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 H₀ 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 |
A standard deviation calculator computes spread around the mean from the values you enter. It shows the formula, live result, supporting metrics, step-by-step work and a chart so you can verify the calculation and cite the method.
Use the labeled fields at the top of the page. Dataset boxes accept comma, space, semicolon, tab and newline separated numbers, including negative values and scientific notation.
Differences usually come from rounding, sample versus population formulas, tail choice, quartile method or whether a z or t critical value is used.
This calculator is for education and checking work. For publication, regulated work or high-stakes decisions, verify results with peer-reviewed statistical software.
The formulas follow NIST/SEMATECH, OpenStax and R stats documentation conventions cited in the references section.