Standard Deviation Calculator
A standard deviation calculator measures how spread out a set of numbers is from their mean. It shows both population standard deviation (dividing by n) and sample standard deviation (dividing by n−1, Bessel’s correction) — a common point of confusion in statistics homework. A small standard deviation relative to the mean indicates the data clusters tightly around the average, while a large one indicates wide spread — the same measurement behind the margin-of-error language used in polling and the grading curves used in some standardized tests.
How it works
- Enter a list of numbers separated by commas or spaces.
- Read the mean, population variance, and population standard deviation.
- For two or more numbers, also read the sample standard deviation.
Formula Population: σ = √(Σ(x−mean)² / n). Sample: s = √(Σ(x−mean)² / (n−1))
Frequently asked questions
Which one should I use — population or sample?
Use population standard deviation when your data IS the entire group you care about; use sample standard deviation when your data is a sample used to estimate a larger population’s spread. Most textbook and real-world statistics problems use sample.
Why does sample standard deviation divide by n−1 instead of n?
Dividing by n−1 (Bessel’s correction) corrects for the fact that a sample’s own mean is closer to its data points than the true population mean would be, which would otherwise systematically underestimate the spread.
What does a standard deviation of 0 mean?
All the numbers are identical — there is no spread.
Why is sample standard deviation not shown for a single number?
It requires dividing by n−1; with only one data point, n−1 = 0, which is undefined.
Is my data uploaded anywhere?
No. Calculation runs entirely in your browser.