Box plot

The box plot (box-and-whisker diagram) summarises a distribution with five numbers, highlighting its centre, spread and any anomalous values.

Definition — Box plot (box-and-whisker diagram)

Graphical summary of a distribution through five numbers:

• $Q_1$ (first quartile): $25\%$ of the data are $\le Q_1$;
• $Q_2$ = median: $50\%$;
• $Q_3$ (third quartile): $75\%$;
• IQR (interquartile range) $=Q_3-Q_1$: robust spread.

The box goes from $Q_1$ to $Q_3$ with a median line at $Q_2$. The “whiskers” extend to the smallest and largest non-outlier values: typically data beyond $Q_1-1,5\,\text{IQR}$ or $Q_3+1,5\,\text{IQR}$ are declared outliers.

Example — Box plot of the marks

On the ordered marks $4,5,6,6,7,7,7,8,9,10$: $Q_1=6$ (third value), $Q_2=7$, $Q_3=8$. IQR $=2$. Whiskers: $4$ (min) and $10$ (max), none is an outlier because $4>6-3=3$ and $10<8+3=11$.

Box plot of the marks: box from $Q_1=6$ to $Q_3=8$, median at $7$, whiskers from $4$ to $10$.

The box plot is particularly useful for comparing several distributions (e.g. the marks of three parallel classes): five numbers per class, side by side, immediately show differences in centre and spread.

Links

Topics: Probabilita
Concepts: Box plot · Iqr · Mediana · Outlier · Quartile
Methods: Box plot
Skills: Interpretare grafico