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Statistics · Cumulative frequency & box plots

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Box plots

What each of the five marks on a box plot means, why the box holds the middle 50% of the data, and why a longer section means the data there is more spread out — never that there is more of it.

Statistics · Cumulative frequency & box plots

Box plots

What each of the five marks on a box plot means, why the box holds the middle 50% of the data, and why a longer section means the data there is more spread out — never that there is more of it.

Why it works

A box plot is not a picture of how many. It is a picture of *where the values sit*. Almost every mark lost on box plots comes from forgetting that one sentence, so it is worth seeing exactly where the five marks on the plot come from.

Start with data, in order. Here are the marks of 2020 students in a test out of 6060, with a bar dropped after every fifth value:

12, 18, 21, 23, 25 ∣ 25, 27, 28, 29, 30 ∣ 30, 32, 33, 34, 35 ∣ 35, 42, 48, 53, 5812,\ 18,\ 21,\ 23,\ 25\ \mid\ 25,\ 27,\ 28,\ 29,\ 30\ \mid\ 30,\ 32,\ 33,\ 34,\ 35\ \mid\ 35,\ 42,\ 48,\ 53,\ 58

Five values is exactly a quarter of 2020, so those three bars cut the data into four equal-sized groups. The bars are the quartiles: lower quartile Q1=25Q_1 = 25, median Q2=30Q_2 = 30, upper quartile Q3=35Q_3 = 35. Add the smallest value 1212 and the largest 5858 and you have the five-number summary: 12, 25, 30, 35, 5812,\ 25,\ 30,\ 35,\ 58.

That definition is the whole reason a box plot can be read at all: the quartiles are the cut points chosen so that the four groups are equal in size. The plot just draws those five numbers on a number line — a box from Q1Q_1 to Q3Q_3 with a line at the median, and whiskers reaching out to the smallest and largest values.102030405060MarkSo the plot has four sections — lower whisker 12→2512 \to 25, left half of the box 25→3025 \to 30, right half of the box 30→3530 \to 35, upper whisker 35→5835 \to 58 — and each one holds a quarter of the data, here 55 students. The box, being the two middle sections, holds the middle 50%50\%; each whisker holds an outer 25%25\%.

The trap, with the numbers in front of you. Look at the lengths. The upper whisker runs from 3535 to 5858: that is 2323 marks wide. The whole box runs from 2525 to 3535: only 1010 marks wide. The upper whisker is more than twice as long as the entire box — so there must be more students up there?

Count them. The upper whisker holds 35,42,48,53,5835, 42, 48, 53, 58 — five students. The box holds 25,27,28,29,30,30,32,33,34,3525, 27, 28, 29, 30, 30, 32, 33, 34, 35 — ten students. The longer section contains half as many students as the shorter one.

Once you have counted it the reason is obvious. Every section holds the same number of values by construction — that is what "quartile" means. The only thing that can differ from section to section is how far along the number line that equally-sized group is stretched. So
  • a short section ⇒\Rightarrow that quarter of the data is bunched together;
  • a long section ⇒\Rightarrow that quarter of the data is spread out.
Length on a box plot measures spread, never frequency. Here the top five students scored 35,42,48,53,5835, 42, 48, 53, 58 — scattered all over — while the middle ten are packed tightly between 2525 and 3535.

That is also the only way to count anything from a box plot. If a question tells you there were 2020 students, then 55 are above Q3Q_3, 1010 are above the median and 1515 are above Q1Q_1 — and that is all. A box plot cannot give you the mean, the mode or any individual value, and it cannot tell you the total unless the question states it.

What the shape tells you. Because the median line splits the box, where it sits reveals symmetry. Median close to Q1Q_1: the lower-middle quarter is squashed and the upper-middle quarter stretched, so the data trails away to the right — positive skew. Median close to Q3Q_3: the tail is to the left — negative skew. Median central with whiskers of similar length: roughly symmetrical. Never assume the median sits in the middle of the box; read where it actually is.

Comparing two box plots. Marks come in pairs, and both must be in context:
  • one comparison of centre — the medians: "the median mark at School A
(4848) is higher than at School B (4141), so students at A scored higher on average";
  • one comparison of spread — the IQRs (or the ranges): "the IQR at A (1212)
is smaller than at B (2020), so A's marks were more consistent."

"A's box is further to the right and thinner" scores nothing: no measure named, no figures, no context. Name the measure, quote the numbers, then say what it means about the thing being measured.

Outliers. When outliers are shown they are crosses drawn beyond the whiskers, and the whisker ends are then no longer the smallest and largest values — they are the most extreme values that are not outliers. The range still uses the true extremes, crosses included. The IQR ignores them completely, which is exactly why the IQR is the better measure of spread once a data set has stray values in it.