One extreme value raises the mean because the mean uses every value’s size. It leaves the median alone, because the median uses only the middle position.
This page is part of choosing the right summary of a data set. It follows equal means but different spread.
Worked example: five daily sales
A stall sells 12, 13, 14, 15 and 16 cups of tea on five days. The sum is 70, so the mean is 14. The median is the middle value, 14.
Now suppose a school event lifts the last day from 16 cups to 46, so the data becomes 12, 13, 14, 15, 46.
- New sum: 12 + 13 + 14 + 15 + 46 = 100, so the new mean is 100 ÷ 5 = 20.
- New median: the ordered list is still 12, 13, 14, 15, 46, and the middle value is 14.
The mean moved by 6, from 14 to 20. The median did not move at all.
Which summary is fairer?
Four of the five days had sales between 12 and 15. A mean of 20 suggests a typical day of 20 cups, which never happened on any of those four days.
The median of 14 describes the typical day well. The 46 is real, so keep it, and note that it was a special event. A good answer names the median as the fairer centre and gives this reason.
What happens to the spread?
The range jumps from 4 to 34. The standard deviation also jumps, because 46 is far from the mean of 20 and its square is large.
The interquartile range copes better with an extreme value in a larger data set. With only five values it moves too: for 12, 13, 14, 15, 46 the quartiles are Q1 = 12.5 and Q3 = 30.5, so it rises from 3 to 18.
So for very small sets, say that the median resists the outlier, and be careful about claiming the same for the spread.
A rule for choosing
Use this decision rule. If the values are close together with no extreme value, the mean and standard deviation are good choices. If there is an extreme value, prefer the median and the interquartile range.
Always check the ordered list before choosing. Practise the choice on invented sets using the descriptive statistics explorer.
Check yourself
The marks 5, 6, 7, 8, 9 have mean 7 and median 7. The last mark is changed to 34. Find the new mean and median, and state which describes a typical mark better.
Answer
The new data is 5, 6, 7, 8, 34. The sum is 60, so the mean is 60 ÷ 5 = 12. The median is still 7.
The median describes a typical mark better, because four of the five marks are 8 or below, and the mean of 12 is pulled up by the single value of 34.
What to study next
Continue with reconstructing a missing observation from a mean and checking plausibility. Return to the cluster overview for the other pages.
For a teacher to question your choice of summary, see online one-to-one Mathematics tuition.