Standard Deviation Calculator
Population and sample standard deviation, variance, and mean from a list of numbers
| Count | 8 |
| Mean | 5.0000 |
| Population variance | 4.0000 |
| Population std. deviation | 2.0000 |
| Sample variance | 4.5714 |
| Sample std. deviation | 2.1381 |
Use population statistics when your numbers ARE the entire group you care about; use sample statistics when they're a sample used to estimate a larger population (sample std. dev. divides by n−1, not n).
Frequently Asked Questions
Standard deviation and variance quantify how spread out a dataset's values are around the mean — and this calculator's distinction between population and sample standard deviation, using slightly different formulas for each, is a genuinely important choice that depends on whether the data represents an entire population or just a sample drawn from a larger one.
Why population and sample standard deviation use different formulas
Population standard deviation is calculated when the dataset represents every member of the group being studied — dividing by the total count (n) directly. Sample standard deviation is calculated when the dataset is only a subset drawn from a larger population, and it uses a slightly adjusted formula — dividing by one less than the count (n−1) rather than n itself — a correction (known as Bessel's correction) that compensates for the fact that a sample's own variability tends to slightly underestimate the true population variability, since the sample's calculated mean is itself derived from the same limited data. Using the wrong version for a given situation produces a systematically biased estimate — population formula on sample data understates true variability, and this distinction, though it might look like a minor denominator tweak, reflects a genuine statistical correction with real practical consequences for how representative the calculated spread actually is.
A worked example
The identical set of numbers, treated once as a complete population and once as a sample drawn from a larger population, produces two slightly different standard deviation values — the sample version is always slightly larger than the population version for the same underlying numbers, reflecting the correction for the sample's tendency to underestimate true population variability.
How this connects to your other statistical calculations
Standard deviation is a direct input to the Confidence Interval Calculator, which uses it alongside sample size to determine how wide a confidence interval needs to be — an inaccurate standard deviation (from using the wrong population-versus-sample formula) propagates directly into an inaccurate confidence interval calculation downstream.
Common mistakes
Using the population standard deviation formula on data that's actually just a sample from a larger population is a common statistical error that systematically understates true variability — whenever the dataset represents a subset rather than the entire group being studied, the sample formula (with its n−1 denominator) is the statistically correct choice, not the population formula.