C1.10 Limits of accuracy

Learning objectives

C1.10.1Give upper and lower bounds for data rounded to a specified accuracy• Give upper and lower bounds for data rounded to a specified accuracy. e.g. write down the upper bound of a length measured correct to the nearest metre. Candidates are not expected to find the bounds of the results of calculations which have used data rounded to a specified accuracy.

A rounded value represents an interval of possible original values. Find the rounding step, take half of it, then subtract and add that half-step to locate the lower and upper boundaries.

$x- rac{r}{2}\le v<x+ rac{r}{2}$

Here $x$ is the stated rounded value, $r$ is one unit at the stated accuracy, and $v$ is the original value. The lower bound is included; the upper bound is excluded because an exact upper-bound value rounds to the next stated value.

Stated accuracy	Rounding step $r$	Half-step	Example interval
nearest kilogram	$1$ kg	$0.5$ kg	$428.5\le m<429.5$ for 429 kg
nearest 5 g	$5$ g	$2.5$ g	$112.5\le m<117.5$ for 115 g
1 decimal place	$0.1$ unit	$0.05$ unit	$76.25\le h<76.35$ for 76.3 m
2 decimal places	$0.01$ unit	$0.005$ unit	$37.835\le h<37.845$ for 37.84 m

Convert the stated accuracy into the variable's unit before halving. If $p$ is in kilograms and 12.4 kg is correct to the nearest 100 g, then $r=0.1$ kg, so $12.35\le p<12.45$ .

For significant figures, use the place value of the final significant digit as the rounding step. For example, 350 correct to 2 significant figures has step 10, giving $345\le v<355$ .

Do not use the full rounding step on each side, and do not write $\le$ at the upper bound. This Core objective asks for bounds of rounded data only; it does not require bounds for results calculated from rounded inputs.

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