Box Plot
A box plot summarizes a dataset using five key values: minimum, Q1, median, Q3, and maximum, displayed as a box with whiskers.
Definition
A box plot (also called a box-and-whisker plot) shows the spread of data using a box and two lines called whiskers: the box covers the middle half of the data, and the line inside the box marks the median. It displays the five-number summary: minimum, $Q_1$ ($25$th percentile), median ($Q_2$, $50$th percentile), $Q_3$ ($75$th percentile), and maximum, with the box spanning $Q_1$ to $Q_3$ (the IQR) and whiskers extending to the most extreme values within $1.5 \times \text{IQR}$ from the quartiles, values beyond this plotted as individual outlier points. Formally, the box plot's outlier rule (flag values beyond $Q_1 - 1.5 \cdot \text{IQR}$ or $Q_3 + 1.5 \cdot \text{IQR}$) corresponds approximately to the range of a normal distribution within about $2.7$ standard deviations of the mean, flagging roughly $0.7\%$ of normally distributed values as outliers, and for skewed data, the box plot's asymmetry reveals the skew direction.
Example
For test scores, the box might stretch from $70$ to $90$ (the middle $50\%$), with the median line at $80$, and whiskers extending to the lowest score ($55$) and highest score ($100$). A five-number summary of $\text{min}=40$, $Q_1=60$, $\text{median}=72$, $Q_3=85$, $\text{max}=95$ gives $\text{IQR}=25$, with outlier fences at lower $= 60 - 1.5(25) = 22.5$ and upper $= 85 + 1.5(25) = 122.5$, so no values fall outside these fences. Comparing box plots of salary distributions across departments, a right-skewed box plot (longer upper whisker and outlier dots on the right) indicates a small number of very high earners pulling the mean above the median.
Key Insight
A box plot gives you a quick summary of a dataset: where the center is (median), how spread out the middle half is (the box), and how far the extremes extend (the whiskers), and it is especially useful for comparing multiple groups side by side. The violin plot is a modern alternative that replaces the box with a kernel density estimate, preserving the five-number summary information while showing the full distributional shape, including multimodality.