Bounds from a rounded measurement
These are rounding-only bounds for positive measurements with halfway values rounded up. Use the stated rounding unit; trailing zeros in an unexplained whole-number reading can be ambiguous
\[\begin{gathered}\text{lower}=x-\frac{u}{2}\\\text{upper}=x+\frac{u}{2}\end{gathered}\]
- State the rounding unit Use the stated precision. A reading such as 640 g alone does not tell you whether it was rounded to 1 g or 10 g.
- Allow half a rounding unit For rounding to the nearest unit u, the rounding uncertainty extends u/2 on each side. This is a bound, not the actual error of every reading \(\displaystyle \text{lower}=x-\frac{u}{2}\)
- Keep the upper boundary open For positive measurements with halfway values rounded up, the lower boundary is included and the upper boundary is excluded \(\displaystyle x-\frac{u}{2}\le t<x+\frac{u}{2}\)
- Separate other errors These bounds model rounding only. A badly calibrated instrument or an inconsistent measuring method can introduce additional error
Worked example
A length is 8.2 cm rounded to the nearest 0.1 cm. Find the possible interval
\[x=8.2\text{ cm},\quad u=0.1\text{ cm}\]
- \(\displaystyle u/2=0.05\text{ cm}\)Use half of the rounding unit
- \(\displaystyle 8.2-0.05=8.15\text{ cm}\)The lower bound is included
- \(\displaystyle 8.2+0.05=8.25\text{ cm}\)The upper bound is excluded
- \(\displaystyle 8.15\le t<8.25\)Here t is measured in centimetres. This interval does not cover additional calibration or method error
Common mistakes
Using a full unit
✕\(\displaystyle 5.6\le t<5.8\)✓\(\displaystyle 5.65\le t<5.75\)
A mass of 5.7 kg rounded to 0.1 kg has half-unit uncertainty 0.05 kg, not 0.1 kgTreating an upper bound as attainable
✕\(\displaystyle P_{\max}=130.2\)✓\(\displaystyle P<130.2\)
For four equal sides each less than 32.55 cm, the perimeter can approach 130.2 cm but cannot equal itDifficulty levels
introIdentify the rounding unit from an explicitly stated recording convention
standardCalculate lower and upper bounds of one- and two-decimal measurements
challengeRecover recorded values or combine bounds for repeated lengths and perimeters
examRecognise the correct pair of bounds and reject common half-unit errors
Questions about this topic
Does 640 g always mean nearest 10 g?+
No. The rounding precision must be stated. Whole-number trailing zeros alone do not establish the resolution
Is half the rounding unit the actual measurement error?+
No. It bounds rounding uncertainty under the stated convention. The actual error depends on the unknown true value, and other measurement errors may exist
Why is there no greatest value below the upper bound?+
Any value below the boundary can be increased a little while remaining below it. The boundary is a strict upper bound, not a value the measurement can attain
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 5: Measurement
In this chapter
5A Length conversion and perimeter: Convert measurement units5A Length conversion and perimeter: Perimeter and area of standard shapesFind an unknown side from a perimeter5A Length conversion and perimeter: Solve a linear equation5A Length conversion and perimeter: Substitute values into a formula5A Length conversion and perimeter: Round measurements to decimal places5B Circumference, sectors and exact π: Circumference and area of a circle5B Circumference, sectors and exact π: Arc length and sector areaPerimeter of a sector5B Circumference, sectors and exact π: Simplify exact expressions involving pi5B Circumference, sectors and exact π: Substitute values into a formula5B Circumference, sectors and exact π: Round measurements to decimal places5C Area formulas and units: Convert measurement units5C Area formulas and units: Perimeter and area of standard shapes5C Area formulas and units: Circumference and area of a circle5C Area formulas and units: Arc length and sector area5C Area formulas and units: Simplify exact expressions involving pi5C Area formulas and units: Substitute values into a formula5C Area formulas and units: Round measurements to decimal placesPerimeter and area of composite shapes5D Composite perimeter and area: Find a side of a right-angled triangle5D Composite perimeter and area: Substitute values into a formula5D Composite perimeter and area: Round measurements to decimal places5E Right-prism surface area: Surface area of cuboidsSurface area of a triangular prism5E Right-prism surface area: Find a side of a right-angled triangle5E Right-prism surface area: Substitute values into a formula5E Right-prism surface area: Round measurements to decimal placesPyramid surface area: Surface area of square pyramids (optional extension)5F Cylinder surface area: Surface area of cylindersSurface area of a cylinder portion5F Cylinder surface area: Substitute values into a formula5F Cylinder surface area: Round measurements to decimal places5G Volume and capacity: Convert measurement units5G Volume and capacity: Volume of right prisms5G Volume and capacity: Capacity of a container5G Volume and capacity: Solve a linear equation5G Volume and capacity: Substitute values into a formula5G Volume and capacity: Round measurements to decimal places5H Cylinder volume and capacity: Volume of cylinders5H Cylinder volume and capacity: Capacity of cylindersVolume of a cylinder portion5H Cylinder volume and capacity: Substitute values into a formula5H Cylinder volume and capacity: Round measurements to decimal placesSmallest unit and limits of accuracy