Box plots, explained with examples
A box plot (box-and-whisker diagram) summarises a set of data with just five numbers. It shows the middle, the spread and any unusual values at a glance, and it makes comparing two groups quick and clear.
On this page
The five-number summary
A box plot is built from five values:
- Minimum: the smallest value (excluding outliers, if they are marked separately).
- Lower quartile, Q1: a quarter of the way through the ordered data.
- Median, Q2: the middle value.
- Upper quartile, Q3: three quarters of the way through.
- Maximum: the largest value (again excluding marked outliers).
The box runs from Q1 to Q3, with a line at the median. The whiskers stretch out to the minimum and maximum. The middle half of the data lies inside the box.
Measures of spread
The range is maximum minus minimum. The interquartile range (IQR) is Q3 − Q1, the spread of the middle 50% of the data. The IQR is usually the better measure, because a single extreme value can make the range very large while barely changing the IQR.
Finding quartiles from a list
Put the data in order first. For n values, a common GCSE method uses positions: the median is at position (n + 1)/2, Q1 at (n + 1)/4 and Q3 at 3(n + 1)/4. If a position falls between two values, take the value halfway between them. Different textbooks use slightly different conventions; follow the one your teacher uses, and show your positions so the method is clear.
Example 1: from raw data
Find the five-number summary of 3, 7, 8, 12, 13, 14, 18, 21, 23, 27, 30.
- n = 11, already in order. Median at position 6: 14.
- Q1 at position 3: 8. Q3 at position 9: 23.
- Minimum 3, maximum 30. IQR = 23 − 8 = 15.
Drawing a box plot
- Draw a scale, evenly spaced, that covers the whole range of the data.
- Mark Q1 and Q3 and draw the box between them.
- Draw a vertical line inside the box at the median.
- Draw whiskers from the box out to the minimum and maximum.
- If outliers are marked, plot them as separate crosses or circles.
Outliers
An outlier is a value that is unusually far from the rest. A common rule is that a value is an outlier if it is more than 1.5 × IQR below Q1 or above Q3. Always use the rule a question gives you.
Example 2: checking for outliers
In the diagram, Q1 = 35, Q3 = 60, and the largest value is 100. Is 100 an outlier?
- IQR = 60 − 35 = 25. 1.5 × IQR = 37.5.
- Upper limit: 60 + 37.5 = 97.5. Since 100 > 97.5, 100 is an outlier, so the whisker stops at the largest value that is not an outlier, 88.
Example 3: reading a box plot
From the diagram, find the median, the IQR and the range of the non-outlier values.
- Median = 48. IQR = 60 − 35 = 25.
- Range of non-outlier values = 88 − 12 = 76.
Example 4: what fraction lies where?
Using the diagram, roughly what fraction of the values are between 35 and 88?
- Q1 to Q3 holds 50%, and Q3 to the upper whisker holds about 25%.
- About 75%, three quarters of the data.
What a box plot cannot show
A box plot hides detail. Two very different data sets can have identical box plots, for example one with values bunched at the quartiles and one spread evenly. It also does not show how many values there are, or whether the data has two peaks. When the shape matters, a histogram or dot plot shows more. Box plots are best for quick comparisons of centre and spread.
Comparing two box plots
A typical question shows two box plots, for example test scores for two classes, and asks you to compare them. A full answer makes two comparisons, each in context:
- Average: compare the medians. “Class A had a higher median score (62) than class B (54), so class A typically scored higher.”
- Spread: compare the IQRs (or ranges). “Class A had a smaller IQR (12) than class B (20), so class A's scores were more consistent.”
Avoid comparing only the maximum or minimum, and always say what the comparison means in context. A comparison without context, such as “A has a higher median”, often loses the mark.
Example 5: a full comparison
Box plots show reaction times. Group P: median 0.31 s, IQR 0.06 s. Group Q: median 0.27 s, IQR 0.11 s. Compare the groups.
- Group Q had a lower median reaction time, so Q typically reacted faster.
- Group Q had a larger IQR, so Q's reaction times were more varied (less consistent) than P's.
Box plots and skew
The shape of a box plot shows symmetry. If the median is in the middle of the box and the whiskers are similar lengths, the data are roughly symmetric. If the median is closer to Q1, with a longer upper whisker, the data are positively skewed. If it is closer to Q3, with a longer lower whisker, they are negatively skewed. The diagram above is roughly symmetric in its box, with a slightly longer upper whisker.
Common mistakes
- Forgetting to order the data before finding quartiles.
- Comparing data without context, or making only one comparison.
- Using the range instead of the IQR when outliers are present.
- Drawing the whisker to an outlier instead of to the largest non-outlier value.
Practice questions
Try each one before opening the answer.
Q1. Q1 = 14 and Q3 = 26. Find the IQR.
Show answer
Q2. Using the previous answer, what is the upper limit for outliers with the 1.5 × IQR rule?
Show answer
Q3. Find the median of 4, 9, 11, 15, 16, 22, 30.
Show answer
Q4. Two box plots have the same median but different IQRs. What does that tell you?
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Frequently asked questions
Are box plots on the Foundation paper?
Box plots are mainly a Higher tier GCSE topic, often combined with cumulative frequency. Foundation papers focus on the median and range.
Are box plots in A level Maths?
Yes. A level Maths statistics uses box plots and the 1.5 × IQR rule for outliers, and they appear in AP Statistics too.
Key takeaways
- A box plot shows minimum, Q1, median, Q3 and maximum.
- IQR = Q3 − Q1, the spread of the middle half.
- Outliers: more than 1.5 × IQR beyond the quartiles.
- Compare medians and IQRs, both in context.
- The median's position in the box shows skew.