative error compares that gap to the true value. Percentage error converts the relative error to a percentage.

   1. Find the absolute error: $\text{absolute error} = \lvert \text{measured} - \text{true} \rvert = \lvert 28 - 27.5 \rvert = 0.5 \text{ cm}$
   2. Find the relative error: $\text{relative error} = \frac{\text{absolute error}}{\text{true value}} = \frac{0.5}{27.5} = \frac{1}{55}$
   3. Convert to a percentage error: $\text{percentage error} = \frac{1}{55} \times 100 \approx 1.82\%$
   4. Answer: The absolute error is $0.5$ cm, the relative error is $\dfrac{1}{55}$, and the percentage error is approximately $1.82\%$.

2. Example 2 · Comparing errors across two measurements

   Maya measures the mass of two objects. For object A, she records 450 g but the true mass is 460 g. For object B, she records 3.2 kg but the true mass is 3.25 kg. Which measurement has the smaller percentage error?

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   Key idea: To compare errors fairly across different measurements, use percentage error, since it is relative to the true value.

   1. Find the absolute error for each object:
      $\text{Object A: } \lvert 450 - 460 \rvert = 10 \text{ g}, \text{Object B: } \lvert 3.2 - 3.25 \rvert = 0.05 \text{ kg} = 50 \text{ g}$
   2. Express both true values in the same unit to check, then calculate percentage errors using consistent units:
      $\begin{gathered}\text{Object A: } \frac{10}{460} \times 100 \approx 2.17\% \\ \text{Object B: } \frac{0.05}{3.25} \times 100 \approx 1.54\%\end{gathered}$
   3. Compare: $1.54\% < 2.17\%$, so Object B has the smaller percentage error.
   4. Answer: Object B has the smaller percentage error ($\approx 1.54\%$ compared with $\approx 2.17\%$).

3. Example 3 · Interpreting measurement error in a real-life context

   A hardware store sells timber planks that are labelled as 2.4 m long. A builder measures 20 planks and finds their actual lengths vary. One plank measures 2.38 m. The store's quality standard requires a percentage error of no more than $1.5\%$. Determine whether this plank meets the standard, and explain what the percentage error means in this context.

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   Key idea: Calculate the percentage error and compare it to the given threshold. The percentage error tells us how significant the measurement gap is relative to the stated length.

   1. Identify the true (labelled) value and the measured value: true value $= 2.4$ m, measured value $= 2.38$ m.
   2. Find the absolute error: $\lvert 2.38 - 2.4 \rvert = 0.02 \text{ m}$
   3. Find the percentage error: $\frac{0.02}{2.4} \times 100 = \frac{2}{240} \times 100 \approx 0.83\%$
   4. Compare with the standard: $0.83\% < 1.5\%$, so the plank meets the quality standard.
   5. Interpret: A percentage error of $0.83\%$ means the plank is only $0.83\%$ shorter than its labelled length, which is within the acceptable tolerance for the builder's purposes.
   6. Answer: The plank has a percentage error of approximately $0.83\%$, which is within the $1.5\%$ standard, so it meets the requirement.

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