POP

C1.10 Limits of Accuracy

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01 — Explanation

What does "correct to the nearest X" actually mean?

Any rounded measurement is hiding a small range of possible true values. For example, a length given as $6\text{ m}$ to the nearest metre could really be anywhere from $5.5\text{ m}$ up to (but not including) $6.5\text{ m}$ .

Core tier note: you only need the bounds of a single rounded measurement — finding bounds of a calculation built from several rounded values is Extended-only.

02 — Explanation

How do I find the upper and lower bounds of a measurement?

Halve the unit of accuracy (not the measurement itself), then add and subtract:

\text{Lower Bound} = \text{Recorded Value} - \frac{u}{2}, \qquad \text{Upper Bound} = \text{Recorded Value} + \frac{u}{2}

Example: a bag of rice has a mass of $2.4\text{ kg}$ , correct to the nearest $0.1\text{ kg}$ .

Unit of accuracy $= 0.1\text{ kg}$ , so half-unit $= 0.05\text{ kg}$ .
Lower bound $= 2.4 - 0.05 = 2.35\text{ kg}$ .
Upper bound $= 2.4 + 0.05 = 2.45\text{ kg}$ .

Common Slip

halving the measurement itself instead of the rounding unit. Always identify what unit the value was rounded to first.

03 — Explanation

How do I write this as an error interval?

State it as a double inequality, with the lower bound included and the upper bound excluded:

\text{Lower Bound} \leqslant x < \text{Upper Bound}

For the rice bag: $2.35 \leqslant m < 2.45$ .

The upper bound is strict ( $<$ , not $\leqslant$ ) because reaching exactly $2.45\text{ kg}$ would actually round up to $2.5\text{ kg}$ , not $2.4\text{ kg}$ .

04 — Worked example

Worked exam-style question

The length of a room is given as $6\text{ m}$ , correct to the nearest metre. Write down the error interval for the true length $L$ .

Unit of accuracy $= 1\text{ m}$ , so half-unit $= 0.5\text{ m}$ .
Lower bound $= 6 - 0.5 = 5.5\text{ m}$ .
Upper bound $= 6 + 0.5 = 6.5\text{ m}$ .
Error interval: $5.5 \leqslant L < 6.5$ .

05 — Explanation

Does the method change if the measurement is rounded to a decimal place instead of a whole unit?

No — the method is identical, you just need to correctly identify the unit of accuracy first. Rounding "to the nearest $0.1$ " or "to 1 decimal place" both mean the unit of accuracy is $0.1$ .

Example: a plank of wood has a length of $3.8\text{ m}$ , correct to 1 decimal place. Find the upper and lower bounds.

Unit of accuracy $=0.1\text{ m}$ , so half-unit $=0.05\text{ m}$ .
Lower bound $=3.8-0.05=3.75\text{ m}$ .
Upper bound $=3.8+0.05=3.85\text{ m}$ .
Error interval: $3.75 \leqslant x < 3.85$ .

Exam Tip

whatever the unit of rounding — whole numbers, tenths, hundredths, or even "nearest 50" — the process is always the same: find the unit of accuracy, halve it, then add and subtract from the recorded value.

06 — Common mistakes

Common mistakes to avoid

Writing $6.49$ as the upper bound instead of $6.5$ — the strict inequality already excludes $6.5$ from rounding back down, so you should still state the value as $6.5$ .
Halving the wrong quantity — always halve the unit of accuracy, not the recorded measurement.
Using $\leqslant$ on both sides. The convention is inclusive lower bound, strict upper bound.