Math & Statistics
Standard Deviation Calculator
Paste a numeric dataset, switch between population and sample statistics, inspect each observation's deviation, review a distribution chart, and export the complete analysis as PDF.
Count0
Mean0
Variance0
Standard deviation0
Minimum0
Maximum0
Distribution analysis
Spread around the mean
Observation distributionFrequency by equal-width interval
| # | Value | Deviation | Squared deviation |
|---|
Decision view
Observations around the mean
Min 0Mean ± SD: 0 to 0Max 0
How to use Standard Deviation Calculator
- Paste the observations separated by commas, spaces, or line breaks and confirm the displayed count.
- Choose population when the values are the complete group; choose sample when they estimate variability in a larger group.
- Read the mean, variance, and standard deviation, then inspect the deviation table and distribution for outliers or clusters.
- Report the selected denominator and units alongside the result so another reader can reproduce the statistic.
Calculator guide
Understanding Standard Deviation Calculator
Standard deviation summarizes how far observations typically sit from their mean. A smaller value indicates tighter clustering; a larger value indicates greater spread in the original units.
Calculation method
How the calculation works
Population variance is the mean squared deviation from the mean; sample variance divides the squared-deviation sum by n - 1. Find the mean, subtract it from every observation, square the deviations, and add them. Divide by n for a population or n - 1 for a sample, then take the square root.
Denominator choice
Population versus sample standard deviation
The same observations produce different results because the variance denominator answers a different statistical question.
Reading spread
What one standard deviation does not reveal
Datasets with the same standard deviation can have different shapes, outliers, clusters, and asymmetry.
Worked situations
Practical examples
- Measure the spread of a complete production batch as a population.
- Estimate variability from a sample of survey observations.
- Identify observations that sit far from the dataset mean.
Better inputs
Useful tips
- Choose population only when the values are the complete group of interest.
- Use sample mode when estimating a larger population from observed data.
- Inspect the deviation table and distribution rather than relying on one summary number.
Before relying on the result
Limitations and common mistakes
- Sample standard deviation requires at least two observations.
- Extreme outliers can substantially increase variance and standard deviation.
- Standard deviation alone does not describe skewness, clusters, or multiple modes.
Reference
Key terms
- Mean
- The sum divided by the number of observations.
- Variance
- Average squared distance from the mean under a selected denominator.
- Sample
- Observed values used to estimate a larger population.
- Population
- The complete set of observations being described.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
What is the difference between sample and population standard deviation?
Population variance divides by n; sample variance divides by n - 1 to estimate population variability from a sample.
Can standard deviation be negative?
No. Squared deviations and their square root cannot produce a negative standard deviation.
What does a standard deviation of zero mean?
Every observation is identical to the mean.
Does a larger standard deviation always mean bad data?
No. It only indicates greater spread; whether that is desirable depends on the context.