Standard Deviation Calculator
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About Standard Deviation
Two classes can average 75 with utterly different stories — one clustered at 70–80, another split between 50s and 95s. The mean can't tell them apart; standard deviation can. It's the number that turns “average” into “average, plus how typical that average actually is.”
Paste up to 500 comma-separated values. You get sample and population standard deviations, both variances, the mean, and the intermediate sums — enough to reproduce every figure on paper, which is exactly what statistics homework graders want to see.
Just the central tendencies (mean, median, mode)? That's the Average Calculator
The Computation, Step by Step
Four mechanical steps:
1. mean = Σx ÷ n 2. deviations: (x − mean)² for each value 3. variance: s² = Σ(x−mean)² ÷ (n−1) [sample] σ² = Σ(x−mean)² ÷ n [population] 4. std dev = √variance
Worked example — the classic dataset 2, 4, 4, 4, 5, 5, 7, 9: mean 5, squared deviations summing to 32, population σ = √(32/8) = 2 exactly, sample s = √(32/7) ≈ 2.14. Same data, two denominators, two honest answers to two different questions.
Sample vs Population
The choice that trips every intro student:
| Situation | Use | Divide by |
|---|---|---|
| Survey of 200 customers (of many) | Sample s | n − 1 |
| Test scores of YOUR whole class, about that class | Population σ | n |
| Lab measurements estimating a true value | Sample s | n − 1 |
| All 12 months of one year's sales, describing that year | Population σ | n |
| Not sure / data represents something larger | Sample s | n − 1 |
The n − 1 (Bessel's correction) exists because using the data's own mean makes deviations systematically too small — dividing by less compensates. With big n the two versions converge anyway.
The 68–95–99.7 Rule
For roughly bell-shaped data, standard deviation becomes a ruler: about 68% of values fall within 1 SD of the mean, 95% within 2, 99.7% within 3. Adult height, measurement errors, and test scores often behave this way — it's why “two standard deviations” is science's favorite threshold for “unusual.”
The honesty clause: skewed data (incomes, house prices, wait times) breaks the rule — the percentages drift and the SD itself gets dragged by the long tail, which is when medians and percentiles describe reality better. Standard deviation is the right spread measure for symmetric-ish data and the input to nearly every statistical test; it is not a universal summary.
Frequently Asked Questions
How do I calculate standard deviation by hand?
Mean first, then each value's squared distance from it, sum those, divide by n−1 (sample) or n (population), square-root. For 2,4,4,4,5,5,7,9: mean 5, squared deviations sum 32, σ = √4 = 2. The calculator shows each intermediate number.
Sample or population — which do I use?
Sample (n−1) whenever your data stands in for something bigger — surveys, experiments, samples by definition. Population (n) only when the list is complete and you're describing exactly it. In doubt, sample: it's the safer default and every stats package's.
Why divide by n − 1 instead of n?
Bessel's correction: deviations are measured from the sample's own mean, which sits artificially close to the data, shrinking the sum. Dividing by n−1 re-inflates the estimate to unbiased size. The effect matters at small n and vanishes as n grows.
What's the difference between variance and standard deviation?
Variance is the squared spread (units squared — dollars², cm²); standard deviation is its square root, back in the data's own units. Variance feeds the math (it adds nicely); SD feeds interpretation. This calculator reports both, both ways.
What does a standard deviation of 2 mean?
Typical values sit about 2 units from the mean — for bell-shaped data, ~68% within mean ± 2 and ~95% within mean ± 4. Whether that's “a lot” depends entirely on scale: SD 2 on test scores near 75 is tight; on values near 5, enormous.
Can standard deviation be zero or negative?
Zero yes — when every value is identical (no spread exists). Negative never: it's a square root of squared quantities. A negative SD on homework means an arithmetic slip, usually a dropped square.
Methodology. This calculator uses standard, peer-reviewed mathematical formulas. It is reviewed and maintained by the Vast Calculators editorial team.
Last updated · July 11, 2026
Results are estimates for general use; verify critical figures independently.