Unordered list
2, 14, 5, 9, 7 has 5 as its third entry, but 5 is not the median.
Before this page: compare numbers, place them in order, and count the values in a list.
The median is the middle value in order
The median divides an ordered list into two equal-sized parts. Put the values in ascending order before choosing the middle.
An odd list has one middle value
- Order the values from least to greatest.
- Count the same number of places from each end.
Median of 9, 4, 7, 12 and 5
Order the values, then select the third value of the five.
- 4, 5, 7, 9, 12The values are now in ascending order.
- 4, 5, 7, 9, 12Two values are before 7 and two are after it.
The median is 7. The original order does not affect it; only the ordered positions matter.
An even list has two middle values
With an even number of values, no single value has equal numbers of values on both sides. Take the mean of the two middle values.
Median of 3, 11, 8 and 6
Order the values, then average the second and third values.
- 3, 6, 8, 11The two middle values are 6 and 8.
- (6 + 8) ÷ 2 = 77 lies halfway between the two middle values.
The median is 7, even though 7 is not in the list. The halfway position can produce a value that was never measured.
Ordering the list matters
Counting positions in the original order can select the wrong value. The position only has meaning after the values have been sorted.
Ordered list
2, 5, 7, 9, 14 has 7 in the middle, so the median is 7.
Use the middle positions to check
For an odd list, the median has the same number of values on each side. For an even list, the two middle values have the same number of values on each side of the pair.
Check 3, 6, 8 and 11
There is one value before 6 and one after 8. The pair is centred in the ordered list, and its midpoint is 7.
Common mistakes
- Choosing the middle entry before ordering In 2, 14, 5, 9 and 7, the third entry is 5. Ordering gives 2, 5, 7, 9, 14, so the median is 7.
- Using one middle value in an even list For 3, 6, 8 and 11, the middle is the pair 6 and 8. Choosing 6 gives the wrong median; (6 + 8) ÷ 2 = 7.
- Adding all the values For 4, 5, 7, 9 and 12, the total 37 is not the median. The median is the middle ordered value, 7.
Summary
- Order the list before choosing a middle position.
- For an odd count, the median is the single middle value.
- For an even count, the median is the mean of the two middle values.
- Check that the chosen value or pair leaves equal numbers of values on each side.