What Is the Mean, Median and Mode?

All three are called averages, which is the root of most confusion about them. They answer the same question, where is the middle, using three different definitions of middle, and on real data those definitions frequently disagree.

The mean

Add everything up and divide by how many there are.

For 4, 6, 7, 9, 14: the total is 40, there are five numbers, the mean is 8.

This is what most people mean by average. It uses every value, which is both its strength and its weakness: every number pulls on the result in proportion to its size, so one very large value moves it a long way.

The median

Sort the numbers and take the middle one.

For 4, 6, 7, 9, 14 sorted, the middle value is 7.

With an even count there is no single middle, so take the mean of the two central values. For 4, 6, 7, 9 the median is 6.5.

The median only cares about order, not magnitude. Changing that 14 to 140 leaves the median at 7 and moves the mean from 8 to 33.2. That insensitivity is the entire point of it.

The mode

The value that appears most often.

For 3, 7, 7, 7, 9, 12 the mode is 7.

A set can have two modes, making it bimodal, or none useful at all if every value appears exactly once. The mode is the only one of the three that works on categories rather than numbers: the most common eye colour, the most popular product size, the most frequent complaint type. A mean shoe colour is meaningless; a modal one is not.

Where they diverge

This is the part that matters in practice.

Take annual salaries at a small company: 24,000, 26,000, 28,000, 30,000 and 400,000.

• Mean: 101,600
• Median: 28,000
• Mode: none, since every value appears once

The mean is higher than four of the five salaries. It is arithmetically correct and it describes nobody in the building. Anyone told the average salary is 101,600 would form a badly wrong impression of what working there pays.

This is exactly why official statistics on income, house prices and wealth quote the median. Those distributions have a long tail at the top, and the mean is dragged along it.

The rule of thumb

Use the median when the data is skewed or contains extremes: income, house prices, waiting times, response times, session lengths, house sizes. Almost anything that cannot go below zero but has no upper limit is skewed this way, and that covers a great deal of real-world data.

Use the mean when values cluster reasonably symmetrically and the total genuinely matters: exam marks, heights, daily temperatures, measurement errors. The mean is also required for most further statistics, since standard deviation and most statistical tests are built on it.

Use the mode for categories, or when the single most common value is the actual question: the most common shoe size to stock, the most frequent cause of a fault, the most popular delivery slot.

How skew shows itself

A quick diagnostic that takes seconds: compare the mean and the median.

• Mean noticeably above median: the data has a tail to the right, meaning a few large values are pulling it up.
• Mean noticeably below median: a tail to the left, a few unusually small values.
• Mean and median close together: roughly symmetrical, and the mean is safe to use.

The size of the gap is itself a measure of skew. If someone reports a mean without a median for something obviously skewed, that gap is what they are not showing you.

The trap in reporting

Because all three are called the average, the word can be used entirely accurately while creating a false impression.

A statement that average earnings rose is compatible with most people earning less, if the rise was concentrated among high earners. A statement that the average house price is some figure tells you little about what a typical house costs in a market with a handful of very expensive properties.

Whenever an average appears in something that matters, two questions are worth asking: which average is this, and what does the other one say? A report giving both is usually being straight with you. One giving a mean for something obviously skewed, without comment, may not be.

Weighted means

One refinement worth knowing. A plain mean treats every value as equally important. A weighted mean does not.

If a module grade is 30 percent coursework and 70 percent exam, the mean of the two marks is wrong. The weighted mean multiplies each by its weight before adding: a 60 in coursework and 70 in the exam gives 0.3 × 60 plus 0.7 × 70, which is 67, not 65.

Anywhere components contribute unequally, from grades to portfolio returns to survey results with unequal sample sizes, the weighted mean is the correct one and the plain mean quietly misleads.

Beyond the middle

The middle alone never describes a dataset fully. Two sets can share a mean and a median and look nothing alike: 49, 50, 51 and 0, 50, 100 both have a mean and a median of 50.

What separates them is spread, measured by range, quartiles or standard deviation. A centre without a spread is half the picture, which is why any competent statistical summary reports both and why a single average in a headline should always prompt the question of what else was measured.