Boxplots and What They Hide

A boxplot is a picture of the five-number summary. Draw a box from Q1 to Q3, with a line inside at the median, then extend whiskers out to the minimum and maximum. The box width is the interquartile range, IQR = Q3 - Q1, and it holds the middle 50% of the data. Each whisker stretches to whatever the most extreme value is on that side, no matter how many points sit out there.

That last fact is the trap. A long whisker tells you the data reaches far, not that many points live there. A whisker could be stretching toward a single lonely value, or toward a dense cluster. The boxplot cannot tell the difference, because it never recorded how many points fall between the quartiles and the extremes, only where those extremes are. Two histograms, one skewed, one bimodal, one perfectly symmetric, can all share the same min, Q1, median, Q3, and max, and so collapse into the exact same boxplot. That is the honest limitation: a boxplot compresses a full distribution into five landmarks, trading shape detail for a quick, comparable summary.

This connects directly to why outliers get special treatment. When a value is flagged as an outlier using the 1.5 x IQR rule, it gets plotted as its own dot, and the whisker stops at the last non-outlier value instead of stretching all the way to the extreme. That shortens the whisker and gives a more honest sense of where the bulk of the data actually ends, which is why modified boxplots are usually preferred over plain ones.