Before this page: calculate the mode, median and mean of a list.
The data decides which average is useful
An average represents a typical value, but different averages preserve different information. Choose the one that answers the question the data raises.
Choose mode for categories, median against extremes, and mean for balanced numerical data
- The mode identifies the most common category, but ignores how the other categories are distributed.
- The median resists an unusually small or large value, but uses only the middle position.
- The mean uses every value, but an extreme value can pull it away from the centre.
Use the mode for categories
Colours, transport types and other names cannot be added, ordered or divided. The mode still identifies the category with the greatest frequency.
Most common travel method
A survey records bus, walk, bus, train, bus and walk.
- bus: three; walk: two; train: one.Bus has the greatest frequency.
- Mode = bus.The mode gives a meaningful category; a mean or median cannot.
Use the median when an extreme would distort the centre
The median depends on position, so one value far from the rest can pull the mean towards it while the median stays near the middle.
Typical house price in thousands
The prices are 120, 125, 130, 135 and 900.
- Median = 130130 is the middle ordered value.
- Mean = 1,410 ÷ 5 = 282The 900 pulls the mean far from the other four prices.
The median of 130 thousand better describes a typical price in this list because the 900-thousand property is far from the other four prices.
Use the mean when the total and every value matter
The mean includes every value and preserves the total through mean × count. It represents these five scores well because they are balanced around 74 and have no extreme value.
Mean score for five tests
The scores are 62, 68, 74, 80 and 86.
- 62 + 68 + 74 + 80 + 86 = 370Every score contributes to the total.
- 370 ÷ 5 = 74The mean score is 74.
Justify the choice with the data
State which average is most appropriate and justify your answer
needs a feature of the data
and the effect it has. Naming an average alone does not justify the choice.
A complete justification
Use the median for house prices because 900 thousand is an extreme value and pulls the mean to 282 thousand, while the median remains 130 thousand.
A claim that the mean is always best because it uses every value is incomplete. Using 900 is an advantage when every pound must count, but a drawback when the aim is to describe a typical house in this list.
Common mistakes
- Calculating a mean for categories Bus, walk, bus, train, bus and walk has mode bus. Adding category names has no mathematical meaning.
- Choosing the mean beside an extreme value For 120, 125, 130, 135 and 900, the mean is 282 thousand and the median is 130 thousand. The extreme price makes the median more representative.
- Choosing the mode because it is always easiest The scores 62, 68, 74, 80 and 86 have no repeated value, but their mean is 74 and uses every score.
Summary
- Use the mode for categories or when the most common value matters.
- Use the median when an extreme value would distort the mean.
- Use the mean for numerical data when every value and the total should affect the result.
- Justify an average by connecting its advantage or drawback to the data and the question.