z=\frac{x-\mu}{\sigma}Variables
- z: standardized score
- x: observed value
- μ: population mean
- σ: population standard deviation
How to use this formula
Expresses how many population standard deviations a value lies from the mean.
Important notes
- A positive z-score lies above the mean; a negative one lies below.
- Use sample statistics only when the method specifically calls for them.
Quick example
If x = 70, μ = 60, and σ = 5, then z = 2.
Applicability, worked calculation, and verification
Domain and applicability
The standard deviation must be positive; interpretation as a normal-distribution percentile additionally requires an appropriate distribution model.
Units
- x, μ, and σ use the same unit
- z is dimensionless
Assumptions and domain checks
- A positive z-score lies above the mean; a negative one lies below.
- For the Z-Score Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For the Z-Score Formula, 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 Z-Score Formula until the sample, event, distribution, independence assumptions, and parameter convention match the problem.
Boundary and special cases
- For Z-Score Formula, check zero, negative, and extreme input values before relying on the result.
- When using Z-Score Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form z=\frac{x-\mu}{\sigma} for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For x = 85, mean μ = 70, and standard deviation σ = 10, z = (85 − 70)/10 = 1.5.
- Subtract the mean from the observation: 85 − 70 = 15.
- Confirm the standard deviation is positive: σ = 10.
- Divide the deviation by the standard deviation: 15/10.
- Report z = 1.5, meaning the value is 1.5 standard deviations above the mean.
Independent verification
Rearranging x = μ + zσ gives 70 + 1.5(10) = 85, recovering the original observation.
Common mistakes
- When copying Z-Score Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- For the Z-Score Formula, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions.
Continue the workflow
Use Z-Score Formula 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
z=\frac{x-\mu}{\sigma}. - 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 Z-Score Formula.
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 Z-Score Formula used for?
Expresses how many population standard deviations a value lies from the mean.
Can I copy this formula as LaTeX?
Yes. Copy z=\frac{x-\mu}{\sigma} 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.