Note · Fundamental concepts
Mean, median and mode: which average to use, and why
The mean adds everything up and divides, the median is the middle value, and the mode is the most common. One outlier can drag the mean far away while the median stays put.
Comments marked the Graphite mark are Graphite’s.
Graphite’s comments.
Contents · 6 sections
The mean, median and mode are three kinds of average, three ways to sum up a list of numbers with one typical value. The mean adds the numbers up and divides by how many there are. The median is the middle value once they are sorted. The mode is the value that appears most often. For evenly spread data they come out close together; when a few values are extreme, they can tell very different stories.
In short
- Mean: add up and divide. It uses every value, so one extreme value can drag it a long way.
- Median: the middle of the sorted list. Half the values are below it and half above, however extreme the ends are.
- Mode: the most common value. It is the only one of the three that works for categories with no numbers, such as the most popular color or blood type.
How do you calculate each one?
Take one small data set: the number of books seven friends read last year.
2, 3, 3, 5, 7, 8, 14
- Mean. Add them: 2 + 3 + 3 + 5 + 7 + 8 + 14 = 42. There are 7 values, so the mean is 42 ÷ 7 = 6.
- Median. The list is already sorted. With 7 values, the middle one is the 4th: 5. Three values sit below it and three above.
- Mode. The value 3 appears twice and every other value once, so the mode is 3.
Three averages, three answers: 6, 5 and 3. None is wrong. They answer different questions: what each friend would read if the total were shared out equally (6), what the friend in the middle read (5), and what the most common number was (3).
Two details trip people up. When the list has an even number of values, there is no single middle, so the median is the mean of the two middle values: for 2, 3, 5, 9 it is (3 + 5) ÷ 2 = 4. And a list can have no mode, if every value appears once, or several, if two values tie for most common. In symbols, the mean of n values x₁ to xₙ is written x̄ = (x₁ + x₂ + … + xₙ) ÷ n, read "x bar"; the NIST/SEMATECH e-Handbook of Statistical Methods gives the formal definitions of all three.
What happens when a billionaire walks into a bar?
Six people are in a bar. Their net worths, the value of what each owns minus what each owes, are:
| Person | Net worth |
|---|---|
| A | $20,000 |
| B | $45,000 |
| C | $80,000 |
| D | $80,000 |
| E | $150,000 |
| F | $225,000 |
Their total is $600,000, so the mean is $600,000 ÷ 6 = $100,000. With six values, the median is the mean of the 3rd and 4th, both $80,000, so it is $80,000. The mode is also $80,000.
Now a billionaire walks in, worth $1,000,000,000. Redo the sums with seven people:
| Average | Before | After the billionaire arrives | Change |
|---|---|---|---|
| Mean | $100,000 | $1,000,600,000 ÷ 7 = about $142.9 million | about 1,430 times larger |
| Median | $80,000 | the 4th of 7 sorted values: $80,000 | none |
| Mode | $80,000 | $80,000 | none |
Nobody in the bar got a cent richer, yet the mean says the typical person there is now worth $143 million. The mean is pulled toward extreme values; the median ignores how extreme they are and only counts how many sit on each side. Statisticians call this robustness: the median is a robust average, the mean is not. Note that six of the seven people are now below the mean, which is a good sign that the mean has stopped describing anyone.
Why is household income reported as a median?
Because incomes look like the bar, only bigger. Most households earn moderate amounts and a small number earn very large ones, so the distribution has a long tail to the right. In a right-skewed distribution like that, the mean is usually above the median, as OpenStax's statistics textbook explains, because the mean gets pulled toward the tail.
The real numbers show the gap. In its report on income in 2024, published in September 2025, the US Census Bureau put median household income at $83,730. Its appendix table for all 134.8 million households puts mean household income at $121,000, about 45% higher. The median household, the one with half of households above it and half below, is the better picture of a typical household. The mean answers a different question: how much each household would have if total income were shared out equally.
The same logic explains why house prices are usually quoted as medians, and why the gap between mean and median is itself a rough sign of inequality. A fuller measure of how unequally income is spread is the Gini coefficient. Figures like GDP per person are means, which is worth remembering when a country's average looks high but most people's incomes don't.
Which average should you use?
A quick way to choose:
- Are the values categories, not numbers? Favorite color, blood type, the brand people buy most: use the mode. You can't add up colors.
- Could a few extreme values distort the picture? Incomes, wealth, house prices, how long patients wait, how many followers accounts have: use the median, and give the mean too if the total matters.
- Is the data roughly symmetric with no wild outliers? Test scores in a large class, adult heights, repeated measurements of the same thing: use the mean. Here the three averages nearly agree, and the mean uses every value, which makes it the most precise and the easiest to combine with other calculations.
- Do you care about the total? If you are budgeting, splitting a bill or working out a team's total output, the mean is the one that multiplies back to the total: mean × count = sum.
- Report the spread as well. An average alone hides how far values stray from it. That is what standard deviation measures, and for symmetric, bell-shaped data, the kind described by the normal distribution, the mean and standard deviation together describe the whole shape.
What do people get wrong about averages?
The most common mistake is to hear "the average" and picture the middle person. In everyday speech "average" usually means the mean, and for skewed data the mean is not where the middle person is. When a news story says the average salary at a company rose, it could be that every worker got a raise, or that one executive's pay doubled. Asking "mean or median?" is often the single most useful question to put to a statistic.
A second mistake is averaging averages. If one class of 10 students averages 80 on a test and another class of 30 averages 60, the average of all 40 students is not 70. It is (10 × 80 + 30 × 60) ÷ 40 = 2,600 ÷ 40 = 65, because the larger class counts for more. This is a weighted mean: each group's mean is weighted by its size.
A third is to treat the mean as a value that must exist. The mean number of children per family can be 1.8 though no family has 1.8 children. The mode is always a value that actually occurs, and so is the median of an odd number of values, which is one reason they are easier to picture; the median of an even number can fall between two values, such as 1.5 children. All three averages, like probabilities, are ways of describing many cases with a single number, and each throws something away.
Check yourself
- Without looking back, explain why the billionaire changed the mean but not the median.
- Why does the Census Bureau's mean household income come out higher than its median?
- Two classes average 80 and 60. Why might the average of all their students not be 70?
Sources
- Income in the United States: 2024 — US Census Bureau, 2025
- Measures of Location, in the e-Handbook of Statistical Methods — NIST/SEMATECH
- Measures of the Center of the Data, in Introductory Statistics 2e — OpenStax, Rice University
- Skewness and the Mean, Median, and Mode, in Introductory Statistics 2e — OpenStax, Rice University
- Median — Wikipedia