Box Plots

A precise guide to five-number summaries, quartiles, IQR fences, skew, and responsible group comparison.

A box plot divides ordered data into four quarter groups. A number line carries a box-and-whisker plot with minimum, quartiles, median, and maximum labelled. Unequal segment lengths reveal spread without implying unequal counts per quartile.

Precise definition

A box plot displays a median, lower and upper quartiles $Q_1,Q_3$, and whiskers under a stated convention. The interquartile range is $IQR=Q_3-Q_1$. In a common modified box plot, potential outliers lie below $Q_1-1.5IQR$ or above $Q_3+1.5IQR$, and whiskers extend to the most extreme non-outlier observations.

Notation and mathematical language

The box spans the middle 50% of ordered data, and its internal line marks the median. Quartile algorithms vary for finite samples, so software or course convention should be stated. A box plot does not show every data value or generally identify the mean.

Conceptual picture

Box length compares central spread; median position and unequal whiskers can suggest skew. Side-by-side plots efficiently compare location, variability, overlap, and flagged observations across groups, but they hide sample size unless annotated.

Fully worked example

Interpretation and application

Box plots summarize waiting times, scores, incomes, and lab measurements. Apparent group differences are descriptive association. Random sampling or experimental assignment and uncertainty analysis determine whether broader or causal conclusions are warranted.