Standard Deviation Calculator
Standard deviation is the square root of variance. Sample variance divides by n - 1. Population variance divides by n.
Choose sample or population first
Enter the values from one dataset. Choose sample or population if the calculator gives that option. Use population when the values are the entire group you care about. Use sample when the values are part of a larger group.
This distinction matters most in statistics classes and reports. For casual data checks, the two results may be close, but they can differ noticeably with small datasets.
Reading spread in the result
Standard deviation measures spread around the mean. A small standard deviation means the values are close to the average. A large standard deviation means the values are more spread out.
It does not explain why the values are spread out. It only describes the dataset you entered. You still need context to interpret whether the spread is good, bad or expected.
Compare the result with the unit and the mean. A standard deviation of 5 can be tiny for salaries, but large for quiz scores out of 20.
Variance and standard deviation
The calculator finds the mean, measures each value distance from the mean, squares those distances, averages them and then takes the square root. Squaring keeps negative and positive distances from canceling each other out.
Population standard deviation divides by n. Sample standard deviation divides by n - 1. The sample version corrects for the fact that a sample usually estimates the spread of a larger population.
Sample standard deviation example
For the values 2, 4, 4, 4, 5, 5, 7 and 9, the mean is 5. The values are not all far from 5, so the standard deviation is moderate. If the list had 30 instead of 9, the spread would be much larger.
The same dataset can produce two slightly different answers depending on sample or population mode. Check the wording of the problem before choosing.
Sample versus population mistakes
The most common mistake is picking sample or population without reading the situation. If the values are all the items of interest, use population. If they are a subset used to estimate a larger group, use sample.
Another mistake is treating standard deviation like average error. It is a spread measure, not a guarantee that each value is that far from the mean.
Standard Deviation Calculator FAQ
Should I use sample or population standard deviation?
Use population standard deviation when your data includes every value in the group you care about. Use sample standard deviation when your data is a sample from a larger group.
If a school problem does not say the full population is included, sample is often the safer choice.
Why does sample standard deviation use n minus 1?
A sample usually underestimates the spread of the larger population. Dividing by n minus 1 corrects that bias in the estimate.
This is why the sample standard deviation is usually a little larger than the population standard deviation for the same values.
In classwork, this is the question that causes the most confusion. The choice is about what the data represents, not about which formula looks easier.
What does a high standard deviation mean?
It means the values are spread out farther from the mean. A low standard deviation means the values cluster more closely around the mean.
High or low is relative. Compare it with the unit, the mean and the purpose of the data.
Can standard deviation be zero?
Yes. Standard deviation is zero when every value in the dataset is the same. There is no spread around the mean.
For example, 5, 5, 5 and 5 has a mean of 5 and a standard deviation of 0.
Is standard deviation affected by outliers?
Yes. Standard deviation squares distances from the mean, so outliers can have a strong effect. One extreme value can make the spread look much larger.
When outliers matter, also look at the median, range and the actual data list.
A graph or sorted list can make the outlier easier to spot before you interpret the number.