Center is only half the story. Two classes can have the same median score but very different spreads. Range and interquartile range measure spread.
Key idea
Range = greatest value − least value. The interquartile range (IQR) = $Q_3 - Q_1$, the spread of the middle half of the data. $Q_1$ is the median of the lower half and $Q_3$ is the median of the upper half.
Why it works
The range uses only the two most extreme values, so one unusual value can stretch it a lot. The IQR leaves out the lowest quarter and the highest quarter of the data, so it describes where the middle half lives, and an outlier hardly changes it.
Worked example: An even number of values
Find the range and IQR of 2, 4, 5, 7, 8, 10, 12, 15.
- Range: $15 - 2 = 13$.
- Split into halves: 2, 4, 5, 7 and 8, 10, 12, 15.
- $Q_1 = (4 + 5) \div 2 = 4.5$ and $Q_3 = (10 + 12) \div 2 = 11$.
- IQR $= 11 - 4.5 = 6.5$.
Answer: Range 13, IQR 6.5
Worked example: An odd number of values
Find the IQR of 3, 5, 6, 8, 9, 11, 20.
- The median is 8, the middle value. Leave it out of both halves.
- Lower half 3, 5, 6: $Q_1 = 5$. Upper half 9, 11, 20: $Q_3 = 11$.
- IQR $= 11 - 5 = 6$. (The range is $20 - 3 = 17$, stretched by the 20.)
Answer: IQR 6
Common mistake
Forgetting to order the data first, or putting the median into one of the halves when there is an odd number of values. Math Ladder leaves the middle value out of both halves, as many U.S. courses do. Quick check: the two halves should have the same number of values.