Free Online Standard Deviation Calculator
Calculate population and sample standard deviation with working.
Use the sample option when your data is a subset of a larger population. Use population when you have every value in the group you are describing.
Step table
| Value (x) | x − mean | (x − mean)² |
|---|---|---|
| Enter a data set to see the working. | ||
About the Standard Deviation Calculator
Standard deviation measures how far a set of values typically sits from its mean. A small value means the data is tightly clustered; a large value means it is spread out. This calculator reports the mean, variance, standard deviation, standard error, count, and sum of squared deviations, and it shows every deviation in a step table so the result is easy to verify.
Population versus sample
The distinction matters. A population includes every member of the group you care about, so the variance divides the sum of squared deviations by n. A sample is a subset used to estimate a larger population, so the variance divides by n − 1. That adjustment, called Bessel's correction, makes the sample estimate slightly larger and less biased. Use the toggle to switch between the two formulas at any time.
The formulas
- Mean: the sum of all values divided by the count.
- Variance: the sum of squared deviations from the mean, divided by n or n − 1.
- Standard deviation: the square root of the variance.
- Standard error: the standard deviation divided by the square root of the count.
The standard error describes how much the sample mean itself is likely to vary from sample to sample. It is not the same as standard deviation, although the two are closely related.
Reading the step table
Each row subtracts the mean from one value and squares the result. Squaring removes negative signs and gives extra weight to observations that are far from the mean. The total row adds those squared deviations, which is the value used in the variance calculation.
Frequently asked questions
Why does my answer differ from another calculator?
Check whether it used the sample formula (n − 1) or the population formula (n). The two results are close for large data sets but can differ noticeably for small ones.
Can I calculate the standard deviation of one value?
The population standard deviation of a single value is 0. The sample standard deviation is undefined because the denominator n − 1 would be zero.
What does a standard deviation of zero mean?
Every value in the data set is identical, so there is no spread around the mean.