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Compare medians and interquartile ranges on shared scales, distinguish IQR from full range and avoid unsupported claims about individuals, means or causes.
Before you startDrawing box plots, medians, IQR and range.
01 / Make two comparisons
Compare the medians, then a relevant spread measure such as IQR.
Quote values and units; explain what each comparison means.
A higher median is not automatically better: longer journeys and higher test scores have different interpretations. A narrower box means a smaller IQR, not necessarily a smaller full range.
Two illustrative journey-time distributions use minutes and the same axis. Whiskers show full observed extrema here.
A median = 20; B median = 26 minutes.
A IQR = 8; B IQR = 12 minutes.
These summaries do not determine every journey time or establish why the groups differ.
02 / Compare typical values
B has the higher median by six minutes
This compares the same location statistic.
A typical journey in B is longer in the median sense
Tie the statement to the quantity being measured.
This does not mean every B journey is longer than every A journey
The groups can overlap substantially.
Median waiting times are 12 minutes in A and 9 minutes in B. Write a comparison.
Give the direction, difference and variable.
B has the smaller median waiting time by 3 minutes, suggesting a shorter typical wait in the median sense.
Two groups both have median score 60. Does that show their distributions are identical?
One location statistic cannot describe all values.
No. Their quartiles, ranges, shapes, sample sizes and individual scores may differ.
03 / Compare the box widths
Pause, replay or seek freely. The notes explain the same idea and stay in view.
IQR = Q₃ − Q₁.
Use numerical differences along the measurement axis, not the decorative thickness of the box.
A has Q₁ = 16, Q₃ = 24; B has Q₁ = 20, Q₃ = 32. Compare their IQRs.
Subtract the quartiles within each group.
A IQR = 8; B IQR = 12. A has a smaller spread between quartiles by 4 units.
A has a smaller IQR of journey times. Can you simply call it more consistent without qualification?
Identify the particular measure of variation.
It is clearer to say A is less variable by IQR, or that the spread between its quartiles is smaller. This need not imply a smaller range or SD.
04 / Distinguish IQR from full range
A has extrema 0 and 50 with quartiles 18 and 22. B has extrema 10 and 30 with quartiles 15 and 25. Compare IQR and range.
Calculate both measures for both groups.
A: IQR 4, range 50. B: IQR 10, range 20. A has the smaller IQR but the larger full range, so specify which measure supports the comparison.
A diagram has whiskers at 5 and 30 and a separate outlier at 45, with no lower outlier. What is its full range?
The outlier remains an observed value.
45 − 5 = 40. The whisker span 25 is not the full range under this convention.
05 / Check comparability
Two separate plots have equally wide boxes, but one axis is in seconds and the other in minutes. Can you conclude equal IQRs?
Convert to common units and read each scale.
No. Compare numerical IQRs after unit conversion; identical drawn widths are not enough.
A box plot for 20 pupils and another for 200 pupils have identical five-number summaries. Must their boxes be different widths to indicate the counts?
A standard box plot need not encode sample size.
No. Standard boxes may have equal decorative width. Sample sizes should be stated separately unless a specific variable-width convention is explicitly explained.
06 / Do not invent information
Unequal whisker lengths may suggest asymmetry, but a box plot does not reveal every cluster or gap. Medians and quartiles cannot establish a particular density curve or an exact standard deviation.
Can a mean be calculated by averaging the five-number summary?
Those five values do not include every observation or frequency.
No. Their average is generally not the dataset mean. Many datasets can share a five-number summary but have different means.
If the upper quartile is 70, must exactly 25% of observations be strictly above 70?
Values can tie at the quartile.
No. Ordered-position summaries do not guarantee an exact strict count when ties occur; conventions and sample size also affect the statement.
07 / Avoid causal and individual claims
Two classes have different median test scores. Does that prove their teachers caused the difference?
Many other factors and selection differences may matter.
No. The plots describe the observed scores. A causal conclusion requires an appropriate study design and further evidence.
Group B has a higher median than A. Does that guarantee a randomly chosen B pupil scores higher than a randomly chosen A pupil?
The individual distributions may overlap.
No. A median comparison does not determine the outcome for any particular pair, nor does it provide that probability without additional distribution information.
08 / Write a supported comparison
Compare median and a named spread measure, check common scales and conventions, and describe what the summaries show. State limits when asked about means, exact counts, individual outcomes or causes.
Section 1 of 8 · Make two comparisons