0580

Averages and Range

Statistics

Splitting the data into quarters

  • The median divides ordered data in half; the quartiles divide it into quarters
  • Split the ordered data in half at the median: the middle of the bottom half is the lower quartile, and the middle of the top half is the upper quartile
  • Where the number of values is odd, leave the median itself out of both halves
  • The interquartile range is the upper quartile minus the lower quartile
  • The IQR measures the spread of the middle half of the data, so it is unaffected by extreme values, which is exactly why it is preferred to the range when one is present
Worked example

Quartiles and interquartile range

Find the median, the quartiles, the range and the interquartile range of 4, 6, 7, 9, 11, 12, 14, 15, 18, 20, 23, 28.

Solution:

  • The data is already in order and there are 12 values
  • The median lies between the 6th and 7th: (12 + 14) ÷ 2 = 13
  • The lower half is 4, 6, 7, 9, 11, 12, whose median is (7 + 9) ÷ 2 = 8, the lower quartile
  • The upper half is 14, 15, 18, 20, 23, 28, whose median is (18 + 20) ÷ 2 = 19, the upper quartile
  • Range = 28 − 4 = 24
  • Interquartile range = 19 − 8 = 11
  • The IQR is much smaller than the range, since it ignores the extremes at both ends