Mean, median and mode: how to calculate and when to use them
Summing up a list of numbers in one value seems simple, but the choice of that value changes the story. The mean, the median and the mode can give very different answers for the same data.
The mean
mean = sum of the values ÷ number of values
Example: 12, 15, 9 and 14
- 12 + 15 + 9 + 14 = 50
- 50 ÷ 4 = 12.5
The mean is 12.5
The median
Sort the values. The median is the one in the middle. With an even number of values, take the mean of the two central ones.
Example with an odd count: 3, 7, 8, 12, 30
- Already sorted
- The middle value is the third
The median is 8, while the mean is 12
Example with an even count: 4, 6, 9, 11
- The central values are 6 and 9
- (6 + 9) ÷ 2 = 7.5
The median is 7.5
The median resists extreme values. For salaries or house prices, a few very high figures pull the mean up but barely move the median.
The weighted average
When some values count more than others, multiply each by its weight, add the results, then divide by the sum of the weights.
Example: 14 with weight 3 and 10 with weight 1
- 14 × 3 + 10 × 1 = 52
- 3 + 1 = 4
- 52 ÷ 4 = 13
The weighted average is 13
Finding the score you need
Three tests at 12, 14 and 16. What is needed on the fourth for a mean of 15?
- Total needed: 15 × 4 = 60
- Already earned: 12 + 14 + 16 = 42
- 60 − 42 = 18
You need 18
Common pitfalls
- Forgetting to sort the values before taking the median.
- Averaging averages from groups of different sizes: that requires a weighted average.
- Dividing a weighted total by the number of values rather than by the sum of the weights.
Worked exercises
Try to find the answer first, then open the solution.
What is the mean of 6, 8 and 10?
Show the solution
- 6 + 8 + 10 = 24
- 24 ÷ 3 = 8
Answer: 8
What is the median of 2, 9, 4 and 7?
Show the solution
- Sorted: 2, 4, 7, 9
- (4 + 7) ÷ 2 = 5.5
Answer: 5.5
Calculate the weighted average of 16 (weight 2) and 11 (weight 3).
Show the solution
- 16 × 2 + 11 × 3 = 65
- 65 ÷ 5 = 13
Answer: 13