Standard Deviation Calculator
Paste or type your numbers (comma, space, or newline separated). Choose population or sample to see σ or s, variance, mean, and step-by-step math.
Your data
Result
Add one more value to compute variability.
Results interpretation
- Small SD → values cluster tightly around the mean (low spread).
- Large SD → values are more dispersed (high spread or outliers).
- Who it’s for: students, researchers, analysts comparing variability across groups, or anyone summarizing spread.
- The familiar 68-95-99.7 rule of thumb applies only when data are roughly normal.
- Prefer sample SD (s) unless you truly have the full population.
How this calculator works
We parse your numbers, compute the mean, then square each deviation from the mean. For a population we divide by N; for a sample we divide by N−1 (Bessel’s correction). The square root of that variance gives standard deviation.
Show formulas
Limitations: SD is sensitive to outliers and assumes numeric, comparable units. Skewed or multimodal data may be better summarized with robust dispersion measures (e.g., IQR).
FAQ
Is 0 a valid standard deviation?
Yes—when all values are identical.
Can I mix commas, spaces, and new lines?
Yes. The input accepts any whitespace or commas as separators.
Which should I report, σ or s?
Use s (sample SD) unless your numbers represent an entire population.
Is SD the same as standard error?
No. SD summarizes data spread; standard error summarizes uncertainty of an estimate.
How many data points do I need?
Two or more. With very small samples, SD is unstable—collect more data when possible.
Use cases & examples
- Population, product weights: 100, 102, 98, 100 → μ = 100; σ ≈ 1.414. Tight process control.
- Sample, quiz scores: 60, 72, 90, 78, 80 → x̄ = 76; s ≈ 11.4. Spread suggests mixed mastery.
- Before/after variability: Compare s across groups to see which treatment stabilizes outcomes.
What this Standard Deviation Calculator does
Standard deviation calculator is a quick way to turn a list of numbers into clear measures of spread: mean, variance, and the standard deviation (population σ or sample s). Paste values, choose the data type, and the result updates instantly with step-by-step math. This article explains what SD means, when to use population vs sample calculations, common assumptions and caveats, and how to apply the results.
Why standard deviation matters
Two datasets can have the same mean but very different spreads. SD captures that spread in original units, which makes it easy to compare consistency across products, students, machines, or experiments. A small SD indicates values cluster near the mean; a large SD signals variability, potential outliers, or a heterogeneous process.
Population vs sample calculations
If your list contains the entire population, divide the sum of squared deviations by Nto get σ². If it’s a sample, divide by N−1 to correct bias—that’s the sample variance s². The square root gives σ or s. Our calculator lets you switch types to see the impact.
Interpreting SD with the 68-95-99.7 rule
For roughly normal data, about 68% of values lie within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD. Use this only as a heuristic; real data can be skewed or heavy-tailed. Always visualize your data and check for outliers.
Limitations & robust alternatives
SD is sensitive to outliers. For skewed distributions, consider the interquartile range (IQR) or median absolute deviation (MAD). When reporting uncertainty of a mean, use the standard error or a confidence interval rather than raw SD.
How to use this calculator effectively
Clean data (consistent units), paste values, select population/sample, and review the steps. If the SD looks large, check the frequency table or a quick plot (not shown here) for outliers. Consider transformations (e.g., log) for multiplicative data before computing SD.