Understanding your result
Standard deviation measures spread in the same units as your inputs. Variance is its square. A result of zero means all values are identical.
To compare the mean with the median and most frequent values, use the Mean, Median & Mode Calculator with the same dataset.
The formula
For a sample, subtract the mean x̄ from each observation, square those differences, add them, and divide by n − 1 before taking the square root. For a complete population, use n as the divisor instead.
The dataset 2, 4, 4, 4, 5, 5, 7, 9
These eight values have a mean of 5. Their squared deviations sum to 32. Population variance is 32 ÷ 8 = 4, so population standard deviation is 2. Sample variance is 32 ÷ 7 ≈ 4.571429, giving a sample standard deviation of approximately 2.138090.
How to use this calculator
- Paste or type your numbers using commas, spaces, semicolons, or new lines.
- Choose Sample if the values represent part of a larger group, or Population if they describe the entire group.
- Calculate to view standard deviation and supporting statistics.
Should you choose sample or population?
Use population standard deviation when your dataset contains every member of the group you want to describe. Use sample standard deviation when your observations are a sample used to estimate spread in a larger group.
The n − 1 divisor corrects the downward bias in the estimate of population variance. It does not make the square root an exactly unbiased estimator of population standard deviation.
Interpreting the amount of spread
A larger standard deviation means observations tend to sit farther from the mean. Its importance depends on the units and context: a spread of 5 means something different for daily temperatures and annual revenue.
Outliers can strongly affect standard deviation because differences are squared. The familiar 68–95–99.7 rule applies approximately to normal distributions; do not assume it fits every dataset.
Assumptions & limitations
What this calculation assumes
- Every observation has equal weight.
- Comma characters separate values rather than mark thousands or decimals.
- Sample results use n − 1; population results use n.
What to keep in mind
- Sample standard deviation needs at least two observations.
- Datasets are limited to 10,000 values with absolute magnitudes up to 1 trillion.
- JavaScript floating-point arithmetic has finite precision, especially for differences smaller than the precision of the original inputs.
Common questions
Can standard deviation be negative?
No. It is the nonnegative square root of variance. Negative input values are allowed.
What is the difference between variance and standard deviation?
Variance averages squared deviations with the selected divisor. Standard deviation is its square root and is expressed in the original units.
Can I enter a single number?
A single value has population standard deviation zero. A sample requires at least two values because its divisor is n − 1.