Skip to content
VirtusAcademy

Quartiles

HigherHigher tier onlyAQAEdexcelOCR

Practise quartiles with this free Higher GCSE Maths worksheet from Virtus Academy. You'll work through finding quartiles and the interquartile range, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. The interquartile range is the upper quartile minus the lower quartile.

Free downloads

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.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Quartiles divide ordered data into four equal parts. The lower quartile has one quarter of the data below it, the median has half, and the upper quartile has three quarters.

The interquartile range is the upper quartile minus the lower quartile. It measures the spread of the middle half of the data, which makes it much less affected by extreme values than the range.

That resistance to outliers is why the interquartile range is often preferred. A single unusually large value changes the range dramatically but leaves the interquartile range almost untouched.

Revision notes

Finding the quartiles

Order the data, then find the median. The lower quartile is the median of the lower half, and the upper quartile the median of the upper half.

For \(n\) values, the lower quartile is at position \(\frac{n+1}{4}\) and the upper at \(\frac{3(n+1)}{4}\).

The interquartile range

Subtract the lower quartile from the upper.

If \(Q_1 = 12\) and \(Q_3 = 28\), the interquartile range is \(16\). A smaller value means the middle half of the data is more tightly clustered.

Why it beats the range

The range uses only the two most extreme values, so one outlier distorts it completely.

The interquartile range ignores the outer quarters entirely, so it describes the typical spread more reliably.

Key points

  • Quartiles divide ordered data into four parts.
  • The lower quartile has a quarter of the data below it.
  • The upper quartile has three quarters below it.
  • Interquartile range is \(Q_3 - Q_1\).
  • It measures the spread of the middle half.
  • It is less affected by outliers than the range.

Worked examples

Example 1

For \(11\) values, find the position of the lower quartile.

Working

\[\frac{11+1}{4}\]use the position formula
\[= 3\]the lower quartile is the 3rd value

Example 2

\(Q_1 = 15\) and \(Q_3 = 37\). Find the interquartile range.

Working

\[37 - 15\]subtract the lower quartile from the upper
\[= 22\]state the interquartile range

Example 3

Explain why the interquartile range is often preferred to the range.

Working

\[\text{The range uses only the extremes}\]one outlier distorts it
\[\text{The IQR uses the middle half}\]so it is less affected by outliers

Common mistakes

  • Forgetting to order the data.

    Quartiles only make sense once the values are in order.

  • Adding the quartiles instead of subtracting.

    The interquartile range is Q₃ minus Q₁.

  • Confusing the interquartile range with the range.

    The range uses the extremes; the IQR uses the quartiles.

  • Using \(\frac{n}{4}\) for the position.

    The formula adds one before dividing.

Exam tips

  • Order the data before doing anything else.
  • Find the median first, then the quartiles of each half.
  • Subtract to find the interquartile range.
  • Mention outlier resistance when comparing measures of spread.

Key terms

Lower quartile
The value with a quarter of the data below it.
Upper quartile
The value with three quarters of the data below it.
Interquartile range
The difference between the upper and lower quartiles.
Outlier
A value far from the rest of the data.

Related topics

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