The five-number summary, drawn

The five-number summary (minimum, $Q_1$, median, $Q_3$, maximum) becomes a picture in four moves:

1. Draw a number line covering the data, with a real scale.
2. Draw a box from $Q_1$ to $Q_3$: it spans the middle 50% of the ordered data.
3. Cut the box with a line at the median. That line is the median, not the mean; the mean appears nowhere on a standard boxplot.
4. Extend whiskers from the box out to the minimum and maximum (outliers get special handling, below).

Take the twelve commute times (in minutes) 4, 12, 14, 14, 16, 18, 20, 22, 28, 30, 30, 32. The median is $\frac{18+20}{2} = 19$, the lower half gives $Q_1 = 14$, the upper half gives $Q_3 = 29$. So the box runs 14 to 29, the cut sits at 19, and the whiskers reach 4 and 32. Every position is computed from the data: nothing on a boxplot is placed by eye.

The four regions of a boxplot (minimum to $Q_1$, $Q_1$ to median, median to $Q_3$, $Q_3$ to maximum) each contain about 25% of the observations. That is forced by what quartiles are: the cut points of the ordered data at the 25th, 50th, and 75th percentiles.

So a region's length is not the amount of data in it. Length is the distance its quarter of the data had to stretch across:

1. A long region means its quarter of the data is spread thin over a wide stretch of values.
2. A short region means its quarter is packed into a narrow stretch, densely.

This inverts the intuition trained by histograms, where tall does mean many. On a boxplot, the longest region is the sparsest, not the most crowded, and the skinny little box in the middle of two sprawling whiskers still holds a full half of the data. Reading length as amount is the single most common boxplot error, and it reverses the message of the plot.

When the data contain outliers by the 1.5 IQR rule (beyond $Q_1 - 1.5 \times \text{IQR}$ or $Q_3 + 1.5 \times \text{IQR}$), the convention changes: each whisker extends only to the most extreme value that is not an outlier, and each outlier is plotted individually as a dot or asterisk.

For the commute data, $\text{IQR} = 29 - 14 = 15$, so the fences sit at $14 - 22.5 = -8.5$ and $29 + 22.5 = 51.5$. Nothing lies beyond either fence, so the whiskers run to the true minimum and maximum, 4 and 32.

Three details are routinely botched. The whisker ends at a data value, never at the fence itself: the fence is a computed threshold that usually is not in the data. The outlier stays in the plot (as its dot) and in every calculation; flagging is not deleting. And the box is untouched by all of this: quartiles are resistant, so one wild maximum changes only the upper whisker and adds a dot.

Skewness is legible on a boxplot: a long upper region (whisker plus upper box) with the median shoved toward the bottom of the box says right skew; the mirror image says left skew. And the mean, though not drawn, can be inferred: if a distribution is roughly symmetric, mean and median sit close; if it is skewed right, the mean is usually greater than the median; skewed left, usually less. The mean chases the tail because the tail's extreme values enter its sum at full strength.

What a boxplot cannot show is how the data are arranged inside each quarter. Five numbers cannot record modes: a two-humped distribution with a gap in the middle can produce exactly the same boxplot as a single smooth mound. Sample size vanishes too: a boxplot of 8 values and one of 8,000 look equally solid. When the number of clusters, gaps, or peaks matters, you need the dotplot, stemplot, or histogram; the boxplot is a summary of a summary, and it inherits the five-number summary's blind spots.