Five numbers, one rectangle
A box plot compresses a dataset into its five-number summary and draws it: a box from the first quartile Q1 to the third quartile Q3, a line inside at the median, and whiskers reaching to the most extreme values that are not outliers. Points beyond the whiskers are drawn individually. In one glance you get the centre, the spread of the middle half, the overall range, the skew (is the median centred in its box?), and the outliers by name.
The twenty commute times (22 to 38 minutes): Q1 = 25.75, median = 27.5, Q3 = 30.25, so the box spans 25.75 to 30.25 with IQR = 4.5. The fences sit at 25.75 − 1.5 × 4.5 = 19 and 30.25 + 1.5 × 4.5 = 37: the 38-minute commute falls outside and is plotted as an outlier dot. The plot tells the story a mean never would: typical days cluster tightly, and then there is that one day.
Why box plots win at comparisons
Place box plots for several groups side by side (same axis) and differences jump out: which group runs higher, which is more variable, which has the ugly outliers. This is the standard first chart of any serious comparison, from drug doses to delivery times by depot. The median lines and box overlaps carry the message; formal tests come after, to check what the picture suggests.
Reading the box like a histogram. Box size is a quartile span, not a count: a tall box means a spread-out middle half, not more data. And box plots hide shape inside the quartiles: a bimodal dataset and a uniform one can draw near-identical boxes. Pair the box plot with a histogram when shape matters.
The 1.5 IQR rule
The fences are a screening device, not a verdict. Tukey chose 1.5 because for normal data it flags roughly 0.7% of points: rare enough to deserve a look, common enough to catch genuine oddities. An outlier is a prompt to investigate (typo? different population? the most interesting observation you own?), never a licence to delete.
Frequently asked questions
What does the box in a box plot represent?
The middle 50% of the data, from the first quartile to the third. The line inside is the median. Half of all values lie inside that box by construction.
How are outliers decided?
Points beyond Q1 - 1.5x IQR or Q3 + 1.5x IQR. It is Tukey's screening rule: flagged points deserve investigation, not automatic removal.
Box plot or histogram: which should I use?
Box plot for comparing groups and spotting outliers compactly; histogram for seeing shape (peaks, gaps, skew detail). Serious reports often show both.
Why is my median line not in the middle of the box?
Skew inside the middle half. A median hugging Q1 means the lower half of the middle group is compressed and the upper half stretched: the plot is reporting asymmetry faithfully.