Mean, Median, and Mode
Each of the three measures of average answers a slightly different question about a dataset.
Mean (arithmetic average)
The mean is calculated by adding all values and dividing by the number of values. For the dataset {4, 7, 3, 9, 2}: add them (25), divide by 5 → mean = 5. The mean uses every value in the dataset and is the most commonly used of the averages. It is sensitive to extreme values — a single very large number can pull the mean far from the typical value.
Median
The median is the middle value when the dataset is arranged in order. For {2, 3, 4, 7, 9}: the median is 4 (middle value). For an even number of values, take the mean of the two middle values. The median is not affected by extreme values — making it better than the mean for skewed data like house prices.
Mode
The mode is the value that appears most often. For {3, 5, 5, 7, 8, 5}: the mode is 5. A dataset can have no mode (all values unique), one mode, or more than one mode (bimodal). The mode is the only average that works for non-numerical data — the most popular shoe size is a mode.
All three are measures of central tendency. Statistics uses them alongside measures of spread to fully describe datasets.
Range and When to Use Each Average
Choosing the right measure depends on the data and the question being asked.
Range
The range is not an average but is usually taught alongside them. It measures the spread of a dataset: largest value minus smallest value. For {2, 3, 4, 7, 9}: range = 9 − 2 = 7. A large range signals high variability; a small range indicates consistency.
When to use mean
Use the mean when data is roughly symmetrical and has no extreme outliers. Test scores, heights, and temperatures typically suit the mean. It makes full use of all the data.
When to use median
Use the median when data is skewed or has outliers. Average house prices, incomes, and waiting times are reported as medians in official statistics — because a few very high values would make the mean unrepresentative of typical experience.
When to use mode
Use the mode for categorical data or when you need the most common value. Retailers track modal shoe or clothing sizes. Election results — which candidate won most votes — is a mode question.
Averages in the Real World
Averages appear constantly in news, government reports, and everyday decisions — and are frequently misused.
Averages in statistics and data
Governments report average income, average house price, and average life expectancy. Understanding whether a reported average is the mean or median determines how to interpret it. The UK median household income and mean household income differ by tens of thousands of pounds — because the mean is pulled up by very high earners.
Averages in sport
Batting averages in cricket, goals per game in football, and points per match all use the mean. Comparing averages across different conditions requires care — a player's mean goals per game means different things in different competitions.
Being misled by averages
Averages can conceal important information. 'Average temperature' in a city tells you nothing about extremes. 'Average salary' may disguise inequality. The statistician's joke — 'put your head in an oven and feet in ice; on average you are comfortable' — captures the limitation. Always ask which average is being used and what it hides. Probability and statistics together build the full toolkit for interpreting data.
Frequently asked questions
- Why does the news often report median house prices instead of mean?
- House prices are skewed — a few extremely expensive properties pull the mean upward. The median is the middle price when all houses are ranked. It better represents what a typical buyer pays. The mean house price in London is higher than the median because a small number of multimillion-pound properties inflate it.
- Can you calculate the mean of non-numerical data?
- No. The mean requires adding values together, which only works for numbers. For categorical data — favourite colours, types of transport, letter grades — you can only calculate the mode (most common value). The median works for ordinal data — categories with a natural order, like satisfaction ratings.
- What is a weighted average?
- A weighted average gives different values different levels of importance. School grade averages are often weighted — coursework and exams may count differently. To calculate: multiply each value by its weight, sum the results, then divide by the total weight. Weighted averages are standard in finance, education, and survey analysis.
- What is the difference between average and mean?
- In everyday language, 'average' usually means the arithmetic mean. In mathematics, 'average' is a broader term covering mean, median, and mode — all measures of central tendency. When someone says 'the average score was 72', they almost certainly mean the mean. Context usually makes clear which is intended.