Box Plots
Every question with a fully worked solution.
- Box Plots - Video - The Five Number Summary, Boxplots, and Outliers Watch
Theory
A box plot (box-and-whisker plot) draws the five-number summary on a scale. This Year 10A maths page shows how to draw a box plot from data, read the median, range and IQR from one, use the fact that each section holds about 25% of the scores, and show outliers as dots.
A box plot shows five numbers on a scale: the minimum, the lower quartile \(Q_1\), the median, the upper quartile \(Q_3\) and the maximum.
The box runs from \(Q_1\) to \(Q_3\), with a line at the median. Its length is the interquartile range. The whiskers run out to the minimum and maximum, so the full length is the range.
The four sections (left whisker, left part of the box, right part of the box, right whisker) each hold about 25% of the scores. A long section means those scores are spread out, not that there are more of them.
An outlier can be shown as a separate dot. The whisker then ends at the most extreme score that is not an outlier.
| Part of the box plot | Shows |
|---|---|
| left end of whisker | minimum |
| left edge of box | \(Q_1\) |
| line in the box | median \(Q_2\) |
| right edge of box | \(Q_3\) |
| right end of whisker | maximum |
| Region | About |
|---|---|
| below \(Q_1\) or above \(Q_3\) | \(25\%\) |
| below or above the median | \(50\%\) |
| between \(Q_1\) and \(Q_3\) | \(50\%\) |
| below \(Q_3\) or above \(Q_1\) | \(75\%\) |
How to draw a box plot
- Order the data and find the five-number summary.
- Check for outliers if asked (\(1.5 \times \text{IQR}\) beyond the quartiles).
- Draw a labelled scale with even steps that covers the minimum to the maximum.
- Draw the box from \(Q_1\) to \(Q_3\), a line at the median, and whiskers to the minimum and maximum (or to the last non-outlier, with outliers as dots).
How to read a box plot
- Read the five values off the scale. Check what each small step on the scale is worth.
- Calculate what is asked: range, IQR, or a percentage of the total.
- Use quarters: each section is about \(25\%\) of the scores.
Order the data and find the five-number summary.
| \(\) | \(\) | \(28,\ 33,\ 35,\ 37,\ 39,\ 40 \ \big|\) \(42,\ 44,\ 46,\ 48,\ 50,\ 61\) |
| \(\text{median}\) | \(=\) | \(\dfrac{40 + 42}{2} = 41\) |
| \(Q_1\) | \(=\) | \(\dfrac{35 + 37}{2} = 36\) |
| \(Q_3\) | \(=\) | \(\dfrac{46 + 48}{2} = 47\) |
Summary: \(\mathbf{28,\ 36,\ 41,\ 47,\ 61}\). Draw a scale that covers \(28\) to \(61\), the box from \(36\) to \(47\) with a line at \(41\), and whiskers to \(28\) and \(61\).
Is \(61\) an outlier? \(47 + 1.5 \times 11 = 63.5\), and \(61 < 63.5\), so no: the whisker goes all the way to \(61\).
(i) the range
(ii) the IQR
(iii) about how many students game for more than \(13\) hours
(iv) about how many game between \(9\) and \(13\) hours.
Read: min \(4\), \(Q_1 = 9\), median \(13\), \(Q_3 = 20\), max \(31\). Each quarter is \(25\%\) of \(80 = 20\) students.
| \(\text{(i) range}\) | \(=\) | \(31 - 4 = \mathbf{27}\text{ h}\) |
| \(\text{(ii) IQR}\) | \(=\) | \(20 - 9 = \mathbf{11}\text{ h}\) |
| \(\text{(iii) above the median}\) | \(=\) | \(50\% \text{ of } 80 = \mathbf{40}\) |
| \(\text{(iv) } Q_1 \text{ to median}\) | \(=\) | \(25\% \text{ of } 80 = \mathbf{20}\) |
The answers are "about" because several students can share the exact value of a quartile.
| \(\text{median}\) | \(=\) | \(\dfrac{24 + 25}{2} = 24.5\) |
| \(Q_1\) | \(=\) | \(21\ \text{(middle of } 4 \ldots 24),\quad Q_3 = 28\) |
| \(\text{IQR}\) | \(=\) | \(28 - 21 = 7\) |
| \(\text{limits}\) | \(=\) | \(21 - 10.5 = 10.5,\) \(28 + 10.5 = 38.5\) |
\(4 < 10.5\), so \(4\) is an outlier. Draw it as a dot. The left whisker ends at \(19\), the smallest score that is not an outlier.
(i) Half the class is taller than \(168\) cm.
(ii) The IQR is \(9\) cm.
(iii) \(25\%\) of the class is shorter than \(171\) cm.
(iv) The tallest student is \(39\) cm taller than the shortest.
Read: \(150, 162, 168, 171, 189\).
| \(\text{(i)}\) | \(\) | \(\text{above the median: } 50\% \Rightarrow \textbf{true}\) |
| \(\text{(ii)}\) | \(\) | \(171 - 162 = 9 \Rightarrow \textbf{true}\) |
| \(\text{(iii)}\) | \(\) | \(\text{below } Q_3 \text{ is } 75\% \Rightarrow \textbf{false}\) |
| \(\text{(iv)}\) | \(\) | \(189 - 150 = 39 \Rightarrow \textbf{true}\) |
A box that is short on one side does not hold fewer people: every section holds about a quarter, however long or short it is drawn.
Common pitfalls
Frequently asked questions
What are the five parts of a box plot?
The minimum, the lower quartile, the median, the upper quartile and the maximum. The box runs from the lower to the upper quartile and the whiskers reach out to the minimum and maximum.
How do you find the IQR from a box plot?
Read the upper quartile at the right edge of the box and the lower quartile at the left edge, then subtract. The IQR is the width of the box.
What percentage of data is in the box of a box plot?
About \(50\%\): the box runs from the lower quartile to the upper quartile, which is the middle half of the data.
How do you show an outlier on a box plot?
Draw the outlier as a separate dot or cross on the scale and end the whisker at the most extreme score that is not an outlier.
Does a longer whisker mean more data?
No. Each whisker holds about a quarter of the scores. A longer whisker means those scores are more spread out.
Why does a box plot sometimes have no whisker?
The whisker joins the minimum to the lower quartile (or the upper quartile to the maximum). If the two values are equal, the whisker has length zero, which happens when several of the lowest scores are the same.