Box plots and the five-number summary

The five-number summary

• Ordering a data set and picking out five values summarises its shape:
  • Minimum — the smallest value.
  • Lower quartile ($Q_1$) — a quarter of the way through the ordered data.
  • Median — the middle value (half way).
  • Upper quartile ($Q_3$) — three quarters of the way through.
  • Maximum — the largest value.
• The median is the measure of centre for an inference. The interquartile range, $\text{IQR} = Q_3 - Q_1$, is the measure of spread.

Box plots

• A box plot draws the five-number summary on a number line.
• The box runs from $Q_1$ to $Q_3$, so it holds the middle 50% of the data. Its width is the IQR.
• The line inside the box marks the median. The whiskers reach out to the minimum and maximum.

• Because the box is the middle 50%, a narrow box means the values are tightly bunched; a wide box means they are spread out.

Dot plots

• A dot plot shows every value as a dot stacked above the number line, so you see individual points and gaps that a box plot hides.
• It is ideal for small samples and for spotting clusters and outliers.

• A good analysis often shows both: dots for the raw values, a box plot beneath for the summary.

Reading centre and spread from a box plot

• Centre: read the median line — a typical value for the group.
• Spread: the IQR is the box width; the range is $\text{max} - \text{min}$ (the whole whisker-to-whisker length).
• Shape: if the median sits in the middle of the box and the whiskers are even, the data is roughly symmetric. If the median is pushed to one side, the data is skewed toward the longer side.

Choosing the display

• Use a dot plot to see every value in a small sample.
• Use a box plot to summarise spread and centre and to compare groups side by side on one scale.