Five-Number Summaries and Box Plots: Quick Revision

A brief note on Five-Number Summaries and Box Plots. Read it before a test. Try each question at the end before you open its answer.

Quartiles and the summary

• Rule: sort; $Q_1$ and $Q_3$ are the medians of the lower and upper halves; leave the median out when the count is odd.
• Five-number summary: minimum, $Q_1$, median, $Q_3$, maximum.
• IQR: $Q_3 - Q_1$, the width of the middle half.
• Software may use another rule and give slightly different quartiles. Say which rule you used.

Outliers

• Fences: $Q_1 - 1.5\,\text{IQR}$ and $Q_3 + 1.5\,\text{IQR}$.
• Outlier: a value beyond a fence. Whiskers stop at the last value inside.
• Plan: check for a mistake; keep a real value; prefer the median and the IQR; report what you did.

Comparing groups

• Put the box plots on one axis. Compare centre, spread, shape and outliers.
• A median above the other group's $Q_3$: at least half of one group beats about three quarters of the other.
• A box plot hides peaks and gaps. Draw a histogram when the shape matters.

A worked example

The nine values $6, 9, 10, 12, 13$, $15, 16, 18, 41$ are recorded.

1. Five numbers. Minimum 6, $Q_1 = \dfrac{9 + 10}{2} = 9.5$, median 13, $Q_3 = \dfrac{16 + 18}{2} = 17$, maximum 41.
2. Fences. IQR $= 7.5$, so the fences are $9.5 - 11.25 = -1.75$ and $17 + 11.25 = 28.25$.
3. Outliers. Only 41. The right whisker ends at 18.

Practice with new numbers

A program to try

The program finds the five numbers, the fences and the outliers of the worked example.

Mistakes to avoid

• Keeping the median in a half when the count is odd.
• One IQR instead of 1.5 IQR for the fences.
• Deleting an outlier without checking it.
• Comparing medians only. Say something about spread too.

Check yourself

1. Find $Q_1$ and $Q_3$ of $2, 5, 6, 9, 11$.

2. $Q_1 = 15$ and $Q_3 = 25$. Find the upper fence.

3. Which is more resistant to an outlier, the mean or the median?

4. Can two different data sets have the same box plot?

Sources

• OpenStax, Introductory Statistics, Chapter 2 (Descriptive Statistics), sections on measures of the location of the data and on box plots.