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Standard Deviation Calculator

Calculate sample or population standard deviation, variance, mean, deviations, squared deviations, and coefficient of variation from a data set.

  • standard deviation
  • sample standard deviation
  • population standard deviation
  • variance calculator
Data type

Status: initial

Results

Awaiting calculation

Calculator guide

Measure how far values spread from the mean

This Standard Deviation Calculator computes the selected sample or population measure and exposes the underlying mean, variance, deviations, squared deviations, and coefficient of variation.

Choose sample or population deliberately

Use population mode when the entered list is the entire group of interest. Use sample mode when the values are observations drawn from a larger population and the n − 1 correction is appropriate.

How standard deviation is calculated

Find the mean, subtract it from every value, square each deviation, add the squares, divide by N or n − 1, and take the square root. Variance remains in squared units; standard deviation returns to the original unit.

Variable explanations

Understand what each input and result means before calculating.

Data values

The numerical observations being summarized.

Mean

The arithmetic centre used to calculate deviations.

Deviation

An observation minus the mean.

Variance

The average squared deviation using the selected divisor.

Standard deviation

The square root of variance, expressed in the data’s original unit.

Reviewed by the Calculator.org.in Editorial Team

Formula behavior, validation cases, explanatory examples, and cited sources are checked before publication. This review supports educational accuracy and is not a substitute for qualified professional advice.

Last reviewed: 2026-08-08

Review process

Formula guide

See the calculation logic, variable definitions, and practical meaning.

Population standard deviation

σ = √[Σ(x − μ)² / N]

  • μ is population mean
  • N is population size

Use when the list contains the full population.

Sample standard deviation

s = √[Σ(x − x̄)² / (n − 1)]

  • x̄ is sample mean
  • n is sample size

The n − 1 divisor corrects estimation bias in sample variance.

Coefficient of variation

CV = |standard deviation / mean| × 100%

  • Mean must be non-zero

CV compares relative spread across data measured on a ratio scale.

Worked examples

Follow realistic inputs through the calculation step by step.

1

Worked example

Population with standard deviation 2

  1. 1Enter 2, 4, 4, 4, 5, 5, 7, 9.
  2. 2Choose Population.
  3. 3Mean is 5, variance is 4, and standard deviation is 2.
2

Worked example

Sample calculation

  1. 1Enter 4, 7, 13, 16.
  2. 2Choose Sample.
  3. 3The n − 1 divisor produces sample variance 30 and standard deviation about 5.4772.
3

Worked example

No spread

  1. 1Enter 6, 6, 6, 6.
  2. 2Every deviation is zero.
  3. 3Variance and standard deviation both equal zero.

Common mistakes

Avoid these common input and interpretation errors.

Mixing sample and population formulas

The denominators differ and can materially change small-data results.

Taking the mean of absolute differences

Standard deviation uses squared deviations, not absolute deviations.

Reporting variance as standard deviation

Take the square root of variance to return to the original measurement unit.

Ignoring units

Standard deviation has the same unit as the observations; variance has squared units.

Frequently asked questions

Quick answers to the questions users ask most often.

What does a high standard deviation mean?
Values are more dispersed around the mean, relative to data with a lower standard deviation in the same unit and context.
Can standard deviation be negative?
No. It is the non-negative square root of variance.
What is the difference between variance and standard deviation?
Variance is the average squared deviation; standard deviation is its square root.
Why does sample variance divide by n − 1?
The correction compensates for estimating the population mean from the same sample.
Can I paste spreadsheet data?
Yes. Use commas, spaces, or line breaks between values.
Does standard deviation prove data are normally distributed?
No. It measures spread and does not by itself identify a distribution shape.

References

Sources used to support the calculator guidance.

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Version history

A transparent record of calculator content updates.

Updated 2026-08-08
  • 1.0.0 · 2026-08-08

    Initial release with sample and population formulas, deviation table, variance, and coefficient of variation.