How to use it
Choose whether your numbers are a sample or an entire population, then paste the data. The working table underneath shows every observation, its deviation from the mean, and that deviation squared, with the column totals at the foot — which is exactly the layout you would produce by hand, and exactly what most coursework asks you to show.
Both answers are always available: whichever formula you select, the other one appears in the fourth readout cell so you can see how much the choice actually changes the result.
Sample or population — the decision that matters
This is the most common error in introductory statistics, and it is worth being precise about, because it propagates into every calculation downstream.
Use the population formula only when your data genuinely is every member of the group you want to describe: the marks of all 28 students in one class when the class is the entire subject of interest, every transaction in a completed month, all fifty states.
Use the sample formula whenever your data is a subset standing in for something larger — which, in practice, is almost always. Survey respondents, measured items from a production run, patients in a trial, one class taken as representative of a year group.
The practical size of the difference shrinks as n grows. At n = 5 the sample SD is about 12% larger than the population SD of the same numbers; at n = 30 it is under 2%; at n = 500 it is negligible. Small samples are exactly where it matters most, and exactly where people are most likely to get it wrong.
A worked example
Take five values: 4, 8, 6, 5, 3.
- Mean: (4 + 8 + 6 + 5 + 3) ÷ 5 = 26 ÷ 5 = 5.2
- Deviations: −1.2, 2.8, 0.8, −0.2, −2.2 — note they sum to zero, always, which is a useful check on your arithmetic
- Squared deviations: 1.44, 7.84, 0.64, 0.04, 4.84
- Sum of squares: 14.8
- Sample variance: 14.8 ÷ 4 = 3.7 → s = √3.7 = 1.9235
- Population variance: 14.8 ÷ 5 = 2.96 → σ = √2.96 = 1.7205
The two answers differ by about 12% on five observations. Paste those five numbers above and the working table reproduces this exactly, row for row.
Variance, standard deviation, standard error
Three related quantities that are constantly confused with one another.
| Quantity | Formula | What it tells you |
|---|---|---|
| Variance | s² | Average squared deviation. Mathematically convenient — variances of independent variables add — but in squared units, so hard to interpret directly. |
| Standard deviation | s = √s² | Typical distance of an observation from the mean, in the original units. This is the one to report. |
| Standard error | SE = s ÷ √n | How much the sample mean itself would vary across repeated samples. Shrinks as n grows. This is what confidence intervals are built from. |
The distinction between the last two catches people out constantly. Standard deviation describes the spread of your data and does not shrink when you collect more of it — a taller sample of people is not a less variable one. Standard error describes the precision of your estimate and does shrink, proportional to √n. Quadrupling your sample size halves the standard error while leaving the standard deviation essentially unchanged.
The empirical rule
For data that is approximately normally distributed, standard deviation converts directly into proportions.
| Interval | Contains about | Outside |
|---|---|---|
| x̄ ± 1s | 68.3% | 1 in 3 |
| x̄ ± 2s | 95.4% | 1 in 22 |
| x̄ ± 3s | 99.7% | 1 in 370 |
This is why "three sigma" became shorthand for a rare event and why control charts use three-standard-deviation limits. It only holds for roughly normal data — on a strongly skewed distribution the proportions are quite different. For any distribution at all, Chebyshev's inequality gives a weaker but universal guarantee: at least 75% of data lies within two standard deviations and at least 89% within three, whatever the shape.
Coefficient of variation
The calculator also reports the CV, which expresses the standard deviation as a percentage of the mean.
This makes spread comparable across variables measured in different units or on wildly different scales. A standard deviation of 5 means something very different on data averaging 20 than on data averaging 5,000. The CV strips out the scale: 25% versus 0.1%.
Two cautions. The CV is meaningless when the mean is near zero, because you are dividing by something tiny, and it is not interpretable at all on data that can go negative or that sits on an interval rather than ratio scale — temperature in Celsius being the classic counterexample.
Frequently asked questions
Which should I use if I am not sure?
Why do my deviations sum to zero?
Can standard deviation be negative?
What does Excel's STDEV do?
Should I remove outliers before calculating?
How many decimal places should I report?
Is my data stored?
Next steps
With a mean and standard deviation you can measure how unusual any single observation is using the z-score calculator, or build an interval estimate for the population mean with the confidence interval calculator — which uses exactly the standard error this page reports. If you have not yet checked the shape of your data, the descriptive summary will tell you whether the mean is a fair centre in the first place.