Sigma Level Calculator
Convert a defect rate or yield into a sigma level, with and without the 1.5 shift.
What this calculator does
The defect rate is converted with Acklam's rational approximation to the inverse standard normal (absolute error below 1.15e-9). The independent check solves the same quantile by bisection on a separately written normal distribution function.
Inputs and what they mean
- What are you starting from?
- — Only the fields for the basis you choose are used; the others are ignored..
- Defects found (optional)
- — Optional. Leave at 0 to exclude it from the result. Used when you start from counts. Defects, not defective units..
- Units inspected (optional)
- — Optional. Leave at 0 to exclude it from the result. Used when you start from counts..
- Opportunities per unit (optional)
- — Optional. Leave at 0 to exclude it from the result. Used when you start from counts. Distinct ways one unit can be defective..
- Defects per million opportunities (optional)
- — Optional. Leave at 0 to exclude it from the result. Used when you start from a DPMO figure..
- Yield (optional)
- — Optional. Leave at 0 to exclude it from the result. Used when you start from a yield percentage..
- 1.5 sigma shift
- — The 1.5 sigma shift is a widely used convention, not a measurement of your process. Both figures are shown either way..
How to use it
- Enter your own figures — the calculator never fills in a rate, price or benchmark for you.
- Press Calculate to see the result.
- Read the formula, variables, assumptions and source below the result before you rely on it.
Formula
DPMO = (Defects ÷ (Units × Opportunities)) × 1,000,000; Z_LT = Φ⁻¹(1 − DPMO ÷ 1,000,000); Sigma level = Z_LT + 1.5
Worked example
233 defects in 500 units with 10 opportunities each
Converted to defects per million opportunities and then to a sigma level.
Reading the result
The headline figure is the main answer. Any breakdown underneath shows the parts that make it up, so you can check the working and see what changes when you adjust an input.
Limitations and assumptions
Results depend entirely on the figures you enter and are rounded for display. They are for general information and education, not professional advice.
Reference: American Society for Quality — Process capability and defect rate measures
Last reviewed:
Common questions
Formula, source and verification
The defect rate is converted to a long-term Z through the inverse standard normal distribution; the sigma level adds the conventional 1.5 sigma shift, which is reported separately from the unshifted Z.
The question it answers: How does the process perform over the long run, and what sigma level does its defect rate imply?
The formula
DPMO = (Defects ÷ (Units × Opportunities)) × 1,000,000; Z_LT = Φ⁻¹(1 − DPMO ÷ 1,000,000); Sigma level = Z_LT + 1.5
- D — Defects
- (count). Defects found, not defective units.
- U — Units
- (count). Units inspected.
- O — Opportunities per unit
- (count). Distinct ways one unit can be defective.
- Z_LT — Long-term Z
- (sigma). Inverse standard normal of the yield; no shift applied.
- Φ⁻¹ — Inverse standard normal
- (—). The quantile function of the standard normal distribution.
Units: Counts, DPMO or a yield percentage in; a sigma level out.
What kind of calculation this is
Deterministic formula. The same inputs always give the same answer. The maths is fixed and does not depend on judgement.
Method
The defect rate is converted with Acklam's rational approximation to the inverse standard normal (absolute error below 1.15e-9). The independent check solves the same quantile by bisection on a separately written normal distribution function.
Assumptions built into the result
- Statistical: The defect rate is treated as the tail area of a normal distribution, which is the convention this measure is built on.
- Mathematical: Opportunities per unit have been counted consistently.
Figures this calculator will never guess for you
- No benchmark, target or 'world class' value is supplied or implied; the result describes only the figures entered.
- No target sigma level is stated; 'six sigma' as an organisational goal is a target, not part of this formula.
- A zero-defect sample is refused rather than converted to an invented sigma level.
Limitations
- A sample with no defects cannot produce a sigma level; the calculator says so instead of showing a number.
- Sigma levels are only comparable between processes when opportunities are counted the same way.
- The 1.5 shift is a convention, not a measurement of the process.
Source and version
- Standard or reference
- Sigma level from defects per million opportunities — ASQ (ASQ quality resources, six sigma measures: a defect rate is converted to a sigma level through the normal distribution, conventionally with a 1.5 sigma long-term shift. The inverse normal itself follows the NIST/SEMATECH e-Handbook definition of the standard normal quantile.)
- Published source
- American Society for Quality — Quality resources
- Formula version
- Version 1
- Verification
- Reviewed against the cited source on
- How much weight the source carries
- Standards or government
- Applies to
- Currency
- The result is a ratio or index, so it does not depend on currency.
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