Standard Deviation Calculator

Mean, variance and both standard deviations.

Sample design decides whether a result can be trusted as much as the arithmetic does: a random sample of a thousand people can describe a country, while a large voluntary response sample describes only itself. Effect size and confidence interval together say more than a p-value alone.

Standard deviation says how far a typical value sits from the mean, and in a normal distribution about 68% of values fall within one standard deviation and 95% within two. The sample formula divides by n−1 because a sample's own mean sits closer to its data than the true mean does.

Statistics in American teaching and research is organised around the normal curve: standard deviation measures spread, a z-score expresses a value in units of it, and a p-value below 0.05 is the conventional line for calling a result significant. Confidence intervals widen quickly as samples shrink, and a result is only as good as the sample it came from.

The 0.05 threshold is a convention, not a law of nature: it means a result this extreme would arise by chance about one time in twenty if there were no real effect.

Sampling error and bias are different problems: a larger sample shrinks the first and does nothing about the second. A confidence interval describes the range the true value plausibly occupies, not the spread of the data.

Standard Deviation Calculator sits with the rest of the library's maths work, and it rests on the same statistical arithmetic the way a textbook or an exam paper does. Its answer is only as good as the assumptions standing behind it.

The quantities it works from — data set and place a value in the set — are the ones a real statistical question actually turns on, and each carries a convention of its own, set out below.

Mathematics is the one subject here that depends on nothing local: the same theorem holds in Ohio and Osaka, and the same fraction carries the same value everywhere. What changes is how results are written — American schooling teaches decimal notation and long division, and business arithmetic runs on fractions of a dollar.

Precision and rounding are where most practical errors creep in.