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Calculates

Average Calculator

Calculate mean, median, mode, range and standard deviation instantly.

Calculated & verified for mathematical accuracy·Reviewed by the Calculates Editorial Team

Your numbers

Separate values with commas, spaces, semicolons or line breaks. A column pasted from a spreadsheet works as-is.

Mean (average)

5

Sum of 40 shared across 8 values.

Count
8
Sum
40
Median
4.5
Mode
4
Minimum
2
Maximum
9
Range
7
Relative spread
42.76%

Standard deviation

Sample (÷ n − 1)
2.1381
Population (÷ n)
2

Use the population figure when these numbers are the entire group you care about. Use the sample figure when they are a subset standing in for something larger — it is always the higher of the two.

The mean (5) and median (4.5) differ noticeably, which means extreme values are pulling the mean. The median is the safer summary here.

About this calculator

Paste or type any list of numbers and this page returns the three things people mean by "average" — mean, median and mode — alongside the figures that describe how spread out the data is. They routinely disagree, and which one you should quote depends on whether your numbers are symmetric or dragged to one side by a few extreme values. Separate your numbers however you like: commas, spaces, tabs or one per line all work.

How to use this calculator

  1. Paste your numbers in

    A column copied straight out of a spreadsheet works without reformatting, as does a comma-separated line typed by hand. Mixing separators in the same list is fine.

  2. Check the count

    The count confirms how many values were actually read. Anything that was not a number is dropped and reported, so a stray label pasted along with the data cannot quietly shift your mean.

  3. Compare the three averages

    When the mean and median are close together, the data is roughly symmetric and either figure is representative. When they diverge, extreme values are pulling the mean and the median is the safer summary.

  4. Choose the right standard deviation

    Use the population figure when your list is the whole group you care about. Use the sample figure when the list is a subset you are using to draw conclusions about something larger.

Formula

Mean = Σx / n Median = middle value of the sorted list (mean of the two middle values if n is even) Mode = the value or values occurring most often Range = max − min Population SD = √( Σ(x − μ)² / n ) Sample SD = √( Σ(x − x̄)² / (n − 1) )
  • Σx is the total of every value and n is how many there are, so the mean is simply the total shared out equally.
  • The median requires the list to be sorted first; the order you typed the values in has no effect on it.
  • A list where nothing repeats has no mode at all, and a list can have several modes if two or more values tie for the highest frequency.
  • The (n − 1) denominator in the sample formula is Bessel’s correction. Deviations are measured from the sample mean rather than the unknown true mean, which understates the real spread, and dividing by one fewer compensates.
  • The sample standard deviation is therefore always the larger of the two, by exactly a factor of √(n / (n − 1)). With 100 values the difference is under half a percent; with five values it is over ten percent.
  • A single observation has no sample standard deviation, because n − 1 is zero and there is nothing to measure variation against.

Worked example

Inputs

  • Values: 2, 4, 4, 4, 5, 5, 7, 9

Calculation

n = 8 Σx = 40 Mean = 40 / 8 = 5 Sorted = 2, 4, 4, 4, 5, 5, 7, 9 Median = (4 + 5) / 2 = 4.5 Mode = 4 (appears three times) Range = 9 − 2 = 7 Squared deviations from the mean of 5: 9, 1, 1, 1, 0, 0, 4, 16 → Σ = 32 Population SD = √(32 / 8) = √4 = 2 Sample SD = √(32 / 7) = √4.571 = 2.1381

The mean is 5 and the median 4.5, close enough that either summarises the data fairly. The population standard deviation is exactly 2, while treating the same eight numbers as a sample raises it to 2.14 — the price of estimating the spread of a wider group from a small subset.

What the result means

  • The mean is the balance point of the data and uses every value, which makes it the most informative summary of a symmetric dataset and the most easily distorted one otherwise.
  • The median is the halfway mark: half your values fall below it. It barely moves when one figure is extreme, which is exactly why incomes and house prices are reported as medians rather than means.
  • The mode is the value you see most often. It is the only average that works on categories as well as numbers, and the only one that can legitimately have no answer.
  • The standard deviation says how far a typical value sits from the mean, in the same units as your data. Roughly two thirds of readings fall within one deviation of the mean when the data is bell-shaped, so a large figure relative to the mean signals that the average alone is hiding a lot.

Related calculations

  • Add one extreme value to a modest list and watch the mean move sharply while the median barely shifts — the clearest demonstration of why the two are reported differently.
  • Compare the sample and population deviations on five values, then on the same data repeated many times, to see the correction shrink as the count rises.
  • Divide the standard deviation by the mean to get the coefficient of variation, which lets you compare the relative spread of two datasets measured in different units.

Frequently asked questions

Which average should I actually use?

Use the mean when your values are roughly symmetric and every one of them should count equally, such as test scores or repeated measurements. Use the median when a few extreme values would distort the picture, which is why salary, property price and wait-time data are almost always reported as medians. Use the mode when you want the most common outcome, such as the shirt size you sell most of, where averaging would produce a size nobody wears.

What is the difference between sample and population standard deviation?

It is a question about what your numbers represent, not about precision. If your list is the complete group you care about — every employee in your company, every match played this season — it is a population and you divide by n. If it is a subset you are using to infer something about a larger group, it is a sample and you divide by n − 1 to correct for the fact that a subset always looks less varied than the whole. When unsure, the sample figure is the conservative choice because it is larger.

Why is the sample standard deviation always bigger?

Because the deviations are measured from the sample mean, which by construction sits at the centre of your particular subset rather than at the true centre of the wider population. That makes the squared deviations systematically too small. Dividing by n − 1 instead of n inflates the answer by exactly the right amount to remove the bias, and the effect fades as the sample grows — negligible at a thousand values, substantial at five.

Can a list of numbers have more than one mode?

Yes. When two or more values tie for the highest frequency, every one of them is a mode, and this calculator lists them all rather than arbitrarily picking the smallest. A dataset with two modes is called bimodal and often means you have accidentally combined two distinct groups — heights of adults of both sexes, for instance, or response times from two different servers.

What does it mean when my data has no mode?

It means no value appears more than once, which is common with continuous measurements recorded to several decimal places. The mode is genuinely undefined in that case, so an empty result is the honest answer rather than an error. If you need a sense of where values cluster, group them into bands first and find the band containing the most readings.

How is the median found when there is an even number of values?

The list is sorted and the two central values are averaged. For 10, 20, 30 and 40 the middle pair is 20 and 30, giving a median of 25 — a figure that need not appear in your data at all. With an odd count there is a single central value and no averaging is needed, which is why an odd-sized list always has a median that is one of its own members.

What counts as a large standard deviation?

There is no universal threshold, because the figure carries the same units as your data — a deviation of 5 is enormous for exam scores out of 10 and trivial for annual salaries. Compare it against the mean instead: the ratio of the two, expressed as a percentage, is the coefficient of variation and is unit-free. Below about 15% suggests a tightly clustered dataset, while anything above 50% means the mean tells you very little on its own.

Does it matter what order I enter the numbers in?

Not at all. The mean and standard deviation are sums, so order is irrelevant by construction, and the list is sorted internally before the median, minimum and maximum are read off. Paste your data in whatever order it happens to be in — the entire result set is identical either way.

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