How this instrument works
Standard deviation is the square root of variance, and that square root matters: variance is expressed in squared units, which nobody thinks in naturally, while standard deviation lands back in the original units of your data. A standard deviation of 2.14 test-score points is something you can picture directly; a variance of 4.57 squared points is not. That is why standard deviation, not variance, is almost always the figure quoted in a report.
This instrument reports two versions from the same data, because the correct one depends on what your numbers represent rather than on the arithmetic itself. The sample standard deviation, s, divides the squared deviations by n − 1 and is the right choice when your data is a subset standing in for a larger group you haven't fully measured. The population standard deviation, σ, divides by the plain count n and is correct only when your data already is the entire group of interest. The two will always sit close together and grow closer as n increases, but for small data sets the difference is real.
For roughly bell-shaped (normal) data, standard deviation earns one more piece of intuition worth knowing: about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three — the empirical rule. It doesn't hold for skewed or heavy-tailed data, but where it applies, a single standard deviation figure tells you almost the whole shape of the distribution at a glance.
- Enter your numbers into Data set — separate them with commas, spaces or new lines in any order.
- Read Sample standard deviation (s) if your data is a subset representing a larger group you haven't fully measured.
- Read Population standard deviation (σ) instead if your data already covers every member of the group you care about.
- Both figures update from the same entered data, so you can compare them directly rather than recalculating by hand.
- Need how precisely your sample's mean estimates the population mean, not how spread out the raw data is? See this site's Standard deviation of the sample mean instrument.
Worked example — the eight quiz scores
Enter 2, 4, 4, 4, 5, 5, 7, 9 into Data set. The mean is 5, and the squared deviations from that mean — 9, 1, 1, 1, 0, 0, 4, 16 — sum to 32.
Divide 32 by n − 1 = 7 and take the square root: Sample standard deviation (s) reads 2.138090 (√4.571428...). Divide the same 32 by n = 8 instead and take the square root: Population standard deviation (σ) reads exactly 2.0 (√4). The two forms use identical raw deviations; they differ only in which count they divide by before the square root.
Questions
Should I quote the sample or population standard deviation?
Quote s, the sample form, whenever your data is a subset of a larger group — a batch of measurements, a survey sample, a handful of trial runs. Quote σ, the population form, only when your data already covers every member of the group you're studying, with nothing left unmeasured. If you're genuinely unsure which applies, the sample form is the more conservative, commonly expected choice.
What's the difference between standard deviation and variance?
Standard deviation is the square root of variance. Variance is expressed in squared units (squared dollars, squared points), which few people can picture directly; standard deviation converts that figure back into the original units, which is why it's the number usually reported while variance stays a step in the underlying formula.
Is 'standard deviation' the same thing as 'standard error'?
No, and mixing them up is a common mistake. Standard deviation (this page) measures how spread out your raw data values are around their mean. Standard error measures how much your sample mean itself would wobble if you repeated the sampling — it's the standard deviation divided by √n. This site's Standard deviation of the sample mean instrument computes that second, related quantity.
What does the empirical rule (68-95-99.7) mean?
For data that is roughly normally (bell-curve) distributed, about 68% of values fall within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% within three. It's a quick way to picture spread once you have a standard deviation figure, but it only applies to roughly symmetric, bell-shaped data — skewed data doesn't follow it reliably.
Why is the sample standard deviation always a bit larger than the population form on the same data?
Because the sample form divides by n − 1, a smaller number than the n used in the population form, and dividing by a smaller number produces a larger result. That inflation is deliberate — it's Bessel's correction, compensating for the slight bias that comes from estimating the mean from the same sample being measured.
How many values does this instrument need?
At least two. A single value has no spread to measure, and the sample formula's n − 1 divisor would become zero, which is undefined — so the instrument prompts for a second number before computing either standard deviation.