s^2=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2Variables
- s²: sample variance
- xᵢ: observation
- x̄: sample mean
- n: sample size
How to use this formula
Measures sample spread using squared deviations and Bessel’s correction.
Important notes
- The denominator n − 1 is used when estimating population variance from a sample.
- Variance is expressed in squared units.
Quick example
For 1, 2, 3, the sample variance is 1.
Applicability, worked calculation, and verification
Domain and applicability
Uses the n − 1 denominator for a sample estimate when n > 1; population variance uses a different denominator.
Units
- variance uses the square of the observation unit
Assumptions and domain checks
- The denominator n − 1 is used when estimating population variance from a sample.
- For the Sample Variance, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For the Sample Variance, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.
- For the Sample Variance, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions.
Do not use this formula when
- Do not use Sample Variance until the sample, event, distribution, independence assumptions, and parameter convention match the problem.
Boundary and special cases
- For Sample Variance, check zero, negative, and extreme input values before relying on the result.
- When using Sample Variance, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form s^2=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2 for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For 2, 4, 6, 8, the mean is 5. Squared deviations sum to 20, so s² = 20/(4 − 1) = 20/3 ≈ 6.667.
- Calculate the sample mean: x̄ = 5.
- Compute deviations −3, −1, 1, and 3, then square them.
- Add the squared deviations: 9 + 1 + 1 + 9 = 20.
- Divide by n − 1 = 3 to obtain approximately 6.667.
Independent verification
The variance is nonnegative, and taking its square root gives a sample standard deviation of about 2.582 in the original unit.
Common mistakes
- When copying Sample Variance, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- For the Sample Variance, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions.
Continue the workflow
Use Sample Variance in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
s^2=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Sample Variance.
Verified against: OpenStax Introductory Statistics
Automated quality check: Passed the core formula indexing gate.
Formula references
- Introductory Statistics 2eOpenStax, Rice University — Reviewed probability and statistics definitions, notation, and formulas.
Frequently asked questions
What is the Sample Variance used for?
Measures sample spread using squared deviations and Bessel’s correction.
Can I copy this formula as LaTeX?
Yes. Copy s^2=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2 or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.