Quartiles and box plots

Averages tell you a typical value; the spread tells you how much the data varies. Quartiles split ordered data into four equal parts, and a box plot shows them at a glance.

Key facts

  • Range = largest − smallest (affected by extreme values).
  • Q1 is a quarter of the way through the ordered data, Q3 three quarters.
  • Interquartile range IQR = Q3 − Q1 (the middle half).
  • Outliers: more than 1.5 × IQR beyond Q1 or Q3.
  • A box plot shows min, Q1, median, Q3, max.

Worked examples

Example 1

Find the median and IQR of 3, 5, 7, 8, 10, 12, 15.

  1. It’s already in order. The median is the middle (4th) value: 8.
  2. Lower half 3, 5, 7: Q1 = 5. Upper half 10, 12, 15: Q3 = 12.
  3. IQR = 12 − 5 = 7

Answer: Median 8, IQR 7

Example 2

Two classes have the same median but class A has IQR 4 and class B has IQR 15. Compare them.

  1. Same median: typical scores are similar.
  2. Class A’s smaller IQR means its middle half are closer together: more consistent.

Answer: Class A is more consistent