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Error Intervals

FoundationHigherAQAEdexcelOCR

Get to grips with error intervals using these Foundation and Higher GCSE Maths practice questions. The worksheet focuses on writing error intervals, and the accompanying mark scheme breaks down each solution clearly. Suitable for AQA, Edexcel and OCR. Download the questions and answers for free. An error interval uses ≤ for the lower bound and < for the upper bound.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

An error interval expresses the range within which a rounded or truncated value must lie. It is written as an inequality using the lower and upper bounds.

For a value rounded to the nearest whole number, the interval extends half a unit either side. So \(7\) to the nearest whole number gives \(6.5 \le x < 7.5\).

Truncation behaves differently. A truncated value has simply had digits removed, so \(7\) truncated means the true value is at least \(7\) but less than \(8\), giving \(7 \le x < 8\). Reading whether a value was rounded or truncated is essential.

Revision notes

Error intervals from rounding

Add and subtract half the rounding unit.

For \(2.4\) to \(1\) decimal place, half of \(0.1\) is \(0.05\), giving \(2.35 \le x < 2.45\).

Error intervals from truncation

Truncation removes digits without rounding, so the value can only be larger than what is shown.

A truncated \(2.4\) gives \(2.4 \le x < 2.5\), because anything from \(2.4\) up to just below \(2.5\) truncates to \(2.4\).

Writing the interval

Use \(\le\) for the lower bound and \(<\) for the upper.

The lower bound is achievable but the upper is not, since a value exactly at the upper bound would round or truncate differently.

Key points

  • An error interval gives the range of possible true values.
  • Rounding extends half a unit either side.
  • Truncation extends upwards only.
  • Use \(\le\) for the lower bound.
  • Use \(<\) for the upper bound.
  • Check whether the value was rounded or truncated.

Worked examples

Example 1

A length is \(15\)cm to the nearest cm. Write the error interval.

Working

\[1 \div 2 = 0.5\]halve the rounding unit
\[14.5 \text{ and } 15.5\]find the bounds
\[14.5 \le x < 15.5\]write the interval

Example 2

A value is \(3.7\) truncated to 1 decimal place. Write the error interval.

Working

\[\text{Truncation removes digits}\]the true value is at least 3.7
\[\text{Anything under 3.8 truncates to 3.7}\]find the upper bound
\[3.7 \le x < 3.8\]write the interval

Example 3

A mass is \(8.2\)kg to 1 decimal place. Write the error interval.

Working

\[0.1 \div 2 = 0.05\]halve the rounding unit
\[8.15 \text{ and } 8.25\]find the bounds
\[8.15 \le x < 8.25\]write the interval

Common mistakes

  • Treating truncation like rounding.

    Truncation extends upwards only, so 3.7 truncated gives 3.7 ≤ x < 3.8.

  • Using ≤ for the upper bound.

    The upper bound is not attainable, so the sign is <.

  • Using the whole rounding unit.

    The interval extends half a unit either side, not a full unit.

  • Rounding the bounds themselves.

    The bounds are exact and should be written in full.

Exam tips

  • Check whether the question says rounded or truncated.
  • Identify the rounding unit before calculating.
  • Use ≤ for the lower and < for the upper bound.
  • Write the interval as a single inequality.

Key terms

Error interval
The range of possible values for a rounded measurement.
Truncation
Removing digits without rounding.
Upper bound
The largest value the measurement could be.
Lower bound
The smallest value the measurement could be.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.