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Box Plots and Quartiles Revision Notes | GCSE AQA Higher Maths

The interquartile range (IQR) = Q3 - Q1, and it measures the spread of the middle 50% of the data. Free GCSE AQA Maths notes and 12 questions answered.

Updated 21 Mar 2026 · Revise Right Team

Box Plots and Quartiles is part of Averages and Spread in GCSE AQA Higher Maths.

4 min

Reading time

11

Lessons in this topic

GCSE AQA Higher

Pathway

Reviewed by Revise Right

Content review

What this lesson covers

The interquartile range (IQR) = Q3 - Q1, and it measures the spread of the middle 50% of the data. Free GCSE AQA Maths notes and 12 questions answered.

  • Identify and calculate the lower quartile, upper quartile, and interquartile range from a data set
  • Construct a box plot from a five-number summary
  • Read and interpret key values from a box plot
  • Compare two distributions using box plots and interquartile range

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Key ideas to keep in view

Use this lesson to stay inside Averages and Spread while connecting the detail back to the wider Statistics topic.

  • Quartiles: Values that split an ordered data set into four equal parts. The lower quartile (Q1) is the median of the lower half; the upper quartile (Q3) is the median of the upper half.
  • Interquartile Range (IQR): A measure of spread equal to Q3 minus Q1. It represents the range of the middle 50% of the data and is less affected by extreme values than the full range.
  • Box Plot: A statistical diagram that displays the five-number summary (minimum, Q1, median, Q3, maximum) on a number line, showing spread and skew at a glance.
  • Five-Number Summary: The five key values needed to draw a box plot: minimum, lower quartile (Q1), median (Q2), upper quartile (Q3), and maximum.

Revision notes

Read the core explanations from Box Plots and Quartiles before testing yourself from memory.

1. Why Spread Matters as Much as the Average

Imagine two classes both score an average of 60% on a test. In Class A, everyone scores between 55% and 65%. In Class B, scores range from 10% to 100%. The averages are identical — but the stories are completely different. That gap is what spread measures, and box plots are the tool that makes spread visible at a glance.

Box plots are built from quartiles — values that divide an ordered data set into four equal-sized groups. Before drawing a single box plot, you need to be confident finding those quartiles. That is where this lesson starts.

2. Quartiles and the Interquartile Range

The median (Q2) splits the data in half. Q1 then splits the lower half, and Q3 splits the upper half. Together they carve the data into four equal portions. The interquartile range (IQR) is the distance between Q1 and Q3, capturing the spread of the middle 50% of values.

Finding quartiles requires a clear method. The steps below show the expected approach for a small data set listed in order.

Worked example: Finding Q1, Median, Q3, and IQR from a Data Set

The times (in minutes) taken by 9 students to complete a puzzle are listed below. Find the median, lower quartile, upper quartile, and interquartile range. 8, 12, 15, 17, 20, 23, 25, 28, 31

  1. 8, 12, 15, 17, 20, 23, 25, 28, 31 — The data is already ordered from smallest to largest. Always order data first — this is where marks are lost if skipped. [M1]
  2. 9 values → median is the $\frac{9+1}{2} = 5$th value Median = 20 — For $n$ values, the median position is $\frac{n+1}{2}$. Here $\frac{10}{2} = 5$, so count to the 5th value. [A1]
  3. Lower half (values below the median): 8, 12, 15, 17 — Exclude the median value itself. The lower half contains the 4 values below position 5.
  4. 4 values → median position = $\frac{4+1}{2} = 2.5$th $Q_1 = \frac{12 + 15}{2} = \frac{27}{2} = 13.5$ — Position 2.5 means average the 2nd and 3rd values of the lower half. [A1]
  5. Upper half (values above the median): 23, 25, 28, 31 $Q_3 = \frac{25 + 28}{2} = \frac{53}{2} = 26.5$ — Same process on the upper half — average the 2nd and 3rd values of that group. [A1]
  6. $\text{IQR} = Q_3 - Q_1 = 26.5 - 13.5 = 13$ — Subtract Q1 from Q3. This tells us the middle 50% of puzzle times span 13 minutes. [A1]

Answer: Median = 20 min, Q1 = 13.5 min, Q3 = 26.5 min, IQR = 13 min

3. Building a Box Plot from the Five-Number Summary

Now that you can find quartiles, you have everything needed to draw a box plot. A box plot requires exactly five values: the minimum, Q1, median, Q3, and maximum. These are called the five-number summary.

The diagram below shows what each part of a box plot represents. The rectangular box spans from Q1 to Q3, with a vertical line marking the median inside it. Horizontal lines called whiskers extend from the box out to the minimum and maximum values.

Reading from the diagram: the box covers Q1 to Q3, so its width equals the IQR (13 minutes here). The left whisker runs from 8 to 13.5, and the right whisker runs from 26.5 to 31. The median line at 20 sits exactly in the centre of the box, since it is 6.5 units from both Q1 and Q3. That suggests the middle 50% of the data is fairly symmetric, even though the whiskers can still show different behaviour at the extremes.

Worked example: Constructing a Box Plot

Draw a box plot for the following data about the heights (cm) of 11 plants: 14, 18, 21, 24, 27, 30, 33, 37, 40, 44, 50 The five-number summary has already been calculated: Min = 14, Q1 = 21, Median = 30, Q3 = 40, Max = 50.

  1. Draw a horizontal axis labelled 'Height (cm)' with a scale covering at least 14 to 50. — The scale must be even and cover the full range from minimum to maximum. Label the axis with units. [M1]
  2. Mark vertical lines at: 14 (min), 21 (Q1), 30 (median), 40 (Q3), 50 (max) — Use a ruler. Each mark must align precisely with the scale — accuracy earns the mark.
  3. Draw a rectangle from Q1 = 21 to Q3 = 40, with a vertical line inside at the median = 30. — The box represents the interquartile range. The median line divides it into two sections. [A1]
  4. Draw a horizontal line from 14 to the left edge of the box (Q1 = 21). Draw a horizontal line from the right edge of the box (Q3 = 40) to 50. — Whiskers connect the box to the minimum and maximum. They show the full range of the data. [A1]
  5. Confirm: box width = $40 - 21 = 19$ cm (IQR). Total range = $50 - 14 = 36$ cm. — A quick sanity check: the box width should equal the IQR you calculated. If it does not, recheck your scale.

Answer: Box plot drawn with whiskers at 14 and 50, box from 21 to 40, median line at 30.

4. Reading and Interpreting Box Plots

Drawing a box plot is one skill; reading one is another. In exam questions, you are often given a completed box plot and asked to extract values or describe what the diagram shows.

Every part of the box plot maps directly to one of the five summary values. The left end of the left whisker is the minimum; the right end of the right whisker is the maximum. The left and right edges of the box are Q1 and Q3. The line inside the box is the median.

5. Comparing Two Distributions Using Box Plots

One of the most common Higher-tier exam tasks is to compare two box plots drawn on the same scale. The examiner wants two things: a comparison of average (using the median) and a comparison of spread (using the IQR or range). You must address both to earn full marks.

Always write your comparison in context — refer to what the data actually represents, not just the numbers. A response that says 'the median is higher' earns fewer marks than one that says 'students in Group A typically scored higher, with a median of 72 compared to 58 in Group B'.

Worked example: Comparing Two Box Plots

Using the two box plots above for Group A and Group B, write a comparison of the two distributions. [4 marks]

  1. Group A median = 72% Group B median = 58% — The median is the measure of average for box plots. Group A has a higher median, so students in Group A typically scored higher. [1 mark for correct values, 1 mark for comparison in context]
  2. Group A: $\text{IQR} = 82 - 60 = 22$ Group B: $\text{IQR} = 70 - 48 = 22$ — Both groups have the same IQR of 22%, so the spread of the middle 50% is identical. This is a useful observation — same spread, different average.
  3. Group A: Range $= 95 - 45 = 50$ Group B: Range $= 88 - 30 = 58$ — Group B has a wider overall range (58 vs 50), meaning its scores are more spread out from the extremes. [1 mark for comparison in context]
  4. Group A students typically scored higher (median 72% vs 58%). Both groups had the same IQR of 22%, but Group B had a wider overall range (58 vs 50), suggesting more variability at the extremes. — A complete comparison addresses average AND spread with context. This is what earns all 4 marks. [A1 for each valid comparative statement in context]

Answer: Group A has a higher median (72% vs 58%). Both groups have the same IQR (22%), but Group B has a wider range (58 vs 50), indicating greater variability overall.

6. Deliberate Practice

The following problems cover the full range of skills from this lesson. Work through each one before checking the solution — this is where the methods become automatic.

7. Exam Technique and Common Mistakes

Box plot questions appear on both questions without a calculator and calculator-allowed exam questions. The arithmetic involved — finding medians and subtracting values — is always manageable without a calculator, so accuracy and method clarity are what separate full marks from partial credit.

Mastering box plots and quartiles means you can both construct a diagram from raw data and extract meaning from one you are given — two distinct skills that exam questions test separately. The IQR is your go-to measure of spread in any statistics question involving box plots, and the five-number summary is the complete toolkit for drawing them accurately.

Frequently asked questions

What is the interquartile range?

The interquartile range (IQR) = Q3 - Q1, and it measures the spread of the middle 50% of the data. With Q1 = 13.5 and Q3 = 26.5, the IQR is 26.5 - 13.5 = 13.

How do you draw a box plot?

Draw a scaled number line and mark the five-number summary: minimum, Q1, median, Q3 and maximum. Draw a box from Q1 to Q3 with a line at the median, then add whiskers out to the minimum and maximum; the box width equals the IQR.

What is the five-number summary?

The five-number summary is the minimum, lower quartile (Q1), median, upper quartile (Q3) and maximum. For plant heights it might be 14, 21, 30, 40 and 50 cm, which are exactly the values needed to draw a box plot.

How do you find the quartiles of a data set?

Order the data, find the median, then find the median of the lower half for Q1 and of the upper half for Q3. For 8, 12, 15, 17, 20, 23, 25, 28, 31, the median is 20, Q1 = (12 + 15) ÷ 2 = 13.5 and Q3 = (25 + 28) ÷ 2 = 26.5.

How do you compare two box plots?

Compare the medians for the average and the IQRs or ranges for the spread, in context. Group A has a median of 72% and Group B 58%, both have an IQR of 22%, but Group B has a wider range of 58 compared with 50.

What is the lower quartile?

The lower quartile, Q1, is the median of the lower half of an ordered data set, the values less than the overall median. One quarter of the data lies at or under it.

What is the upper quartile?

The upper quartile, Q3, is the median of the upper half of an ordered data set, the values more than the overall median. Three quarters of the data lies under it.

How do you find the position of the median?

The median position for n ordered values is (n + 1) ÷ 2. For 9 values, (9 + 1) ÷ 2 = 5, so the median is the 5th value.

How do you find the quartiles of 9 values?

For 9 ordered values such as 8, 12, 15, 17, 20, 23, 25, 28, 31, the median is the 5th value, 20. The lower half 8, 12, 15, 17 gives Q1 = (12 + 15) ÷ 2 = 13.5, and the upper half gives Q3 = (25 + 28) ÷ 2 = 26.5.

What do the whiskers on a box plot show?

The whiskers on a box plot extend from the box to the minimum and maximum values, showing the full range of the data. The left whisker ends at the minimum and the right whisker ends at the maximum.

What does the box in a box plot show?

The box in a box plot spans from the lower quartile to the upper quartile, so its width is the interquartile range, the spread of the middle 50% of the data. The line inside the box marks the median.

What does it mean if two box plots have the same IQR but different medians?

Two box plots with the same IQR but different medians have the same spread in the middle 50% but a different typical value. For example, medians of 72% and 58% with both IQRs 22% means the first group typically scored higher with the same spread.

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