Statistical landmarks are the handful of numbers that summarise a data set. They are quick to compute, and the interesting question is never how but which: an average that includes one enormous value can describe a group that contains nobody at all.
The six landmarks
| Landmark | How to find it | Value here |
|---|---|---|
| Minimum | smallest value | 3 |
| Maximum | largest value | 14 |
| Range | maximum minus minimum | 11 |
| Mode | value appearing most often | 7 |
| Median | middle value once sorted | 7 |
| Mean | add them all, divide by how many | 8 |
The mean, worked out
3 + 5 + 7 + 7 + 8 + 12 + 14 = 56 56 / 7 = 8
Finding the median
Sort the values first, always. With an odd count the median is the middle value. With an even count it is the mean of the two middle values.
Median of an even sized set
data: 2, 5, 6, 9 the two middle values are 5 and 6 (5 + 6) / 2 = 5.5
Which one to use
The mean uses every value, which is a strength and a weakness. One extreme value drags it a long way, and then it stops describing a typical case.
Seven salaries in a small firm, in thousands: 31, 33, 34, 36, 38, 40, 299.
The mean is 73. The median is 36. Six of the seven people earn less than the mean, so the median is the honest summary of what a person there earns.
| Use | When |
|---|---|
| Mean | values are fairly even, and every value should count |
| Median | there are extreme values, as with incomes or house prices |
| Mode | the data is categories rather than numbers, such as favourite colour |
| Range | you want to describe spread, not a typical value |
Common mistakes
Taking the median without sorting
The middle of the list as written means nothing. Sort first.
Assuming there is exactly one mode
A set can have two modes, or none at all when every value appears once.
Saying average when you mean mean
In everyday speech average usually means the mean, but median and mode are averages too. In a statistics answer, name the one you used.
Check yourself
Find all six landmarks for 4, 4, 6, 10, 11.
Minimum 4, maximum 11, range 7, mode 4, median 6, mean 35 / 5 = 7.
Median of 12, 3, 8, 5.
Sorted: 3, 5, 8, 12. The middle two are 5 and 8, so the median is 6.5.
A set has a mean of 20 and five values, four of which are 15, 18, 22 and 25. Find the fifth.
The total must be 100. The four given add to 80, so the fifth is 20.
House prices in a street are mostly near 200 thousand, with one mansion at 4 million. Which summary should a buyer be given?
The median. The mean is pulled far above anything actually for sale on that street.
See also
Sources
- Statistical landmarks as taught in United States middle grades mathematics. Data on this page is invented for illustration.