Standard Deviation Calculator

Standard deviation measures how spread out a set of numbers is around its mean. A small value means the data clusters tightly; a large one means it is dispersed. It is the most useful single companion to an average, because a mean on its own hides the difference between a consistent process and a wildly variable one that happens to average out. This calculator takes numbers separated by commas, spaces or new lines and returns both the sample and population standard deviation, along with the mean, variance and count. Which of the two you want depends on whether your numbers are the entire group you care about or a sample drawn from a larger one — a distinction that changes the divisor in the formula and is explained below, because choosing wrongly biases the result.

How this is calculated

Find the mean. Subtract it from each value and square the differences. Sum those squares. Divide by n for the population standard deviation, or by n − 1 for the sample. Take the square root.

Worked example on 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4, 16, summing to 32. Population variance is 32 ÷ 8 = 4, so the population standard deviation is 2. Sample variance is 32 ÷ 7 ≈ 4.571, giving a sample standard deviation of about 2.138.

The deviations are squared rather than taken as absolute values for two reasons: it prevents positive and negative deviations cancelling, and it makes the resulting measure differentiable, which is what allows standard deviation to underpin most of statistics.

Sample or population — which divisor to use

Use the population formula, dividing by n, only when your numbers are the complete group you are describing: every employee in a company, every item in a finished batch, all twelve months of a year.

Use the sample formula, dividing by n − 1, when your numbers are a subset used to estimate something about a larger group. This is the more common case, and it is the default in most statistical software.

The n − 1 adjustment is called Bessel's correction. It exists because a sample's own mean sits closer to the sample's values than the true population mean does, which systematically understates the spread. Dividing by a smaller number compensates. The effect is large for small samples and negligible for large ones — at n = 5 the correction raises the result by about 12%, at n = 100 by about 0.5%.

Reading a standard deviation

For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. This is the empirical rule, and it is what makes standard deviation intuitively meaningful rather than just a number.

It is expressed in the same units as your data, which makes it directly interpretable — a standard deviation of 2 kg means something concrete in a way that variance, in kilograms squared, does not. For comparing spread between datasets in different units, divide the standard deviation by the mean to get the coefficient of variation.

Standard deviation is sensitive to outliers, because squaring makes distant values dominate the sum. A single extreme value can inflate it substantially, which is worth checking before drawing conclusions from a spread that looks surprisingly large.

How to use the standard deviation calculator

Paste your numbers. Commas, spaces or new lines all work, so data copied from a spreadsheet column pastes directly.
Read both standard deviations. Sample and population are shown together. Pick the one matching whether your data is a subset or the whole group.
Check the mean and count. Both are shown alongside. If the count is not what you expected, a separator in your input was not parsed as you assumed.

Frequently asked questions

Sample vs population standard deviation?

Population (σ, dividing by n) when you have every member of the group; sample (s, dividing by n−1) when your data is a sample estimating a larger population — the n−1 corrects the bias.

Why divide by n − 1 instead of n?

This is Bessel's correction. A sample's own mean sits closer to its values than the true population mean does, which understates the real spread. Dividing by a smaller number compensates. The effect is about 12% at n = 5 and under 1% at n = 100.

What does a standard deviation of 2 actually mean?

It is in the same units as your data. For roughly bell-shaped data, about 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three — so a mean of 10 with a standard deviation of 2 puts most values between 6 and 14.

What is variance and how does it relate?

Variance is the standard deviation squared — the average squared deviation from the mean, before taking the root. It is mathematically convenient but hard to interpret, because its units are squared. Standard deviation exists to bring the measure back into the units of the original data.