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Learn Index · Data Management and Statistics Grade 9–10

Box-and-whisker plots

A box-and-whisker plot shows the minimum, Q1, median, Q3 and maximum.

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A picture of spread

A box-and-whisker plot (box plot) summarizes data with five numbers, the five-number summary:

Value Meaning
Minimum smallest value
Q1 lower quartile (25% below)
Median middle value (50% below)
Q3 upper quartile (75% below)
Maximum largest value

Drawing one

Data: 4, 7, 9, 10, 12, 15, 18, 20, 25

  1. Order the data (done).
  2. Median: 12.
  3. Q1: median of 4, 7, 9, 10 → 8.
  4. Q3: median of 15, 18, 20, 25 → 19.
  5. Min 4, max 25.

On a number line:

  • Draw a box from Q1 (8) to Q3 (19).
  • Draw a line inside the box at the median (12).
  • Draw whiskers from the box out to 4 and 25.

What it shows

  • Each section holds about 25% of the data.
  • The box holds the middle 50%: its width is the interquartile range (IQR = 19 − 8 = 11).
  • The overall length shows the range (25 − 4 = 21).
  • Long sections mean data is spread out there; short sections mean it’s bunched together.

Shape

Box plot looks like Distribution
median in the middle, whiskers equal symmetric
median closer to Q1, long right whisker skewed right
median closer to Q3, long left whisker skewed left

Outliers

A common rule: values below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR are outliers. They’re shown as separate dots, and the whisker stops at the most extreme non-outlier.

From the data: 1.5 × 11 = 16.5. Fences: 8 − 16.5 = −8.5 and 19 + 16.5 = 35.5. No outliers.

Comparing groups

Box plots are great side by side.

Example: Two classes’ test scores:

Min Q1 Median Q3 Max
Class A 45 62 70 78 95
Class B 55 68 74 80 88
  • Class B has a higher median.
  • Class B’s scores are less spread out (smaller range and IQR).

Worked examples

Example 1: Five-number summary: 2, 5, 9, 14, 20. IQR? 14 − 5 = 9.

Example 2: What percent of data lies between Q1 and the maximum? 75%.

Example 3: A box plot’s median is near Q3 and the left whisker is long. Shape? Skewed left.

Common mistakes

  • Not ordering the data first.
  • Thinking a longer section contains more data. Each quarter has the same number of values; a longer section means they’re more spread out.
  • Plotting on an uneven scale.

Try it yourself

Data: 3, 6, 7, 8, 10, 12, 13, 15, 22

  1. Find the five-number summary.
  2. Find the IQR.
  3. Is 22 an outlier?

Answers: 1) 3, 6.5, 10, 14, 22 2) 7.5 3) No: upper fence = 14 + 11.25 = 25.25.

Quick recap

  • A box-and-whisker plot shows the five-number summary.
  • The box spans Q1 to Q3 (the IQR); each part holds 25% of the data.
  • Use box plots to compare spread, centre and outliers.