Reading boxplots
Read numerical positions accurately and qualify percentage interpretations
\[\mathrm{IQR}=Q_3-Q_1\]
- Landmarks The line inside the box is the median; box edges are quartiles
- Range Use the full data extremes, including separately plotted observations if shown
- Approximate proportions Quartile positions suggest approximate quarter shares. Ties and sample size affect exact counts
- Limits A boxplot does not generally determine the mean, mode, sample size or original data
Worked example
An ordinary boxplot has minimum 2, Q1 5, median 9, Q3 14 and maximum 18 minutes
\[(2,5,9,14,18)\]
- \(\displaystyle \mathrm{IQR}=14-5=9\)The middle-half spread is 9 minutes
- \(\displaystyle \mathrm{range}=18-2=16\)The full span is 16 minutes
- \(\displaystyle 100\%-75\%=25\%\)About a quarter lies above Q3 under the usual approximate interpretation. This is not an exact count claim
Common mistakes
Using whisker width as frequency
✕\(\displaystyle \begin{gathered}\text{longer whisker}\\\Rightarrow\text{more observations}\end{gathered}\)✓\(\displaystyle \begin{gathered}\text{longer whisker}\\\Rightarrow\text{wider value interval}\end{gathered}\)
Quartile positions concern order. The width concerns spreadAssuming the mean equals the median
✕\(\displaystyle \mathrm{mean}=9\)✓\(\displaystyle \mathrm{median}=9\)
The mean cannot be recovered from this summary aloneDifficulty levels
introRead the median and a quartile
standardCalculate range and IQR, then interpret an approximate proportion
Questions about this topic
Why say approximately 25%?+
Ties, finite samples and different quartile conventions can change exact counts below a boundary
Can I count observations from box width?+
No. Numerical width measures spread, not frequency
Does the median split the box into equal lengths?+
No. Each side can have a different numerical width
What if there are outlier points?+
They are still observations. Include the extreme plotted points when finding the full data range
Can two different data sets have the same boxplot?+
Yes. Five summary values discard much of the original detail
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