LaTeX
\sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2Variables
- N: population size
- μ: population mean
- xᵢ: observations
How to use this formula
Measures average squared deviation from the population mean.
Important notes
- Use N, not N−1, for a complete population.
Quick example
For 1,2,3, μ=2 and σ²=2/3.
Applicability, worked calculation, and verification
Assumptions and domain checks
- For the Population Variance, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For the Population Variance, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.
- For the Population Variance, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions.
Worked example
For 1,2,3, μ=2 and σ²=2/3.
Common mistakes
- When copying Population Variance, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- For the Population Variance, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions.
Continue the workflow
Use Population 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
\sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Introductory Statistics 2eOpenStax, Rice University — Reviewed probability and statistics definitions, notation, and formulas.
Frequently asked questions
What is the Population Variance used for?
Measures average squared deviation from the population mean.
Can I copy this formula as LaTeX?
Yes. Copy \sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^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.