Paste or type your numbers separated by commas or spaces, choose sample or population, and get the standard deviation instantly — along with the mean, variance, and sum of squared deviations.
How the standard deviation calculator works
Standard deviation answers one question: on average, how far do the values sit from the mean? The calculator finds the mean of your numbers, measures each value's distance from it, squares those distances (so negatives don't cancel positives), averages the squares, and takes the square root to get back to your original units. That last step is what makes standard deviation so readable — if your data is in dollars, the spread is in dollars too.
The only decision you have to make is the toggle: population if your list is the whole group, sample if it's a slice of something bigger.
The formula
Population: σ = √( Σ(x − μ)² ÷ n )
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )
Here x is each value, μ (or x̄) is the mean, n is how many values you have, and Σ means "add them all up." The only difference between the two formulas is the divisor: n for a population, n − 1 for a sample.
Worked example
Take the data set 4, 8, 6, 5, 3, 7 (n = 6).
Mean = 33 ÷ 6 = 5.5. The deviations from the mean are −1.5, 2.5, 0.5, −0.5, −2.5, 1.5; squared and summed they give 2.25 + 6.25 + 0.25 + 0.25 + 6.25 + 2.25 = 17.5.
Sample: variance = 17.5 ÷ 5 = 3.5, so s = √3.5 ≈ 1.87.
Population: variance = 17.5 ÷ 6 ≈ 2.9167, so σ = √2.9167 ≈ 1.71.
Why n − 1? Bessel's correction, briefly
A sample's mean is calculated from the sample itself, so the sample values are — by construction — as close as possible to that mean. They'd be a little farther, on average, from the true population mean you never got to see. Dividing by n would therefore lowball the spread every time. Dividing by n − 1 corrects that bias, which is why it's called an unbiased estimator. The practical rules: measuring the whole group and only describing it? Divide by n. Using a sample to generalize beyond it? Divide by n − 1. And notice the correction matters most when n is small — at n = 6 the two answers differ by about 9%, but at n = 1,000 they're nearly identical.
Frequently asked questions
Should I use sample or population standard deviation?
Use population if your numbers are the entire group you care about (every student in your class, every order this month). Use sample if your numbers are a subset used to estimate a larger group (a survey of 50 customers standing in for all customers). When in doubt, sample is the safer default — most real data sets are samples.
What does standard deviation actually tell me?
It's the typical distance between a data point and the mean, in the same units as your data. A small standard deviation means the values cluster tightly around the average; a large one means they're spread out. For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean and about 95% within two.
What is the formula for standard deviation?
Subtract the mean from each value, square each difference, add them up, divide by n (population) or n−1 (sample), then take the square root. In symbols: σ = √(Σ(x − μ)² ÷ n) for a population, and s = √(Σ(x − x̄)² ÷ (n − 1)) for a sample.
Why divide by n−1 instead of n for a sample?
Because a sample's values sit closer to their own mean than to the true population mean, dividing by n systematically underestimates the spread. Dividing by n−1 — called Bessel's correction — inflates the result just enough to make it an unbiased estimate of the population variance.
Can standard deviation be negative or zero?
Never negative — it's built from squared differences, so the smallest it can be is zero. A standard deviation of exactly zero means every value in your data set is identical, so there's no spread at all.