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The five numbers from last stop are on Mia's pad: 21, 24, 29, 35 and 58 minutes. The newsletter has room for a picture, not a table, so the five have to become a drawing on a single scale.
Where do the two ends of the box go?
Box Plots and Comparing Distributions
Draw a box plot from a five-figure summary, and compare two distributions by their medians and their spread in words about the situation.
What a learner can do afterwards
- Draw a box plot from minimum, quartiles, median and maximum
- Compare two box plots in a sentence about the context, not only the numbers
- Spot a skewed distribution from where the median sits inside the box
1 · Read
Last stop left you holding five numbers about the race: 21 for the fastest time, 24 and 35 for the two quartiles, 29 for the median and 58 for the slowest. Those five together are called the five-figure summary, and they are enough to draw the whole field on one line. The drawing is a box plot. It fits on a poster, and two of them side by side answer questions that two lists of times never could.
To draw it, rule a scale along the bottom with equal gaps for equal amounts, the way you did for a scatter graph. Draw a box from the lower quartile to the upper quartile. Rule a line inside the box at the median. Then draw a whisker from each end of the box out to the smallest and largest values. That is the whole thing. About half the runners are inside the box, with about a quarter along each whisker.
Now the point of drawing them. Last year the same course gave 26, 32, 36, 40 and 50. Its median is 36 against this year's 29, so the field came home about seven minutes faster this year. Its interquartile range is 40 - 32 = 8 against this year's 35 - 24 = 11, so last year's middle half was tighter. Say it as a sentence about running, not about numbers: this year's field was faster but more strung out.
Where the median sits inside the box tells you about the shape. The junior race gave 12, 14, 16, 24 and 30. Its box runs from 14 to 24 with the median at 16, so 2 units of box sit below the median and 8 above. The juniors bunched tightly near the front, and the same-sized quarter behind the median was spread over four times as much time. Skew is named for the side the tail is on, so this one is skewed to the right. A long half below the median means skewed to the left, and a centred median means a fairly even spread.
One warning. A box plot draws five numbers, so five numbers is all it knows. It cannot tell you how many runners there were, and it cannot show a gap or a crowd inside the box. Two races could hand you the same picture with 11 runners in one and 400 in the other. Whenever you compare two plots, say how many were in each, because the picture will not say it for you.
A box plot draws a five-figure summary: a box from the lower quartile to the upper quartile, a line inside at the median, and whiskers out to the smallest and largest values. Half the data sits inside the box and a quarter in each whisker. Compare two by their medians for position and their interquartile ranges for spread, and say the answer as a sentence about the situation. A median off-centre in its box means the data is skewed towards the longer side.
2 · Watch
3 · Play
Take it off screen
Print a worksheetA4 with an answer key page for grown-ups. No screen, no internet.
Where it sits
Learn first
This opens up
Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
Then practise
24 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.