Summary: This standard deviation calculator computes sample and population SD with live steps, formulas and a chart. It accepts labeled numeric inputs, works offline through file:// and includes source-backed explanations for students, analysts and researchers.
Standard deviation calculator for sample and population SD with variance, mean, deviations, formulas, chart and step-by-step working. The calculator works offline, updates instantly and includes a worked example, plain-text formula, MathML, references and structured data.
Default values are loaded. Click any field and edit it; results and chart update automatically.
| Input or setting | Result or interpretation | Use this when |
|---|---|---|
| 2,4,4,4,5,5,7,9 | sample s about 2.138 | spread around the mean |
| 10,10,10 | standard deviation = 0 | spread around the mean |
| 1,2,3 | sample s = 1 | spread around the mean |
sample SD: s = sqrt(sum((x_i - mean)^2) / (n - 1)); population SD: sigma = sqrt(sum((x_i - mu)^2) / N)Sample standard deviation divides squared deviations by n minus 1. Population standard deviation divides by N.
2,4,4,4,5,5,7,9
sample s about 2.138; population sigma = 2
This page uses the shared statistics core for distribution functions, quantiles and exact integer counting where needed. The formula is shown in plain text so screen readers and search engines can parse it reliably.
Use Load example in the calculator to reproduce this reference result.
{
"tool": "Standard Deviation Calculator",
"input": "2,4,4,4,5,5,7,9",
"output": "sample s about 2.138; population sigma = 2",
"formula": "sample SD: s = sqrt(sum((x_i - mean)^2) / (n - 1)); population SD: sigma = sqrt(sum((x_i - mu)^2) / N)"
}| Calculator | Example input | Expected output |
|---|---|---|
| Standard Deviation Calculator | 2,4,4,4,5,5,7,9 | sample s about 2.138; population sigma = 2 |
What does p = 0.03 mean? If the null hypothesis and model assumptions were true, a result at least this extreme would occur about 3% of the time. The American Statistical Association cautions that a p-value alone does not measure effect size, practical importance or the probability that Hâ‚€ is true.3
For most classroom and professional reports, pair the calculator result with the question you are answering. A mean or median summarizes location, but spread explains consistency. A confidence interval estimates plausible values, while a hypothesis test evaluates compatibility with a null model. Regression and correlation describe association, so they should be reported with a chart and residual or outlier review. When a result is statistically significant, still ask whether the effect is large enough to matter in the real setting.
| Statistic | Small | Medium | Large | Use |
|---|---|---|---|---|
| Cohen's d | 0.2 | 0.5 | 0.8 | t-test effect size |
| Cramér's V | 0.1 | 0.3 | 0.5 | chi-square association |
| |r| | 0.10 | 0.30 | 0.50 | correlation strength |
| R² | 0.01 | 0.09 | 0.25 | variance explained |
Five steps. (1) Find the mean of your data. (2) Subtract the mean from each value to get deviations. (3) Square each deviation. (4) Add up those squared deviations and divide — by n for the population SD, by n − 1 for the sample SD. (5) Take the square root. Quick example with {2, 4, 4, 4, 5, 5, 7, 9}: mean = 5, squared deviations sum = 32, sample variance = 32/7 ≈ 4.57, sample SD ≈ 2.14. Population SD would use 32/8 = 4, giving σ = 2.
Population SD (σ) uses every member of the population and divides the sum of squared deviations by N. Sample SD (s) uses a subset and divides by n − 1 instead of n — that's Bessel's correction. The minus-one tweak compensates for the fact that using the sample mean (instead of the unknown population mean) tends to underestimate spread. In practice, you almost always have a sample, so n − 1 is the default. The correction matters most for small samples; with very large n, σ and s look nearly identical.
Most calculators accept a list of numbers separated by commas, spaces, or newlines. Paste your data, then choose between sample and population SD — that toggle matters. Sample SD divides squared deviations by n − 1; population SD divides by n. If your data represents a sample drawn from a larger group (almost always the case), pick sample SD. If your data covers the entire population (every student in your class, say), pick population SD. The output gives you the spread of values around the mean in the original units.
This is Bessel's correction. When you use the sample mean instead of the unknown population mean, the squared deviations tend to be slightly smaller than they would be if you used the true mean. Dividing by n − 1 inflates the variance just enough to give an unbiased estimate of the population variance. With n = 10, dividing by 9 instead of 10 makes a noticeable difference. With n = 1000, the effect is tiny. The correction matters most for small samples and disappears as n grows.
Square the standard deviation: variance = SD². Going the other way, take the square root: SD = √variance. So if SD = 4, variance = 16. If variance = 25, SD = 5. The two are mathematically tied — they describe the same spread, just in different units. SD lives in the original data's units (e.g. centimetres), while variance is in squared units (square centimetres), which often makes SD more intuitive to interpret. Both are computed from the same squared deviations from the mean.
A high SD means your data points are spread far from the mean — high variability. Low SD means values cluster tightly around the mean. But "high" is relative: an SD of 5 is huge if the mean is 10, and small if the mean is 10,000. Compare SD with the mean using the coefficient of variation (CV = SD/mean) for a fair sense of relative spread. In practical terms: high SD in test scores means the class has very mixed performance; low SD means students performed similarly.
Excel offers two main functions. Use STDEV.S(range) when your data is a sample drawn from a larger population — this divides by n − 1. Use STDEV.P(range) when your data is the entire population — this divides by n. For example, STDEV.S(A1:A20) computes the sample SD of values in cells A1 through A20. Older versions had STDEV (equivalent to STDEV.S) and STDEVP (equivalent to STDEV.P) — these still work but are deprecated. Always pick the right one for your context; the difference can matter, especially with small datasets.
Standard deviation is highly sensitive to outliers because the formula squares every deviation from the mean. Squaring magnifies the impact of values far from the centre — one extreme point can inflate the SD dramatically. Take {2, 4, 6, 8, 10}: SD ≈ 2.83. Replace 10 with 100, and SD jumps to about 38.7. That's why SD is unreliable for skewed data with outliers. In those cases, switch to the IQR or median absolute deviation, both of which ignore extreme values and give a more robust picture of spread.