Mean, Median, Mode and Range
How to calculate and choose between the common measures of average and spread.
Averages summarise a set of numbers with one value. Each type of average answers a different question.
Definitions
- Mean: add all values, then divide by how many there are.
- Median: the middle value when the data is in order. If there are two middle values, take their mean.
- Mode: the most frequent value.
- Range: largest value minus smallest value.
Worked example
Data: 4, 7, 7, 9, 13.
- Mean: (4 + 7 + 7 + 9 + 13) ÷ 5 = 40 ÷ 5 = 8.
- Median: the middle of five ordered values is the third, 7.
- Mode: 7 appears twice, so the mode is 7.
- Range: 13 − 4 = 9.
Even number of values
Data: 2, 5, 6, 10.
- Median: the middle two values are 5 and 6, so the median is (5 + 6) ÷ 2 = 5.5.
Which average to use
- Use the mean when values are fairly symmetric and no extreme values distort the data.
- Use the median when there are extreme values (outliers). For example, in incomes, a few very high values can raise the mean while the median stays typical.
- Use the mode for categories, such as the most common shoe size.
Effect of an outlier
Data: 3, 4, 4, 5, 40.
- Mean = 56 ÷ 5 = 11.2, which is larger than four of the five values.
- Median = 4, which describes the typical value better.
Common mistakes
- Forgetting to order the data before finding the median.
- Dividing by the wrong number when finding the mean.
- Saying "no mode" when more than one value is tied. A data set can have more than one mode.
Quick recap
Mean is the balance point, median is the middle, mode is the most common, and range measures spread.