Standard Deviation Calculator
Show steps
| x | x − x̄ | (x − x̄)² |
|---|
How to use
- Paste or type your numbers. Commas, spaces, semicolons and line breaks all work as separators.
- Read the sample standard deviation if your numbers are a sample from a larger group, or the population value if they are the whole group.
- Open Show steps to see each value, its distance from the mean and the squared distance.
How it works
Mean: x̄ = Σx ÷ n. Each value's deviation is x − x̄, and the deviations are squared and added up.
Population variance σ² = Σ(x − x̄)² ÷ n, and the population standard deviation σ is its square root.
Sample variance s² = Σ(x − x̄)² ÷ (n − 1). Dividing by n − 1 (Bessel's correction) avoids underestimating the spread of the larger population. s is its square root.
FAQ
Should I use sample or population standard deviation?
Use population when your data covers every member of the group, such as all students in one class. Use sample when the data is a subset you want to generalise from, which is the usual case in surveys and experiments.
Why is the sample value larger?
It divides by n − 1 instead of n. With small data sets the difference is noticeable, and it shrinks as n grows.
Why is the sample standard deviation missing for one number?
With a single value n − 1 is zero, so the sample formula is undefined. The population value is 0.