Mean calculator that also tells you how much to trust the mean
Enter your numbers and get the sample mean, the sample standard deviation and the standard error of the mean, each with its working. The median is shown alongside, because the gap between the mean and the median is what tells you whether the average is a fair summary.
For variance, mode and range as well, the full statistics solver reports everything at once.
Add everything up, share it out equally
The arithmetic mean is the total divided by the count. It answers one specific question: if the total were shared out equally among all the values, how much would each get? That framing explains both its usefulness and its weakness.
The usefulness is that the mean carries the total. Multiply it by n and you recover the sum, which no other average does. If you need the total cost, the total mass, or the total anything, the mean is the average that connects to it.
The weakness is that every value pulls with its full weight, so one extreme number moves it a long way. Four salaries of about 33,000 and one of 900,000 give a mean of 206,600, a figure nobody in the room earns. The median handles that case better.
This page also reports the standard error of the mean, which most calculators leave out. The mean of a sample is an estimate of the population's mean, and the standard error says how far off that estimate is likely to be.
Sample mean
x̄ = Σx / n
Add every value, divide by how many there are. The bar over the x marks it as a sample mean rather than a population one.
Sample standard deviation
s = √(Σ(x − x̄)² / (n − 1))
Typical distance from the mean. The n − 1 rather than n is Bessel's correction, which keeps the estimate unbiased.
Standard error of the mean
SEM = s / √n
How far this sample's mean is likely to sit from the true population mean.
Why √n and not n
4× the data halves the error
Precision improves with the square root of the sample size, so gains get expensive quickly.
How to find the mean, and then how far to trust it
Include duplicates and negatives exactly as they appear. The sum is what the mean is built from, and one mis-keyed entry shifts everything downstream.
That count is n. The result is the sample mean, written x̄. For 4, 8, 10, 14 the sum is 36 and n is 4, so the mean is 9.
Subtract the mean from each value, square the differences, add them, divide by n − 1, and take the square root. Dividing by n − 1 rather than n corrects for having used the same data to estimate the mean.
SEM = s / √n. This is not how spread out the data is; it is how uncertain the mean itself is. The two are routinely confused and they answer different questions.
If the mean sits well above the median, the data has a long right tail and large values are pulling it. If the two agree closely, the distribution is roughly symmetric and the mean is a fair summary.
The mean, the spread, and the uncertainty
A straightforward mean
4, 8, 10, 14- Add: 4 + 8 + 10 + 14 = 36.
- Count: n = 4.
- Divide: 36 / 4 = 9.
- The median is also 9 here, so the data is symmetric and the mean describes it fairly.
Mean, standard deviation and standard error
2, 4, 4, 4, 5, 5, 7, 9- Sum = 40, n = 8, so the mean is 40 / 8 = 5.
- Deviations from the mean: −3, −1, −1, −1, 0, 0, 2, 4. Squared and summed: 32.
- Sample variance is 32 / 7 ≈ 4.571, so s ≈ 2.138.
- Standard error is 2.138 / √8 ≈ 0.756, so the true mean plausibly sits around 5 ± 1.5 at two standard errors.
When the mean misleads
30, 32, 35, 36, 900- Sum = 1033, n = 5, so the mean is 206.6.
- Four of the five values are below 40. The mean describes none of them.
- The median is 35, which does describe the middle of the group.
- The mean is not wrong. It correctly reports the total shared equally, and is simply answering a question nobody asked here.
Standard deviation and standard error are not the same
Standard deviation describes the data: how far a typical individual value sits from the mean. It does not shrink as you collect more data, because the spread of the population is whatever it is.
Standard error describes the estimate: how far your calculated mean is likely to sit from the true population mean. It does shrink as you collect more data, because more observations pin the average down. Quoting one where the other is meant is among the most common errors in reported statistics, and it makes results look far more precise than they are.
Which divisor to use
If your numbers are the entire population, divide the squared deviations by n. If they are a sample drawn from a larger population, divide by n − 1. Almost all real data is a sample, so n − 1 is the usual choice and the one this page uses.
The correction matters most when n is small: with 5 values, dividing by 4 instead of 5 raises the variance by a quarter. With 500 values the difference is negligible. The full statistics solver reports both so you can see the gap.
Mean calculator FAQ
How do you calculate the mean?
Add every value together and divide by how many values there are. For 4, 8, 10 and 14 the sum is 36 and the count is 4, so the mean is 9. Duplicates and negative values are included exactly as they appear.
What is the sample mean?
The mean of a sample rather than a whole population, written x̄. The arithmetic is identical; the name signals that it is an estimate of the population mean rather than the population mean itself, which is why a standard error is usually quoted with it.
What is the standard error of the mean?
SEM = s / √n, where s is the sample standard deviation and n is the sample size. It estimates how far your calculated mean is likely to be from the true population mean. It shrinks as you collect more data, but only with the square root, so quadrupling the sample halves the error.
What is the difference between standard deviation and standard error?
Standard deviation describes how spread out the individual data values are, and does not shrink with more data. Standard error describes how uncertain your estimate of the mean is, and does shrink with more data. Confusing the two makes results look far more precise than they are.
Why divide by n − 1 instead of n?
Because the mean you subtract was calculated from the same data, which makes the deviations slightly too small. Dividing by n − 1, called Bessel's correction, compensates and gives an unbiased estimate of the population variance. Use n only when your data is the entire population.
When should you use the median instead of the mean?
When the data is skewed or contains outliers. The mean of 30, 32, 35, 36 and 900 is 206.6, which describes none of the five values, while the median of 35 describes the middle one exactly. Incomes, house prices and waiting times are usually reported as medians for this reason.
Is this mean calculator free?
Yes, with no account and no limit. It shows the sum, the division, the sample standard deviation and the standard error, and reports the median alongside so you can see whether the data is skewed.
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