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Quartiles and Percentiles

FoundationHigherAQAEdexcel

Practise Quartiles and Percentiles for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. Quartiles and percentiles divide ordered data into equal-sized parts to describe its spread.

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

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Quartiles divide ordered data into four equal parts, and percentiles divide it into a hundred.

The lower quartile is at position \(\frac{n+1}{4}\), the median at \(\frac{n+1}{2}\), and the upper quartile at \(\frac{3(n+1)}{4}\), where \(n\) is the number of values. The data must be in order before any position is used.

The \(k\)th percentile is at position \(\frac{k(n+1)}{100}\). So the 25th percentile is the lower quartile and the 90th percentile is the value below which 90 per cent of the data lies. If a position is not a whole number, take the value partway between the two neighbouring values.

Revision notes

Quartile positions

Lower quartile \(Q_1\) at position \(\frac{n+1}{4}\).

Median \(Q_2\) at position \(\frac{n+1}{2}\). Upper quartile \(Q_3\) at position \(\frac{3(n+1)}{4}\). Always order the data first.

Percentiles

The \(k\)th percentile is at position \(\frac{k(n+1)}{100}\).

The 25th percentile is the lower quartile, the 50th is the median, and the 75th is the upper quartile. The 90th percentile is the value below which 90 per cent of the data lies.

Non-whole positions

If a position works out as, say, 4.5, take the value halfway between the 4th and 5th values.

For position 4.25, take a quarter of the way from the 4th value towards the 5th. Interquartile range and interpercentile range are both found by subtraction.

Key points

  • Quartiles divide data into four equal parts.
  • Lower quartile position is (n+1) ÷ 4.
  • Upper quartile position is 3(n+1) ÷ 4.
  • Percentiles divide data into a hundred parts.
  • The kth percentile is at k(n+1) ÷ 100.
  • Order the data before finding any position.

Worked examples

Example 1

There are 11 values in order. Work out the position of the lower quartile. [2 marks]

Working

\(\frac{n+1}{4} = \frac{11+1}{4}\)substitute into the position formula
\(= 3\), so the lower quartile is the 3rd valuework out the position

Example 2

There are 19 values. Work out the position of the upper quartile. [2 marks]

Working

\(\frac{3(n+1)}{4} = \frac{3 \times 20}{4}\)substitute into the position formula
\(= 15\), so the upper quartile is the 15th valuework out the position

Example 3

A quartile position works out as 6.5. Explain how to find its value. [2 marks]

Working

The position lies between the 6th and 7th valuesstate where the position falls
so take the value halfway between them, which is the mean of those two valuesstate how to find it

Common mistakes

  • Forgetting to order the data first.

    All quartile positions assume the data is in ascending order.

  • Using n instead of n+1.

    The position formulae use n+1 for a list of individual values.

  • Giving the position as the answer.

    The position tells you which value to read; the value is the answer.

  • Rounding a non-whole position.

    Take the value partway between the two neighbours instead.

Exam tips

  • Order the data before doing anything else.
  • Use n+1 in the position formulae.
  • Read the value at that position, not the position itself.
  • Interpolate for non-whole positions.

Key terms

Quartile
A value dividing ordered data into quarters.
Percentile
A value dividing ordered data into hundredths.
Position
Where in the ordered list a value sits.
Interpolate
To find a value between two neighbouring values.

Related topics

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