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Formula
Mean = (x₁ + x₂ + … + xₙ) ÷ n
- x₁ … xₙ
- the values in your list
- n
- how many values there are
Mean, median and mode
“Average” usually means the mean: add every value and divide by how many there are. But there are three common ways to describe the center of a set of numbers, and each answers a slightly different question.
- Mean. The total shared out equally. If the values were poured together and split evenly, each would get the mean.
- Median. The middle value once the list is sorted. Half the values are at or below it, and half are at or above.
- Mode. The value that occurs most often. It’s the only one of the three that also works for categories, such as the most common shoe size sold.
How to use the calculator
- Type or paste your numbers into the box. Commas, spaces, semicolons and line breaks all work as separators.
- Remove thousands separators first. Type 1250, not 1,250.
- Read the mean at the top and the other statistics below it. The working line shows the sum divided by the count.
If you’re comparing two averages, such as this year’s mean score against last year’s, the percentage change calculator expresses the gap as a percentage.
How to calculate an average by hand
Take the list 9, 4, 7, 13, 7.
- Sort it: 4, 7, 7, 9, 13. Sorting first makes the median, mode and range easy to read off.
- Mean: add the values, 4 + 7 + 7 + 9 + 13 = 40, then divide by the count of 5. The mean is 8.
- Median: with 5 values, the middle one is the 3rd. The median is 7.
- Mode: 7 appears twice and every other value once, so the mode is 7.
- Range: the largest minus the smallest, 13 − 4 = 9.
For an even number of values there’s no single middle value, so average the two middle ones. That’s the only extra step.
Why the mean and median can differ
The mean takes the size of every value into account, so a single extreme value can move it a long way. The median only cares about order, so it barely moves.
Take the list 2, 3, 3, 4 and 48. The sum is 60, so the mean is 60 ÷ 5 = 12, which is larger than four of the five values. The median is 3, which is much more typical of the list. When the mean is well above the median, the data has a few large values pulling it up. When the mean is below the median, a few small values are pulling it down.
The table shows how the three measures line up, or don’t, depending on the shape of the data.
| Data | Shape | Mean | Median | Mode |
|---|---|---|---|---|
| 10, 12, 14, 14, 16, 18 | Symmetric | 14 | 14 | 14 |
| 10, 11, 11, 12, 13, 45 | One high outlier (right-skewed) | 17 | 11.5 | 11 |
| 2, 15, 16, 17, 17, 18 | One low outlier (left-skewed) | 14.166667 | 16.5 | 17 |
In the symmetric list all three agree. A single high value drags the mean above everything except the outlier itself, while a single low value drags it below most of the list. The guide to mean, median and mode explains skew and outliers in more depth.
Spread: range, minimum and maximum
An average on its own doesn’t say how spread out the values are. The lists 49, 50, 51 and 0, 50, 100 both have a mean and median of 50, but they describe very different situations. The range, which is the maximum minus the minimum, gives a quick sense of that spread. The average calculator shows it alongside the minimum and maximum.
The range depends only on the two most extreme values, so a single outlier can inflate it. If you need a measure of spread that reflects every value, statisticians use the standard deviation, which this calculator doesn’t compute.
Which average to use
| Situation | Usually the right choice | Why |
|---|---|---|
| Test scores, temperatures, daily sales | Mean | Values are fairly balanced and the total matters |
| Incomes, house prices, wait times | Median | A few very large values would distort the mean |
| Shoe or clothing sizes, survey choices | Mode | You want the most common option, not a value in between |
| Ratings on a 1–5 scale | Median or mode | The gaps between ratings aren’t necessarily equal |
Averaging percentages
A list of percentages can be averaged like any other numbers. Enter 82, 91 and 76 without the % sign and the mean is 83, meaning 83%. That simple mean is right when each percentage comes from the same total, such as three tests each marked out of 100.
When the totals differ, a simple mean gives the small group too much weight. Add the raw counts instead, or use a weighted average with each group’s size as its weight. The pass-rate example under common mistakes below shows how large the gap can be.
Real-life uses
Averaging grades and test scores
Your grade in a class where every test counts equally is the mean of the scores, so you can paste the scores straight into the average calculator. If some components count more than others, such as a final exam worth half the grade, you need a weighted average instead (see the FAQ below). A median score is useful for a teacher who wants to know how a typical student did when a few students scored very low.
Incomes and home prices
Government statistics and real estate reports usually quote median household income and median sale price. A handful of very expensive sales can raise the mean price sharply while the median, the price of the home in the middle, stays representative of what most buyers paid. The second worked example below shows the effect with six incomes.
Business and personal tracking
Averaging daily sales, monthly electricity bills or weekly running distances smooths out day-to-day noise. Comparing the mean with the median tells you whether a few unusual days are distorting the picture.
Inventory and product choices
A shoe store ordering stock cares about the mode, the size that sells most, rather than a mean size of 8.857 that no customer wears.
Tips and common mistakes
- Averaging averages. If one class of 20 has a pass rate of 90% and another of 60 has a pass rate of 75%, the combined pass rate isn’t 82.5%. Out of 80 students, 63 passed, which is 78.75%. Combine the raw totals instead, or weight each average by its group size.
- Averaging speeds or rates. Driving the same distance at 30 mph and then at 60 mph gives an average speed of 40 mph, not 45, because you spend twice as long at the slower speed.
- Forgetting zeros. A day with zero sales is a real value. Leaving it out raises the mean.
- Reporting only the mean. For skewed data, give the median too, and mention the range so readers know how spread out the values are.
- Pasting formatted numbers. Currency symbols and thousands separators must be removed first, or the calculator will flag them.
Glossary
- Outlier: a value far from the rest of the data, such as 45 in the second row of the table above.
- Skew: a lopsided distribution with a long tail on one side. Right-skewed data has a tail of high values, and its mean sits above the median.
- Weighted average: an average where some values count more than others.
- Range: the gap between the largest and smallest values.
For percentages of a total rather than averages, use the percentage calculator. More tools are on the math calculators page.
Worked examples
Average of test scores
Six test scores: 72, 85, 85, 90, 64 and 78.
Inputs: Numbers = 72, 85, 85, 90, 64, 78
Mean = 474 ÷ 6 = 79
Answer: Mean 79, median 81.5, mode 85 (n = 6).
Household incomes with one outlier
Annual incomes of six households on one street, one of which is far higher than the rest.
Inputs: Numbers = 42000 48000 51000 55000 61000 250000
Mean = 507,000 ÷ 6 = 84,500
Answer: Mean 84,500, median 53,000, no mode (n = 6).
Shoe sizes with two modes
Shoe sizes of seven people, where two sizes tie for most common.
Inputs: Numbers = 7, 8, 8, 9, 9, 10, 11
Mean = 62 ÷ 7 = 8.857143
Answer: Mean 8.857143, median 9, modes 8, 9 (n = 7).
Assumptions and limitations
- Separate values with commas, spaces, semicolons or new lines. Don't use thousands separators. If an entry like 1,250 looks like one, the calculator asks you to remove the comma rather than guessing.
- Up to 10,000 values can be entered. Negative numbers and decimals are allowed.
- The mean, median, sum, range, minimum and maximum are shown to up to 6 decimal places.
- If no value appears more than once, there is no mode. If several values tie for the most appearances, all of them are listed as modes, from smallest to largest.
- The range is the maximum minus the minimum.
- Every value carries equal weight. The calculator doesn't compute weighted, geometric or harmonic averages.
Frequently asked questions
Should I use the mean or the median?
Use the mean when values are fairly evenly spread and every amount should count, such as total spending divided among people. Use the median when the data is skewed or has outliers, such as house prices or incomes, because one extreme value can pull the mean a long way.
How do you calculate an average?
Add all the values and divide by how many there are. For 12, 15, 15, 18 and 21 the sum is 81 and there are 5 values, so the average is 81 ÷ 5 = 16.2. In everyday use "average" means this arithmetic mean. The median and mode are also averages, because each describes a typical value, which is why the calculator shows all three.
How is the median found when there's an even number of values?
The values are sorted and the two in the middle are averaged. For 3, 5, 8 and 10, the middle pair is 5 and 8, so the median is 6.5.
Why does it say there's no mode?
The mode is the most frequent value. If every value appears exactly once, none is more common than the others, so there's no mode to report. The opposite can also happen. If two or more values tie for the most appearances, each is a mode, so 7, 8, 8, 9, 9, 10, 11 has two modes, 8 and 9.
How do you calculate an average percentage?
If every percentage is out of the same total, such as tests each marked out of 100, average them like any other numbers. If the totals differ, add the raw scores and divide by the combined total. Scores of 45 out of 50 (90%) and 60 out of 100 (60%) make 105 out of 150, which is 70%, not the 75% a simple average of the two percentages gives.
How do I calculate a weighted average?
Multiply each value by its weight, add the products, and divide by the total of the weights. If homework (20%) averages 92, quizzes (30%) average 81 and the final (50%) is 74, the grade is 0.2 × 92 + 0.3 × 81 + 0.5 × 74 = 79.7.
Can I paste a column from a spreadsheet?
Yes. Values copied from a single column arrive on separate lines, which the calculator reads as separate numbers. Make sure the cells aren't formatted with thousands separators or currency symbols first.