The range is the largest value minus the smallest. The interquartile range is the third quartile minus the first quartile, and it measures the spread of the middle half of the data.
This lesson opens dispersion of ungrouped data. The next lesson, finding variance and standard deviation, measures spread from the mean.
What is the quartile routine?
- Write the values in order, smallest first.
- Find the median Q2.
- Split the list into a lower half and an upper half.
- Q1 is the median of the lower half, and Q3 is the median of the upper half.
Worked example 1: an even number of values
Data: 12, 15, 9, 20, 18, 14, 11, 16. Sorted: 9, 11, 12, 14, 15, 16, 18, 20 (8 values).
The range is 20 − 9 = 11. The median is (14 + 15) ÷ 2 = 14.5. The lower half is 9, 11, 12, 14, so Q1 = (11 + 12) ÷ 2 = 11.5. The upper half is 15, 16, 18, 20, so Q3 = (16 + 18) ÷ 2 = 17.
The interquartile range is 17 − 11.5 = 5.5.
Worked example 2: an odd number of values
Data: 3, 5, 6, 8, 9, 12, 15 (7 values, already sorted). The median is the 4th value, 8.
Leaving the median out, the lower half is 3, 5, 6 and Q1 = 5. The upper half is 9, 12, 15 and Q3 = 12. The range is 15 − 3 = 12 and the interquartile range is 12 − 5 = 7.
When one value is extreme
Change the 15 to 45 in the second example: 3, 5, 6, 8, 9, 12, 45. The range becomes 42, a jump of 30. The quartiles are still Q1 = 5 and Q3 = 12, so the interquartile range is still 7.
This is why the interquartile range is preferred when an extreme value is present. The range reacts to the single large value, and the interquartile range ignores it.
The slips to avoid
The first slip is forgetting to sort, which makes the median wrong and everything after it. The second is taking Q1 as a quarter of the largest value instead of the median of the lower half.
A third is finding the interquartile range as Q3 + Q1. Write the subtraction on the page before filling in the numbers. Keep a note of the slip you make in the mistake log and paper-error review.
Check yourself
Find the range and interquartile range of 7, 2, 10, 4, 9, 5.
Answer
Sorted: 2, 4, 5, 7, 9, 10 (6 values). Range = 10 − 2 = 8.
Median = (5 + 7) ÷ 2 = 6. Lower half 2, 4, 5 gives Q1 = 4. Upper half 7, 9, 10 gives Q3 = 9.
Interquartile range = 9 − 4 = 5.
What to study next
Continue with finding variance and standard deviation. Check your quartiles on invented data using the descriptive statistics explorer.
For a teacher to go through both the even and odd cases with you, see online one-to-one Mathematics tuition.