Constructing an ordinary boxplot
Boxplots summarise centre and spread; they do not display every observation
\[\mathrm{IQR}=Q_3-Q_1\]
- State the convention Use medians of halves, retaining repeats. For odd n omit only the single middle observation
- Use one linear scale Place the five features at their numerical positions. Equal distances on the axis represent equal numerical differences
- Keep the median distinct The median line need not be centred geometrically inside the box
Worked example
Times in minutes are 4, 6, 8, 10, 14, 18, 20, 24. Find the five-number summary
\[\begin{gathered}4,6,8,10\\14,18,20,24\end{gathered}\]
- \(\displaystyle \mathrm{median}=(10+14)/2=12\)Average the two central observations
- \(\displaystyle Q_1=(6+8)/2=7\)Median of the lower half
- \(\displaystyle Q_3=(18+20)/2=19\)Median of the upper half
- \(\displaystyle (4,7,12,19,24)\)Minimum, Q1, median, Q3, maximum. Draw the box from 7 to 19, median at 12 and whiskers to 4 and 24.
Common mistakes
Using extremes as box edges
✕\(\displaystyle \mathrm{box}:4\text{ to }24\)✓\(\displaystyle \mathrm{box}:7\text{ to }19\)
The extremes determine the whiskers, not the boxForcing the median to the box centre
✕\(\displaystyle \mathrm{median}=(7+19)/2=13\)✓\(\displaystyle \mathrm{median}=(10+14)/2=12\)
Use the central observations, not the midpoint of the quartilesDifficulty levels
introMatch a summary to a plot
standardCalculate summaries from odd and even data
challengeUse dot/stem plots and read back a summary
examRecognise a plot or one of its features
Questions about this topic
Can a box or whisker have zero length?+
Yes. Repeated observations can make summary values coincide
Must the median be halfway across the box?+
No. Its numerical position comes from the data
Are quartile conventions universal?+
No. State the convention; these questions use medians of halves
Do whiskers always show the minimum and maximum?+
In the ordinary boxplots here, yes. Modified boxplots can plot unusual values separately
Does matching a diagram prove I can draw one?+
No. Also practise drawing on paper and check the scale and all five features
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 9: Probability and statistics
In this chapter
Finite events, probability assignments and complementsDescribe and order probability values9B Venn diagrams and tables: Venn diagrams, two-way tables and set notation for two eventsComplete and use a two-way probability tableCompound events, the addition rule and mutual exclusivity9C Set notation: Event notation and shaded regionsFinite sets and set operationsBuild two-step sample spaces with and without replacementUse two-step probability treesExperimental probability from observed frequenciesDesign and interpret a probability simulationFind the mean and the median, and choose between themWork backwards from a mean or medianMode, bimodal and no-mode data, and the effect of an outlierConstruct an ordered stem plot from raw dataConstruct and read back-to-back stem-and-leaf plotsDescribe the shape of a distributionGroup numerical data into class intervalsConstruct a histogram from a frequency tableRead values from a histogramFind the range, the quartiles and the IQRProduce a five-number summary and construct a boxplotRead values and percentages from a boxplotCompare a numerical variable across groupsSurveys and statistical investigations