What standard deviation measures
Standard deviation shows how spread out numbers are around their mean. You find the mean, take each value's squared distance from it, average those, and square-root the result. For 2, 4, 4, 4, 5, 5 the mean is 4 and the population standard deviation is 1 — a small spread.
The squaring step stops positive and negative distances from cancelling out, and the final square root returns the answer to the original units. The value under the square root is the variance, which the variance calculator reports on its own.
How to use this calculator
Enter your numbers in the value fields and leave the rest blank. The result shows the population standard deviation (σ, dividing by n), the sample standard deviation (s, dividing by n − 1), and the mean.
All the values must be in the same unit, and each observation should be entered separately rather than averaged into groups first. Adding a constant to every value leaves the standard deviation unchanged. Once you have it, the z-score calculator tells you how unusual any single value is.
Population vs. sample
Use the population formula when your data is the whole group, and the sample formula (n − 1) when it's a sample standing in for a larger population — sample SD is slightly larger. For just the average, see the average calculator.
The n − 1 adjustment exists because a sample's own mean sits closer to its data than the true population mean does, which makes the raw spread an underestimate. The difference shrinks as the sample grows — noticeable with five values, negligible with five hundred.
Frequently asked questions
- How do I calculate standard deviation?
- Find the mean, then the squared difference of each value from it, average those (÷ n for population, ÷ n−1 for sample), and take the square root.
- What is the difference between population and sample SD?
- Population divides by n and assumes you have all the data; sample divides by n − 1 and is used when your data is a sample of a bigger group.
- What is variance?
- Variance is the standard deviation squared — the average squared distance from the mean before taking the square root.
- What does a small standard deviation mean?
- The values are clustered close to the mean. A large standard deviation means they're spread out widely.
- Can the standard deviation be zero or negative?
- It can be zero, which happens only when every value is identical, but it can never be negative. The formula averages squared distances and then takes a square root, and neither step can produce a value below zero.