Standard Deviation Calculator

Find the standard deviation and variance of a data set.

Standard Deviation Calculator

Enter your numbers separated by commas or spaces, choose population or sample, then select Calculate.

Numbers

Separate numbers with commas or spaces.

Type

Choose Population when your numbers are the entire group being studied (e.g. all students in one specific class); choose Sample when they're only part of a larger group being estimated (e.g. 10 students selected from a whole school).

How this is calculated

Standard deviation measures how spread out the numbers are from the mean.

Population formula:

σ = \sqrt{\frac{{Σ (x - μ)}^{2}}{N}}

Sample formula:

s = \sqrt{\frac{{Σ (x - x̄)}^{2}}{n - 1}}

Show calculation explanation

Calculate a result to see this explained with your own numbers.

Advanced: margin of error

Margin of error:

ME = \frac{1.96 \times σ}{\sqrt{n}}

This estimates how far the average might be from the true group average, at 95% confidence — a separate estimate, not part of the standard deviation calculation itself. Population uses 1.96; Sample uses a slightly larger Student's t value for small samples, converging toward 1.96 as the sample grows.

Examples

For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5, and the population standard deviation is exactly 2. For that same data set, the margin of error at 95% confidence is about 1.386 (population) or 1.788 (sample, using the larger Student's t critical value appropriate for this sample's size).

About this calculation

This uses the standard, universally taught population and sample standard deviation formulas from introductory statistics.

Frequently asked questions

What is standard deviation?

Standard deviation is a single number that summarizes how spread out a set of values is around their mean. A small standard deviation means the values cluster close to the mean; a large one means they're spread out further.

What is variance, and how is it different from standard deviation?

Variance measures spread by taking each value's difference from the mean, squaring it, and combining those squared differences into one number. Standard deviation is the square root of variance, which returns that spread measure to the data's original unit, making it easier to interpret directly.

What do the symbols in the formulas mean?

Σ means: add these up, e.g. 4 + 5 + 6 + 7 = 22. N: number of observations in the Population — not their sum. n: number of observations in the Sample — not their sum. μ: Population mean. x̄: Sample mean. σ²: Population variance. s²: Sample variance. σ: Population standard deviation. s: Sample standard deviation.

What is a population?

A population is every member of the specific group you're studying — for example, every student in one particular class. When your numbers cover the whole group, use "Population."

What is a sample?

A sample is a smaller group selected from a larger population to estimate something about that larger group — for example, 10 students picked from a school of 500 to estimate the whole school's average grade. When your numbers are a subset like this, use "Sample."

How do I know whether to choose population or sample?

Ask what your numbers represent, not how many there are. The same values can be either: if 4, 5, 6, 7 are the ages of every child in one specific family, that's a population — you already have everyone. If they're the ages of four students picked from a class of 30 to estimate the class's typical age, they're a sample.

Why does the sample formula use one less observation (n − 1) in its denominator?

A sample is being used to estimate the spread of a larger population. Dividing by n − 1 instead of n corrects for a sample's natural tendency to underestimate that spread. It also means at least 2 numbers are required for "Sample" — with only one, n − 1 would be zero.

What does "margin of error" mean here?

It's an estimate, at 95% confidence, of how far the average might be from the true average of the whole group. For Population, this multiplies the standard deviation by 1.96 — appropriate since the population's spread is exactly known. For Sample, it uses a slightly larger multiplier from the Student's t-distribution instead, sized to the sample's degrees of freedom (n − 1) and converging toward 1.96 as the sample grows, because a sample's standard deviation is only an estimate of the true spread. A smaller margin means the data more tightly pins down the average; it does not mean the data itself has less spread.

Standard Deviation Calculator