What does the line inside the box represent?
Middle value.
The line shows the median.
Box Plots
Overview
A box plot is a compact way to show the distribution of a data set.
It is built from five important values: the minimum, lower quartile, median, upper quartile and maximum.
Five-number summary = minimum, lower quartile, median, upper quartile, maximum
Box plots are useful because they show both the centre and the spread of the data, and they make it easy to compare two groups.
What you should understand after this topic
- Understand what each part of a box plot represents
- Draw a box plot from the five-number summary
- Read median and quartiles from a box plot
- Calculate range and interquartile range
- Compare two box plots clearly
Key Definitions
Minimum
The smallest value in the data set.
Lower Quartile (\(Q_1\))
The value one quarter of the way through the data.
Median
The middle value in the data set.
Upper Quartile (\(Q_3\))
The value three quarters of the way through the data.
Maximum
The largest value in the data set.
Interquartile Range
The difference between upper quartile and lower quartile.
Key Rules
Use five values
A box plot is based on the five-number summary.
The box shows the middle 50%
The box goes from \(Q_1\) to \(Q_3\).
The line shows the median
The vertical line inside the box is the median.
The whiskers show the ends
The whiskers run out to the minimum and maximum.
Key Formulas
Range
\( \text{Range} = \text{Maximum} - \text{Minimum} \)
Interquartile Range
\( \text{IQR} = Q_3 - Q_1 \)
How to Solve
What is a box plot?
A box plot is a diagram that summarises a set of data using five important values. It gives a quick picture of the centre and spread of the data.
Box plots are often used after finding averages and spread, especially the median, range and interquartile range.
The five-number summary
To draw a box plot, you need the five-number summary:
Minimum, lower quartile, median, upper quartile, maximum
Exam tip: Label the number line carefully before plotting the five values.
Minimum
Start of the left whisker.
Lower Quartile
Left side of the box.
Median
Line inside the box.
Upper Quartile
Right side of the box.
Maximum
End of the right whisker.
How to draw the box plot
Draw the box plot in stages so each part is placed accurately.
Draw a number line with a suitable scale.
Mark the minimum and maximum values.
Mark the lower quartile and upper quartile, then draw a box between them.
Draw a line inside the box at the median.
Join the minimum to the box and the box to the maximum with whiskers.
Why this matters: A clear scale is essential because every part of the box plot is read from the number line.
How to read a box plot
You can read the five-number summary directly from the diagram. From that, you can also work out the range and interquartile range.
\( \text{Range} = 30 - 8 = 22 \)
\( \text{IQR} = 22 - 12 = 10 \)
The interquartile range is the spread of the middle 50% of the data.
Exam tip: Use the quartiles for IQR, not the minimum and maximum.
What the box plot tells you
A box plot helps you describe both the typical value and how spread out the data is.
Median
Shows the centre of the data.
Range
Shows the total spread of the data.
Interquartile Range
Shows the spread of the middle 50% of the data.
Skew / shape clues
Uneven box or whisker lengths can suggest the data is not spread evenly.
Comparing box plots
When comparing two box plots, mention both the median and the spread. This makes your comparison more complete.
Exam tip: A strong comparison usually needs two comments: one about average and one about spread.
Median
Which group has the higher typical value?
Spread
Which group has the bigger range or interquartile range?
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on reading values from box plots.
Edexcel
A box plot has lower quartile 14 and upper quartile 23.
Find the interquartile range.
Edexcel
A box plot has minimum value 5 and maximum value 29.
Find the range.
Edexcel
The box plot shows the distribution of some test scores.
Write down the median score.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on quartiles, range and spread.
AQA
A box plot has a lower quartile of 18 and an upper quartile of 32.
Find the interquartile range.
AQA
A box plot has a minimum value of 12 and a maximum value of 47.
Find the range.
AQA
Explain what the box represents in a box plot.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising comparison of two box plots.
OCR
The two box plots show the times, in seconds, taken by two groups to complete a puzzle.
Compare the distributions of the times for group A and group B.
OCR
Box plot A has median 18. Box plot B has median 24.
Which group has the higher typical value? Give a reason for your answer.
OCR
Box plot A has interquartile range 6. Box plot B has interquartile range 11.
Which group is more spread out in the middle 50%? Give a reason for your answer.
Exam Checklist
Step 1
Identify the five-number summary.
Step 2
Use a clear and accurate scale.
Step 3
Draw the box from \(Q_1\) to \(Q_3\) and the median inside it.
Step 4
When comparing plots, mention both median and spread.
Most common exam mistakes
Median confusion
Reading a quartile instead of the median.
IQR mistake
Using maximum and minimum instead of quartiles.
Weak comparison
Comparing only one feature when two are needed.
Scale error
Plotting values unevenly on the number line.
Common Mistakes
These are common mistakes students make when interpreting and drawing box plots in GCSE Maths.
Mixing up quartiles and the median
Incorrect
A student labels the median as a quartile or places quartiles incorrectly.
Correct
A box plot has five key values: minimum, lower quartile (Q1), median, upper quartile (Q3) and maximum. The median is the line inside the box, not one of the quartiles.
Using an incorrect scale
Incorrect
A student draws or reads the number line with uneven or incorrect intervals.
Correct
The number line must use a consistent scale. Incorrect scaling leads to inaccurate plotting and reading of values.
Not understanding the box
Incorrect
A student thinks the box represents all the data.
Correct
The box shows the middle 50% of the data, between Q1 and Q3. The lines (whiskers) show the spread of the remaining data.
Comparing only the range
Incorrect
A student decides which dataset is better based only on the range.
Correct
When comparing box plots, consider both the median (central value) and the spread (range or interquartile range). The median often gives more useful information.
Reading values inaccurately
Incorrect
A student estimates values incorrectly from the plot.
Correct
Read values carefully from the scale and align them precisely with the plotted points. Small errors can lead to incorrect answers.
Try It Yourself
Practise interpreting quartiles, medians and spreads using box plots.
Foundation
Foundation Practice
Identify key values from box plots.
A box plot has lower quartile 10 and upper quartile 30. Find the IQR.
Upper − lower.
30 − 10 = 20.
What does the box represent?
Between quartiles.
Box = Q1 to Q3.
Minimum = 5, maximum = 45. Find the range.
Max − min.
45 − 5 = 40.
What does the lower quartile represent?
Quarter.
Lower quartile = 25%.
Lower quartile = 8, upper quartile = 18. Find IQR.
Subtract.
18 − 8 = 10.
Which value is the median?
Middle.
Median is the middle value.
Minimum = 12, maximum = 60. Find range.
Subtract.
60 − 12 = 48.
What do the whiskers show?
Ends of data.
Whiskers show extremes.
Lower quartile = 20, upper quartile = 50. Find IQR.
Subtract.
50 − 20 = 30.
Higher
Higher Practice
Interpret and compare box plots using medians and spread.
Two box plots have the same median but different IQR. What does this show?
Think variation.
Different IQR = different spread.
Lower quartile = 15, upper quartile = 35. Find IQR.
Subtract.
35 − 15 = 20.
Which box plot is more consistent?
Less spread = consistent.
Smaller IQR = more consistent.
Minimum = 10, maximum = 70. Find range.
Subtract.
70 − 10 = 60.
If the median is closer to the lower quartile, what does this suggest?
Think symmetry.
Median position shows skew.
Lower quartile = 25, upper quartile = 55. Find IQR.
Subtract.
55 − 25 = 30.
A student compares only medians. What is missing?
Think variation.
Must consider spread.
Minimum = 5, maximum = 95. Find range.
Subtract.
95 − 5 = 90.
Which data set has greater variation?
Think spread.
Larger IQR = more variation.
Lower quartile = 40, upper quartile = 70. Find IQR.
Subtract.
70 − 40 = 30.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is a box plot?
A summary of data using quartiles.
What is the median?
The middle value.
Why use box plots?
To compare distributions.