Standard Deviation Calculator
Calculate the standard deviation and variance of a data set, both population and sample.
- Population Std. Deviation (÷N)4.899
- Sample Std. Deviation (÷N−1)5.2372
Dividing by N−1 instead of N makes the sample estimate a little wider — 5.2372 vs 4.899 — to correct for only having a sample, not the whole population.
- Mean18
- Sample Std. Deviation5.2372
- Sample Variance27.4286
ƒShow your work
σ = √(Σ(x − mean)² / N)
About the Standard Deviation Calculator
Standard deviation answers a question the mean alone can't: not just where a data set is centered, but how tightly or loosely the values are clustered around that center. Two classes could both average 75% on a test, but one class might have everyone scoring between 70-80% while the other ranges from 40-100% — the mean is identical, but the standard deviation tells you those are very different distributions.
This shows up anywhere consistency matters as much as the average: a teacher comparing how tightly grouped test scores are, a quality-control check on manufactured parts, an athlete tracking whether their times are becoming more consistent race to race, or a student working through the formula step by step for a statistics class. This calculator reports both population and sample standard deviation, since which one is correct depends on whether your numbers are the entire group of interest or just a sample drawn from it.
How it’s calculated
The core formula is σ = √(Σ(x − mean)² / N): for every value, subtract the mean, square that difference (so negative and positive deviations don't cancel out), add up all the squared differences, divide by the count, and take the square root at the end to bring the units back in line with the original data.
The only difference between population and sample standard deviation is the denominator: population divides by N (the full count), while sample divides by N−1. That N−1 is called Bessel's correction, and it exists because a sample's own mean is calculated from the same data being measured, which makes the sample slightly less spread out than the true population — dividing by one fewer point corrects for that bias.
In practice, use sample standard deviation unless your data genuinely is the entire population you care about (every student in a specific class, every unit actually produced) rather than a subset representing something larger.
Frequently asked questions
Should I use population or sample standard deviation?
Use sample standard deviation (dividing by N−1) whenever your data is a subset of a larger population, which covers the vast majority of real situations — surveys, experiments, measurements. Use population standard deviation only when your data set literally is every member of the group you're studying.
What counts as a 'good' standard deviation?
It depends entirely on context and the scale of the data — there's no universal threshold. What matters is comparing it to the mean or to another data set's standard deviation on the same scale; a standard deviation of 5 is tiny for data averaging 1,000 but huge for data averaging 10.
How is variance related to standard deviation?
Variance is standard deviation squared (or equivalently, standard deviation is the square root of variance). Variance is used in some further statistical calculations, but standard deviation is usually easier to interpret because it shares the same units as the original data.
Why does the sample standard deviation come out larger than the population one?
Because it divides by N−1 instead of N — dividing the same sum by a smaller number produces a larger result. This built-in inflation compensates for the tendency of a sample to understate the true spread of the full population it was drawn from.
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