When someone says 'the average', they usually mean the mean, but there are at least three different averages and they can give very different answers for the same data. Choosing the wrong one is one of the easiest ways to be misled by a statistic, and one of the easiest ways to mislead. This guide explains each measure with a small example, shows when they disagree and introduces standard deviation, which describes how spread out the data is.
The mean
The mean adds up all the values and divides by how many there are. For the numbers 2, 4, 4, 4, 5, 5, 7, 9 the sum is 40 and there are 8 values, so the mean is 40 / 8 = 5.
The mean uses every value, which makes it a good summary when the data are fairly balanced. But it also means one extreme value can move it a lot.
The median
The median is the middle value when the numbers are put in order. With an odd number of values, it is the one in the middle. With an even number, it is the average of the two middle ones.
For 2, 4, 4, 4, 5, 5, 7, 9 the two middle values are 4 and 5, so the median is 4.5.
Because the median depends only on position, a very large or very small value at either end does not change it. That makes it a better summary for skewed data such as salaries or house prices.
The mode
The mode is the value that appears most often. In the same list, 4 appears three times and every other value fewer, so the mode is 4. A data set can have no mode, one mode or several.
The mode is the only one of the three that works for categories that are not numbers, such as the most common shoe size or the most popular colour.
When they disagree: salaries
Take seven monthly salaries: 30, 32, 35, 38, 40, 45 and 250 (in thousands). The sum is 470, so the mean is 470 / 7 = about 67. The median is the fourth value, 38.
If a company advertises that 'the average salary is 67,000', that is technically true, but six of the seven people earn less than that. The one very high salary pulls the mean up. The median of 38,000 describes the typical employee far better. Whenever a distribution has a long tail on one side, the mean and the median will differ, and the gap itself is a sign of skew.
Standard deviation: measuring spread
Two data sets can have the same mean and look nothing alike. Standard deviation measures how far the values typically sit from the mean.
Steps: find the mean; subtract it from each value and square the result; add up the squares; divide by the number of values (for a whole population) or by one less than that (for a sample); then take the square root.
For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5. The squared differences are 9, 1, 1, 1, 0, 0, 4 and 16, which add up to 32. For a whole population the variance is 32 / 8 = 4 and the standard deviation is 2. If the numbers are a sample from a larger group, divide by 7 instead: the variance is about 4.57 and the standard deviation is about 2.14.
What standard deviation tells you
A small standard deviation means values cluster close to the mean. A large one means they are spread out. Two classes may both average 70 on a test, but if one has a standard deviation of 5 and the other 20, the second has far more variation between students.
For data that follow a bell-shaped (normal) distribution, about 68 percent of values fall within one standard deviation of the mean, about 95 percent within two and about 99.7 percent within three. That rule is an approximation and does not hold for strongly skewed data.
Choosing the right measure
Use the mean for symmetric data without extreme values, such as heights of adults or measurement errors. Use the median when the data are skewed or when outliers are possible, such as income, rent or waiting times. Use the mode for categories or when you want the most common outcome. Report the spread with it, because an average alone hides how much the values vary.
If a number seems too good, ask which average it is and what the data look like. A median and a mean that are far apart are a signal to look at the distribution.
Frequently asked questions
Why divide by n − 1 for a sample? A sample tends to underestimate the spread of the full population, and dividing by n − 1 corrects most of that bias. This is called Bessel's correction.
Can the mean be a value that is not in the data? Yes. The mean of 1 and 2 is 1.5, which is not in the data.
Is the median always better than the mean? No. For well-behaved, symmetric data the mean uses more information and is usually preferred.
What is an outlier? A value far from the rest. It is worth checking whether it is a mistake or a genuine, important observation before deciding to keep or remove it.
Check your own data
Use our statistics calculators to find the mean, median, mode and standard deviation of your own list of numbers, and compare them. Looking at all of them together tells you more than any single average.