Standard Deviation Calculator
This professional standard deviation calculator lets you paste or type numbers to instantly compute sample and population standard deviation, variance, mean, median, and other descriptive statistics. It is designed for students, analysts, scientists, and anyone who needs precise, accessible, and mobile‑friendly statistical results.
Calculator
Results
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Data Source and Methodology
Authoritative source: NIST/SEMATECH e-Handbook of Statistical Methods (2012, updated). Section: “Measures of Variability”. National Institute of Standards and Technology. Direct link.
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.
The Formula Explained
Population standard deviation
σ = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2 }
where μ is the population mean and N is the population size.
Sample standard deviation
s = \sqrt{ \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 }
where x̄ is the sample mean and n is the sample size. The n−1 term is Bessel’s correction.
Additional statistics
\text{Mean }(\bar{x}) = \frac{1}{n}\sum x_i, \quad \text{Range} = \max(x) - \min(x)
\text{SEM} = \frac{s}{\sqrt{n}}, \quad \text{CV} = \frac{\text{SD}}{|\bar{x}|}
Glossary of Variables
- Data set: The list of numeric observations to analyze.
- Sample: Use when your data are a subset of a larger population; uses n−1 in the denominator.
- Population: Use when your data comprise the entire population; uses N in the denominator.
- Mean (x̄/μ): Average of all values.
- Variance (s²/σ²): Average of squared deviations from the mean.
- Standard deviation (s/σ): Square root of variance, in the same units as the data.
- Median: Middle value (or average of two middle values) when data are ordered.
- Range: Difference between maximum and minimum.
- Coefficient of variation (CV): Relative dispersion: SD divided by absolute mean (unitless).
- Standard error of the mean (SEM): Estimated uncertainty of the sample mean, s/√n.
Worked Example
How It Works: A Step-by-Step Example
Consider the data set: 2, 4, 4, 4, 5, 5, 7, 9.
- Compute the mean: \bar{x} = (2+4+4+4+5+5+7+9)/8 = 40/8 = 5
- Compute squared deviations and sum: \sum (x_i - \bar{x})^2 = 32
- Population variance and SD: \sigma^2 = 32/8 = 4,\quad \sigma = \sqrt{4} = 2
- Sample variance and SD: s^2 = 32/7 \approx 4.571428,\quad s \approx 2.138090
You can load this example into the calculator with the “Load example” button to verify the results.
Frequently Asked Questions (FAQ)
When should I choose sample vs population?
Choose sample when your data represent a subset of a larger population; choose population when you have every member of the population. If unsure, use sample.
Why does sample SD use n−1?
Using n−1 (Bessel’s correction) corrects the bias in estimating the population variance and SD from a sample.
What separators are supported for input?
Commas, spaces, tabs, semicolons, and new lines. Multiple separators and blank lines are handled gracefully.
Do you handle scientific notation?
Yes. Inputs like 1e3, 2.5e-2, and -3.2E+1 are parsed as valid numbers.
Can I sort the data without changing results?
Yes. Sorting affects display order only. Statistics are order-invariant.
What if some tokens are not numbers?
They are ignored, and we show a note indicating how many were skipped so you can correct your data if needed.
How is rounding handled?
Enter the number of decimal places to round the display. Internal calculations use double precision and are then rounded for presentation.
Formula (LaTeX) + variables + units
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Population standard deviation σ = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2 } where μ is the population mean and N is the population size. Sample standard deviation s = \sqrt{ \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 } where x̄ is the sample mean and n is the sample size. The n−1 term is Bessel’s correction. Additional statistics \text{Mean }(\bar{x}) = \frac{1}{n}\sum x_i, \quad \text{Range} = \max(x) - \min(x) \text{SEM} = \frac{s}{\sqrt{n}}, \quad \text{CV} = \frac{\text{SD}}{|\bar{x}|}
- No variables provided in audit spec.
- Direct link — itl.nist.gov · Accessed 2026-01-19
https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc35.htm
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
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- Confirm sources are authoritative and relevant to the calculator methodology.