Math
Average calculator
Three different averages that answer three different questions. Which one is right depends on the shape of the data.
Three averages, three questions
| Measure | Question it answers |
|---|---|
| Mean | If the total were shared equally, how much each? |
| Median | What is the typical value in the middle? |
| Mode | Which value occurs most often? |
The mean uses every value, which makes it efficient and sensitive to outliers in equal measure. The median only cares about position, so a single extreme value cannot drag it. The mode is the only one that works on categories rather than numbers.
When the mean misleads
Income is the standard example. Add one billionaire to a room of a hundred ordinary earners and the mean income jumps enormously while the median barely moves. The mean is now technically correct and practically useless as a description of what people in the room earn.
This is why income and house price statistics are usually reported as medians. Any distribution with a long tail — wealth, waiting times, city populations, response times — has this property, and the median is the more honest summary.
A useful diagnostic: if the mean sits well above the median, the data is skewed to the right by large values. If it sits well below, there is a tail of small ones.
Quartiles and the interquartile range
Q1 is the value below which a quarter of the data falls; Q3 is the three-quarter point. The gap between them, the interquartile range, describes the spread of the middle half while ignoring the extremes entirely.
It is the basis of the standard outlier rule: values more than 1.5 × IQR below Q1 or above Q3 are conventionally flagged as outliers. That is a convention rather than a law — it flags roughly 0.7% of a normal distribution — but it is a reasonable first pass.
The geometric mean
For rates of change, the ordinary mean gives the wrong answer. An investment gaining 50% then losing 50% has an arithmetic mean return of 0%, but you finish with 75% of what you started with. The geometric mean — multiply the growth factors and take the nth root — correctly gives about −13.4% per period.
Use it for anything that compounds: returns, growth rates, ratios. Use the arithmetic mean for quantities that add, such as heights, weights and test scores.
Common questions
Can a data set have more than one mode?
Yes. If two or more values tie for the highest frequency, the set is bimodal or multimodal, and the calculator lists all of them. A genuinely bimodal distribution often means two different populations have been mixed together, which is worth investigating rather than averaging away.
What if no value repeats?
Then there is no mode. This is common in continuous measurements, where exact repeats are unlikely. For such data the mode is only meaningful after grouping the values into intervals.
How do I calculate a weighted average?
Multiply each value by its weight, add those products, and divide by the sum of the weights. This is what a GPA does, weighting each grade by credit hours, and what a portfolio return does, weighting each holding by its share.
Which average should I report?
Report the median for skewed data and the mean for roughly symmetric data. Reporting both is better still — the gap between them is informative in itself, and hiding it is how misleading statistics get made.