Standard Deviation Calculator

Calculate the standard deviation, variance, and mean of a data set.

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Standard Deviation

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Variance—

Mean—

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How it works

Standard deviation is one of the most fundamental statistics, measuring how much individual values in a dataset deviate from the average. A low standard deviation indicates values cluster close to the mean; a high standard deviation signals wide spread. The choice between population and sample standard deviation depends on your data. If you have data for every member of a group (e.g., all students in a class), use population standard deviation. If your data is a sample drawn from a larger group (e.g., 50 students surveyed from all students in a country), use sample standard deviation. The sample formula uses n−1 in the denominator (Bessel's correction) to produce an unbiased estimate of the true population standard deviation. In practice, standard deviation is used everywhere: quality control (is a manufacturing process consistent?), finance (how volatile is a stock?), education (how spread are test scores?), and science (how precise are measurements?). Combined with the mean, standard deviation defines the normal distribution — roughly 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three (the 68-95-99.7 rule).

Formula

σ = √(Σ(x − μ)² / N) for population; s = √(Σ(x − x̄)² / (n−1)) for sample

Squaring each deviation (x − mean) removes negative signs so distances above and below the mean do not cancel, and it penalizes large gaps more heavily. Averaging those squares gives the variance, and taking the square root returns the answer to the original units. Sample data divides by n−1 instead of N because using the sample mean slightly understates spread, and Bessel's correction offsets that bias.

Source:Introductory Statistics, OpenStax

We review all our tools from time to time to ensure accuracy.

Worked example

Data Set:2, 4, 4, 4, 5, 5, 7, 9Type:population

For the dataset {2, 4, 4, 4, 5, 5, 7, 9}, the mean is 5. Subtracting the mean and squaring each value, then averaging those squares gives a variance of 4. The square root of 4 is 2, so the population standard deviation is 2.

How to use this calculator

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Frequently asked questions

What is the difference between population and sample standard deviation?

Population standard deviation (σ) is used when your dataset contains every member of the group you're studying. It divides the sum of squared deviations by N. Sample standard deviation (s) is used when your dataset is a subset of a larger population. It divides by n−1 (Bessel's correction) to avoid underestimating the true population standard deviation. In most real-world research scenarios, you're working with a sample, so use n−1.

What does a standard deviation of 0 mean?

A standard deviation of exactly zero means every value in your dataset is identical — there is no variation whatsoever. For example, the dataset {5, 5, 5, 5} has a standard deviation of 0. In practice, a very low standard deviation (close to zero) indicates very little variability in your data, which could mean a highly consistent manufacturing process, very similar test scores, or stable measurements.

How do I interpret standard deviation in real life?

Standard deviation tells you the 'typical' distance of data points from the average. For example, if test scores average 75 with a standard deviation of 10, most students scored between 65 and 85. In finance, a stock with high standard deviation is more volatile. In manufacturing, low standard deviation means consistent product quality. The 68-95-99.7 rule states that for normally distributed data, about 68% of values fall within one standard deviation of the mean.

Why is standard deviation preferred over variance?

Standard deviation is preferred because it's expressed in the same units as the original data, making it intuitively interpretable. Variance is expressed in squared units, which can be difficult to relate to the original measurements. For example, if you measure heights in centimeters, the variance would be in cm², but standard deviation remains in cm. Standard deviation is also more directly used in statistical tests, confidence intervals, and the normal distribution.

Can standard deviation be negative?

No, standard deviation can never be negative. It's calculated as the square root of variance, and variance is a sum of squared values (which are always non-negative) divided by a positive number. The minimum possible standard deviation is 0, which occurs only when all values in the dataset are identical. If you ever get a negative result, there's an error in the calculation — always double-check your inputs.