What's the difference between mean, median and mode?
All three are types of average, but they measure different things: the mean is the total shared out evenly, the median is the middle value, and the mode is the most frequent. Understanding which one to use — and when — is one of the most practically useful bits of maths you'll ever learn.
The mean (also called the arithmetic average) is what most people picture when they hear the word "average". Add all the numbers together and divide by how many there are. Simple and widely understood, but easily skewed by extreme values.
The median is the middle value when all numbers are sorted in order. If there's an even count of numbers, it's the average of the two middle values. The median ignores outliers entirely — it only cares about position, not magnitude.
The mode is the value that appears most often. It's the only average that makes sense for non-numerical data (like the most popular shoe size in a shop), and it's particularly useful when you need to know what's most common rather than what's typical.
Why is the median often more useful than the mean?
Because it ignores outliers, the median often describes "typical" better than the mean — half the values sit above it and half below. Consider UK house prices: the Office for National Statistics reports a mean that is significantly higher than what most buyers actually pay, because a small number of multi-million-pound London properties drag the mean upward. The median gives a fairer picture of what a typical buyer pays.
The same logic applies to salaries. When the ONS reports "average salary" in its Annual Survey of Hours and Earnings (ASHE), they use the median — not the mean — precisely because a small number of very high earners would inflate the mean far beyond what most workers actually take home. To see how a salary translates after tax, use the take-home pay calculator.
The mode matters most when you're looking at categories or frequencies. A shoe retailer doesn't need to know the mean shoe size of their customers — they need to know the most popular size so they know what stock to order. If you need to express one of these figures as a proportion of a total, the percentage calculator handles that in one step.
What do mean, median and mode look like for a real data set?
Take eight test scores: the mean (74.125) and median (74.5) come out close because the data is fairly symmetric, while the mode (71) picks out the only repeated value.
Example data set
Eight students sat a test. Their scores were: 45, 62, 71, 71, 78, 82, 89, 95
Mean: (45 + 62 + 71 + 71 + 78 + 82 + 89 + 95) ÷ 8 = 593 ÷ 8 = 74.125
Median: Sorted data is already in order. With 8 values, the median is the average of the 4th and 5th values: (71 + 78) ÷ 2 = 74.5
Mode: 71 appears twice; all other scores appear once. Mode = 71
Range: 95 − 45 = 50
Notice how the mean (74.125) and median (74.5) are close in this fairly symmetric dataset. In a dataset with extreme outliers, they would be much further apart — which is exactly when the choice between them matters most.
How do outliers affect your average?
A single outlier can pull the mean a long way while barely moving the median — the core reason to prefer the median for skewed data. The mean is sensitive to outliers and the median is not. Imagine you have the salaries of five colleagues: £25,000, £27,000, £28,000, £30,000, and £32,000. The mean is £28,400 and the median is £28,000 — very close. Now a sixth person joins the team on £500,000. The mean jumps to £107,000, which is wildly unrepresentative of what anyone actually earns. The median barely moves — it shifts to £29,000.
This is why the ONS always reports median pay when describing UK earnings. A handful of hedge fund managers and FTSE 100 chief executives would otherwise push the national mean far above what the typical worker takes home.
Which average should you use?
Use the mean for symmetric data, the median for skewed data with outliers, and the mode when you need the most common value. Use the mean when your data is reasonably symmetric and has no extreme outliers — for example, the heights of a group of adults, or the time it takes to complete a task. The mean uses every data point and is the most statistically powerful when used appropriately.
Use the median when your data has outliers or is skewed — house prices, salaries, wealth, waiting times. It gives a better sense of what's "typical" in the dataset.
Use the mode when you need to know the most common value — most popular product size, most frequent response in a survey, most common number of items in an order.
What is standard deviation in simple terms?
Standard deviation tells you how spread out the numbers are from the mean — low means tightly clustered, high means widely scattered. The standard deviation is exactly that measure of spread. A low standard deviation means the values cluster tightly around the mean — they're similar to each other. A high standard deviation means the values are widely scattered. This calculator uses the population standard deviation (σ), which is appropriate when your numbers represent the complete dataset rather than a sample from a larger population.
For example, in the test scores dataset (45–95), the standard deviation is around 15.7. This means most scores fall within about 15–16 marks of the mean of 74.125. If the standard deviation were 2, you'd know the scores were all clustered very closely together — a much more consistent class.
What does the range tell you?
The range is the highest value minus the lowest — the quickest measure of spread, but very sensitive to a single extreme value. The range is simply the highest value minus the lowest value. It's the quickest way to describe how spread out the extreme values are. However, range is very sensitive to outliers — a single unusually high or low value will change it dramatically, even if all the other values are clustered together. For a more robust measure of spread, standard deviation is usually more informative.
What does UK ONS data show about mean vs median pay?
ONS earnings data consistently shows median pay below mean pay, because high earners pull the mean upward — the gap between them is a rough measure of income skew. The ONS Annual Survey of Hours and Earnings (ASHE) consistently shows this meaningful gap between median and mean pay in the UK.
In 2024, full-time employees earned a median of around £37,500 per year, but the mean was noticeably higher — because high earners pull the mean upward. The spread between these two figures tells you something important about income inequality: the wider the gap between median and mean, the more skewed the distribution is toward the top.
A worked example: one big house on a small street
Six houses on a street sold this year for £240,000, £255,000, £260,000, £270,000, £285,000 and £1,200,000. The mean is the total, £2,510,000, divided by six: about £418,300. The median is the middle of the sorted list, halfway between £260,000 and £270,000: £265,000.
The mean is more than £150,000 higher than five of the six sales, because one large house pulls it up. The median is much closer to what a typical house on the street actually costs. That's why house prices and incomes are usually reported as medians: a few very high values can make the mean misleading.
When the mean is right
The mean is the right choice when every value matters to the total, for example when working out average monthly spending to set a budget, or the average score across equally weighted tests. If the groups are different sizes, use a weighted mean, which multiplies each value by its weight before dividing. The mode, the most common value, is most useful for categories, such as the most common shoe size or the most popular product.