Calculation details
Planning estimate only, not financial, tax, or legal advice. Verify assumptions and current rules before making decisions.
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Use this result
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What does this calculator estimate?
For [2, 4, 4, 4, 5, 5, 7, 9], the population standard deviation is about 2.00 and the sample standard deviation about 2.14. Population divides by n; sample by n - 1.
- Population SD divides by n; sample SD by n - 1 (Wikipedia).
- SD measures typical distance from the mean in the data's units.
- A low SD means values cluster tightly around the mean.
What standard deviation tells you
Standard deviation is the typical distance of values from the mean — the go-to measure of spread and risk. In a normal distribution, ~68% of values fall within ±1 SD, ~95% within ±2.
Limitations to watch for
Sample vs population matters: use ÷ (n−1) when estimating from a sample. SD is sensitive to outliers. It assumes a roughly symmetric distribution for the ±68/95 interpretation.
How to use it in practice
Use SD to compare consistency (portfolio risk, process quality, test scores) and to spot unusual values. Report it with the mean — the pair describes both center and spread.
['Enter the numbers.', 'Choose population or sample.', 'Read mean, variance, and standard deviation.']
How this calculator works
Formula
Standard deviation = √(Σ(xᵢ − mean)² ÷ n) for a population, or ÷ (n − 1) for a sample. It measures how spread out values are around the mean.
Worked example
For [2, 4, 4, 4, 5, 5, 7, 9]: population SD ≈ 2.00, sample SD ≈ 2.14.
Assumptions to verify
- At least two numbers are provided.
- The mode (population/sample) matches the context.
- Values are numeric.
Frequently asked questions
What is standard deviation?
The typical distance of values from the mean — how spread out the data is.
How is it calculated?
Square each deviation from the mean, average (n or n−1), and take the square root.
Population or sample?
Sample (n−1) when your data estimates a larger population — the usual research case.
What does ±1 SD mean?
About 68% of values in a normal distribution fall within one standard deviation of the mean.
How is it different from variance?
SD is the square root — in the original units, so easier to interpret.
What is it used for?
Risk (finance), process consistency (quality), and statistical inference.
Why is my SD affected by outliers?
Squaring the deviations gives extreme values extra weight.
Cite this tool
BoringToolsKit. “Standard Deviation Calculator.” boringtoolskit.com/standard-deviation-calculator/ (reviewed 2026-08-25). Free to reference in articles, syllabi, and answer posts with a link.
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