Statistics · step 4 of 13

Boxplots, histograms & quartiles

The five-number summary — minimum, Q₁, median, Q₃, maximum — is what a boxplot draws: a box spanning the interquartile range with whiskers to the extremes (or to the outlier fences). A histogram instead shows the shape by binning, and the two are complementary: boxplots compare groups compactly, histograms reveal modality that a boxplot completely hides.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Sort and find the medianThis splits the data into a lower and an upper half.
Find Q₁ and Q₃ as medians of the halvesFor odd n, conventions differ on whether to include the median in each half — state which you use.
Compute the IQR and the fencesIQR = Q₃ − Q₁; fences at Q₁ − 1.5·IQR and Q₃ + 1.5·IQR.
Draw or read the plotBox from Q₁ to Q₃ with the median marked; whiskers to the most extreme non-outlier points; outliers plotted individually.

Worked example

Give the five-number summary of 3, 7, 8, 12, 14, 18, 21.

n = 7, already sorted. Median is the 4th value: 12.
Lower half (excluding the median): 3, 7, 8 — its median is Q₁ = 7.
Upper half: 14, 18, 21 — its median is Q₃ = 18.
Min = 3, max = 21, IQR = 18 − 7 = 11.

Answer. Five-number summary: 3, 7, 12, 18, 21, with IQR 11 and fences at −9.5 and 34.5 — no outliers.

Where marks get dropped

These are the specific errors that cost credit on boxplots, histograms & quartiles questions — QED's rubric penalises each of them separately.

Switching quartile conventions midway. Whether the median is included in each half changes Q₁ and Q₃ for odd n, so pick one and say so.
Reading a boxplot as showing modality. A bimodal and a uniform data set can produce identical boxplots — that is what histograms are for.
Assuming equal box lengths mean equal counts. Each quartile always contains 25% of the data; the box WIDTH shows spread, not frequency.

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Boxplots, histograms & quartiles — frequently asked questions

What does the box actually contain?

The middle 50% of the data, from Q₁ to Q₃. The line inside is the median, not the mean.

How do I choose histogram bin widths?

Too few bins hide structure, too many produce noise. Common rules are √n bins or Sturges’ rule, but trying several is standard practice.

Why do software packages disagree on quartiles?

There are at least nine defined methods for interpolating quartiles. They agree for large n and can differ noticeably for small samples.

The rest of Statistics

Describing data, distributions, estimation and hypothesis tests. Each subtopic below has its own method, worked example and mark-losing traps.

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