Did you know? Two box plots can look almost identical even when one represents seven pupils and the other eleven. A box plot is a picture of selected positions in a distribution, not a tally of people. When a learner decides that equally wide boxes must mean equal group sizes, the learner has mistaken a visual convention for evidence.
To compare box plots with unequal group sizes, read the medians, lower and upper quartiles, interquartile ranges and plotted extremes on a common scale, while checking each group’s sample size separately. Compare what is shown, then state what is hidden: individual scores, exact frequencies and explanations for any differences. A box plot does not prove which group was taught better or whether every pupil improved.
Below is an original, entirely fictional quiz example that parents, pupils and teachers can reproduce by hand. The numbers are not actual school results and make no claim about an examination cohort. The learning goal is to read an unfamiliar graph responsibly rather than memorise one set of quartile positions.
What a box plot tells us—and what it leaves out
The box extends from the lower quartile Q1 to the upper quartile Q3. Their difference, Q3 − Q1, is the interquartile range or IQR, a measure of the spread of the central part of the ordered data. A line inside the box marks the median. Depending on the plotting convention, whiskers may extend to minimum and maximum values or to the most extreme observations inside specified outlier fences.
The US National Institute of Standards and Technology’s box-plot guide explains comparison of location and variation and notes a subtle issue: plot widths may be arbitrary or proportional to sample size. If the caption does not define a variable-width convention, do not read the number of pupils from how wide the box looks. Write “n = 7” and “n = 11” on the figure instead.
Begin with a diagnosis, not a formula
Ask a student to evaluate three statements: “Both boxes are equally wide, therefore both groups have the same number of students”; “Group B’s median is higher, therefore every child in B scored higher”; and “A longer whisker always identifies a marking error.” All three are unsupported without further information. The specific gap is one of interpretation: the student has not yet separated a numerical summary from raw-data counts, paired comparisons or causal explanations.
A useful follow-up is: “If I hide the group-size labels, what can you still know?” The median and IQR may remain readable; the number of pupils does not. We can use this loss-of-information test every time a chart reduces a long dataset to a simple shape.
Worked example: two unequal groups taking the same fictional quiz
Imagine a 15-mark quiz completed under comparable conditions. Group A has 7 pupils with ordered scores 6, 7, 8, 8, 9, 10, 12 marks. Group B has 11 pupils with ordered scores 5, 7, 7, 8, 8, 9, 9, 10, 10, 11, 14 marks. Every value is invented solely to demonstrate the calculation.
We must state the quartile convention. Here, with an odd number of observations, we use the median of each half, excluding the central median. Some software uses another percentile method, especially for small samples. A pupil should follow the convention specified by the teacher, examination question or software and should not silently switch methods partway through the comparison.
Calculate Group A without skipping a step
- Seven observations mean the median is the fourth value: 8 marks.
- Below the median: 6, 7, 8. Their median gives Q1 = 7 marks.
- Above the median: 9, 10, 12. Their median gives Q3 = 10 marks.
- IQR = Q3 − Q1 = 10 − 7 = 3 marks.
- Minimum = 6, maximum = 12, range = 12 − 6 = 6 marks.
Check that Q1 ≤ median ≤ Q3: 7 ≤ 8 ≤ 10. The box therefore runs from 7 to 10, with its median line at 8. Under a minimum-to-maximum whisker convention, the whiskers reach 6 and 12. Anyone can independently reconstruct that plot from the list.
Calculate Group B using Group B’s own positions
- Eleven observations mean the median is the sixth value: 9 marks.
- Five values below that middle observation: 5, 7, 7, 8, 8. Their median gives Q1 = 7 marks.
- Five values above it: 9, 10, 10, 11, 14. Their median gives Q3 = 10 marks.
- IQR = 10 − 7 = 3 marks, equal to Group A’s IQR.
- Minimum = 5, maximum = 14, range = 14 − 5 = 9 marks.
The calculations show a striking combination: both boxes extend from 7 to 10 marks, although their medians differ by one mark and their extreme values differ. The unequal group sizes did not stop the quartile intervals from matching. That is why a similar-looking box cannot certify equal group sizes or identical datasets.
Write the comparison in language that survives checking
A correct statement is: “In these fictional samples, Group B has a higher median (9 versus 8 marks); both have an IQR of 3 marks; the total observed range is larger in Group B (9 versus 6 marks).” That compares centre and spread using the same units. It does not turn these small groups into representatives of entire schools or claim that one method of teaching caused their scores.
The absolute sample sizes still deserve mention. Group A contains seven observations and Group B eleven. Small samples can yield unstable summaries, and different selection procedures or quizzes would weaken comparability further. A descriptive difference is a legitimate observation, but a claim about a population or a treatment effect needs an appropriate study design, not just two rectangles.
The count trap: how many pupils achieved at least ten marks?
In Group A, exactly two marks meet or exceed 10: 10 and 12. The proportion is 2 ÷ 7 ≈ 28.6%. In Group B, the qualifying marks are 10, 10, 11 and 14: four out of eleven, or 4 ÷ 11 ≈ 36.4%. The calculation uses the original score lists. An ordinary box plot does not reliably provide this exact threshold count; it contains selected quantiles, not a full frequency table.
Even if two graphs share Q1 and Q3, their internal piles of repeated scores may differ. A box can conceal ties, clusters, gaps and distinct shapes. If a question asks for exact numbers above a threshold, seek the raw values, a count table or a suitable cumulative frequency representation. Do not invent frequencies by visually splitting the box into four equal counts of children.
Why box widths and whiskers need captions
A standard classroom box plot often uses the same visual width for all groups. A variable-width design can scale width with sample size, but that must be explained. With whiskers, one convention goes directly to the minimum and maximum; another stops at the farthest points within fences based on 1.5 times the IQR. The NIST outlier reference describes these fences and cautions that surprising observations require investigation, not automatic deletion.
For both invented groups, Q1 = 7, Q3 = 10 and IQR = 3. The illustrative inner fences are 7 − (1.5 × 3) = 2.5 and 10 + (1.5 × 3) = 14.5 marks. Every listed score, including B’s 14, is inside the fences. The result agrees with the raw values: no listed score is flagged by this particular rule. A teacher should still explain the plotting convention rather than assuming every software package uses the same one.
Mean and median are not interchangeable
A child might infer that Group B’s higher median guarantees a higher mean. That is not a universal rule. Here, Group A’s scores sum to 60 and its mean is 60 ÷ 7 ≈ 8.57 marks. Group B’s scores sum to 98 and its mean is 98 ÷ 11 ≈ 8.91 marks. The means happen to have the same direction as the medians in this particular example, but we had to calculate them. Most box plots do not display a mean unless an extra marker is defined.
A median identifies the centre by ordered position, while a mean uses the magnitude of every value. A very high or low observation can affect the mean strongly. When a pupil chooses a summary, the correct question is “What decision or comparison is this measure meant to support?” rather than “Which statistic makes my group look stronger?”
A five-stage method pupils can reuse in any statistics task
- Identify the variable, units, population described and the common axis scale.
- Find the stated sample sizes; when absent, flag the missing information rather than guessing.
- Calculate or read Q1, median, Q3 and IQR using the correct convention.
- Check whether whiskers show full extremes or values inside outlier fences, and read any separate dots.
- Write one supported comparison of centre, one of spread, and one explicit limitation.
Progressive practice: guided, paired, independent
Guided. Draw the two boxes from the computed five-number summaries on one 0–15 mark axis. Label the groups n = 7 and n = 11. Ask a classmate to verify every endpoint against the ordered score lists. This is the move from arithmetic to a trustworthy representation.
Paired. Student A writes, “Every pupil in B scored more because its median is higher.” Student B must rewrite it as a defensible descriptive comparison and explain why an individual claim cannot follow. Swap roles for, “The boxes have equal IQRs, so the groups have identical scores.” The goal is to identify the exact inferential jump, not merely to say “wrong”.
Independent. Give the learner a new pair of box plots, one for 9 children and one for 30, with a visibly different median and IQR. Ask for an interpretation without a sentence starter. Then hide the n labels and ask the learner to identify which facts are no longer known. If they can state what the picture omits without prompting, they have transferred the method.
What adults should look for when checking progress
Instead of marking only the final median, check four observable skills: accurate quartile construction, attention to sample size and plotting rules, defensible numerical comparisons, and honest restraint about unobserved individuals or causes. A pupil may calculate well yet still overclaim. That is a reasoning gap, not a multiplication error. Parents can ask, “What facts are invisible in this graph?” and see whether the child brings up individual values, counts and context.
A helpful follow-up is Why a Higher Class Average Does Not Prove Every Student Improved. For another graph where shape can mislead, read How to Read Unequal-Width Histograms. For more advanced subject practice, see Secondary 4 G3 Mathematics Tutor Bukit Timah: Box Plots and IQR. Each link addresses a distinct part of data interpretation.
Limits of the model
A box plot is a compact summary, not an audit trail of the sample design. It cannot tell whether marking was consistent, whether any pupils were absent, whether the quiz covered the intended skill or whether the difference is meaningful beyond those observed. Both groups might have used different revisions or started from different prior levels. Label hypothetical data as hypothetical, and use real conclusions only when the supporting evidence exists.
Frequently asked questions
Can I compare box plots when the group sizes differ?
Yes, for careful descriptive comparisons on a common variable and scale. State the group sizes and limitations. Do not infer that the difference is statistically significant simply because the medians differ.
Why does a calculator give slightly different quartiles?
Different software and teaching materials may use different percentile conventions, especially for small samples. This article explicitly excludes the central value when finding quartiles for an odd sample. Follow the prescribed convention and disclose it.
Are all the scores within a box exactly the same fraction of pupils?
The box runs between approximate percentile cut-points of the ordered data. Ties, small samples and quartile rules mean it should not be read as a literal count of a precise number of pupils occupying every painted quarter.
Does a higher median prove that one teaching method works better?
No. It describes the observed centre. Causal claims need stronger comparative evidence, including how participants were selected and whether other differences could explain the pattern.
The wider educational aim
The useful skill is knowing how far the evidence reaches. A pupil who can calculate, question, interpret and check independently can take that habit into surveys, science, economics and public claims. Continue through the eduKate Education Hub and the Core Aim learning hub. The progression from diagnosis to independent verification is also modelled in the immutable Clementi Mathematics teaching reference. The conclusion is simple: a good graph answer says what the figure shows and what it cannot show.