Paste or type a list of numbers to see how spread out they are. You get the sample and population standard deviation, the variance, the mean, the count and the standard error.
Results are estimates for planning. Check the figures before you make a financial or health decision.
How the Standard deviation calculator works
The calculator follows these steps:
- Find the mean, the total divided by the count.
- Subtract the mean from each number and square the result.
- Add up those squares to get the sum of squared deviations.
- Divide that sum by n to get the population variance, or by n − 1 to get the sample variance.
- Take the square root of each variance to get the standard deviation.
The standard error of the mean is the sample standard deviation divided by the square root of n. It shows how precisely the sample mean estimates the true mean.
Use the sample figure when your numbers are a sample drawn from a larger group, which is the usual case. Use the population figure only when your list is every value in the group. A single number has no sample standard deviation, so that row shows a dash.
Separate numbers with commas, spaces or new lines. The tool accepts up to 5,000 numbers.
Worked example
Using the default values in the calculator above:
| Input | Value |
|---|---|
| Numbers | 2, 4, 4, 4, 5, 5, 7, 9 |
Sample standard deviation (s): 2.13809
| Population standard deviation (σ) | 2 |
|---|---|
| Sample variance (s²) | 4.571429 |
| Population variance (σ²) | 4 |
| Mean | 5 |
| Count | 8 |
| Standard error of the mean | 0.755929 |
Tips and common mistakes
Check the list for typing slips first, since one extra zero can change the standard deviation a great deal. A large value compared with the mean means the numbers are widely spread; a small one means they cluster near the mean.
Standard deviation is in the same units as your data, whereas variance is in squared units. If you need the median, mode or range as well, the average calculator gives them. Do not compare standard deviations of data measured on very different scales without thinking about the means.
Frequently asked questions
What is the difference between sample and population standard deviation?
Population divides by n and is for a complete group. Sample divides by n − 1 and is for a sample from a larger group, which corrects for the sample tending to understate the spread.
What does a standard deviation of zero mean?
Every number in the list is identical, so there is no spread at all.
Which one should I use?
Use the sample standard deviation unless you really have every value in the group you care about. Most real data, such as survey results or measurements, is a sample.