Upper and Lower Bounds Using Error Intervals

📋 Spec coverage: GCSE Edexcel Mathematics (1MA1) This article covers:

• Edexcel N15 - Rounding and error intervals: using inequality notation to give error intervals for rounded values.
• Edexcel N16 - Limits of accuracy: identifying and interpreting upper and lower bounds.

Tier: Foundation and Higher. These skills can be assessed on Papers 1, 2 or 3.

To find the lower and upper bounds, take half of the rounding unit. Subtract it from the rounded value for the lower bound and add it for the upper bound.

$$Lower\; bound=rounded\; value-\frac{rounding\; unit}{2}$$ $$Upper\; bound=rounded\; value+\frac{rounding\; unit}{2}$$

This assumes the value was rounded normally rather than truncated.

Writing an error interval

An error interval shows every possible original value. If the lower bound is $L$, the upper bound is $U$, and the original value is $x$, write:

$$L\le x<U$$

The lower bound is included, but the upper bound is not. This is because a value exactly equal to the upper bound would round up to the next number.

Worked example

A length is $7.4cm$, correct to the nearest $0.1cm$.

Half of the rounding unit is:

$$0.1÷2=0.05$$

Therefore:

$$Lower\; bound=7.4-0.05=7.35$$ $$Upper\; bound=7.4+0.05=7.45$$

The error interval is:

$$7.35\le x<7.45$$

So the lower bound is $7.35cm$ and the upper bound is $7.45cm$.

Quick second example

A mass is $320g$, rounded to the nearest $10g$. The rounding unit is $10g$, so half is $5g$. The possible values run from $315g$ up to, but not including, $325g$:

$$315\le x<325$$

Notice that the displayed value does not have to be a decimal. The same half-unit method works for whole numbers, decimal places and measurements.

Common exam mistake

Do not write $x\le 7.45$. The upper bound is not included, so you need the strict inequality $x<7.45$.

Also check the rounding unit carefully. For example, rounding to the nearest $10$ gives a half-unit of $5$, not $0.5$.

Completeness check: This covers finding both bounds, writing the error interval with correct inequality notation, and interpreting which endpoints are included.