Standard Deviation Calculator
Calculate sample and population standard deviation, variance and mean from a data set, with the formula and calculation steps explained.
About the Standard Deviation Calculator
This standard deviation calculator computes both the sample standard deviation (s) and the population standard deviation (σ) for any data set, along with both variances, the mean and the count. Paste numbers separated by commas, spaces or new lines and the results update live.
Standard deviation measures how spread out data is around the mean: a small value means observations cluster tightly, a large one means they scatter widely. It is the backbone of statistics — grading curves, quality control, finance risk measures and lab reports all rely on it.
The tool also spells out the calculation steps with your actual numbers: find the mean, square each deviation, sum them, divide by n − 1 (sample) or n (population), and take the square root. That makes it useful not just for answers but for checking homework working line by line.
How to Use the Standard Deviation Calculator
- 1Enter at least two numbers, separated by commas, spaces or new lines.
- 2Read the sample SD, population SD, variances and mean instantly.
- 3Review the step-by-step breakdown to see how the result was derived.
- 4Use sample SD for a subset of data, population SD when you have every value.
Frequently Asked Questions
What is the difference between sample and population standard deviation?
Population SD divides the summed squared deviations by N and applies when your data is the entire group of interest. Sample SD divides by n − 1 instead (Bessel's correction), compensating for the fact that a sample's spread slightly underestimates the population's. If your data is a sample drawn from something larger — which is the usual case — use the sample formula.
What is the formula for standard deviation?
For a population: σ = √(Σ(x − μ)² ÷ N), where μ is the population mean. For a sample: s = √(Σ(x − x̄)² ÷ (n − 1)), where x̄ is the sample mean. Variance is simply the value before the square root. This calculator computes both versions from the same data so you can quote whichever your assignment requires.
What does a standard deviation value actually tell me?
It is the typical distance of a data point from the mean, in the same units as the data. For roughly bell-shaped data, about 68% of values fall within one SD of the mean, 95% within two, and 99.7% within three — the empirical rule. An SD of 5 on test scores averaging 70 means most students scored between 65 and 75.
Why is variance different from standard deviation?
Variance is the average of squared deviations, so its units are squared (rupees², cm²), which is awkward to interpret. Standard deviation is the square root of variance, restoring the original units. Statisticians work with variance because it has cleaner mathematical properties; practitioners report standard deviation because it is directly comparable to the data.
Can standard deviation be zero or negative?
It can be zero — but only when every value in the data set is identical, so there is no spread at all. It can never be negative, because it is the square root of a sum of squared (hence non-negative) deviations. If you compute a negative SD by hand, an arithmetic slip happened, usually a sign error in the deviations.