\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_iVariables
- x̄: sample mean
- xᵢ: ith observation
- n: number of observations
How to use this formula
Calculates the arithmetic average of n observed values.
Important notes
- Every observation contributes equally.
- The sample mean is sensitive to extreme values.
Quick example
The mean of 2, 4, and 9 is 5.
Applicability, worked calculation, and verification
Domain and applicability
Applies to quantitative observations with sample size n > 0; sensitivity to extreme values should be considered for skewed data.
Units
- the sample mean has the same unit as the observations
Assumptions and domain checks
- Every observation contributes equally.
- For the Sample Mean, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For the Sample Mean, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.
- For the Sample Mean, 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 Mean until the sample, event, distribution, independence assumptions, and parameter convention match the problem.
Boundary and special cases
- For Sample Mean, check zero, negative, and extreme input values before relying on the result.
- When using Sample Mean, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For the sample 2, 4, 6, 8, the mean is (2 + 4 + 6 + 8)/4 = 20/4 = 5.
- Add all sample observations to obtain a total of 20.
- Count the observations: n = 4.
- Divide the total by the sample size: 20/4.
- Report the sample mean as 5 in the original measurement unit.
Independent verification
The deviations from 5 are −3, −1, 1, and 3, which sum to zero as deviations from an arithmetic mean should.
Common mistakes
- When copying Sample Mean, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- For the Sample Mean, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions.
Continue the workflow
Use Sample Mean 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
\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i. - 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 Mean.
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 Mean used for?
Calculates the arithmetic average of n observed values.
Can I copy this formula as LaTeX?
Yes. Copy \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i 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.