How sigma level is calculated
A sigma level is a way of writing a defect rate as a z-value. If a fraction \(p\) of opportunities is defective, the long-term z-value is the point on the standard normal distribution with that much area beyond it:
DPMO is \(10^6 p\), yield is \(1 - p\), and \(\Phi^{-1}\) is the inverse standard normal CDF. The conversion is one-sided: it treats all defects as lying beyond a single limit. That matches the published Six Sigma tables, where 6σ corresponds to 3.4 DPMO.
The 1.5σ shift, honestly
Motorola's Six Sigma program assumed that a process studied over a short period would drift by about 1.5 standard deviations over the long term. Sigma levels are therefore quoted as short-term capability: a “6σ” process is expected to deliver the long-term defect rate of a 4.5σ one-sided tail, 3.4 DPMO. Without the shift, 6σ would mean about 0.001 DPMO one-sided, or 0.002 two-sided.
The 1.5σ figure is a convention, not a law of nature. Some processes drift much less, such as a well-controlled laser process with closed-loop power monitoring. Others drift more, such as tool wear without compensation. Critics note that the shift makes the scale hard to interpret and can hide poor long-term performance. For engineering decisions, it is clearer to quote measured long-term performance directly, as Ppk or observed PPM, and to state which convention any sigma level uses.
| Sigma level | DPMO (1.5σ shift) | Yield (1.5σ shift) | DPMO, one-sided, no shift | PPM, ±Zσ two-sided, centered |
|---|---|---|---|---|
| 1 | 691,462 | 30.85 % | 158,655 | 317,311 |
| 2 | 308,538 | 69.15 % | 22,750 | 45,500 |
| 3 | 66,807 | 93.32 % | 1,350 | 2,700 |
| 4 | 6,210 | 99.379 % | 31.7 | 63.3 |
| 5 | 233 | 99.9767 % | 0.287 | 0.573 |
| 6 | 3.4 | 99.99966 % | 0.00099 | 0.0020 |
DPU, DPO, and first-pass yield
If defects land on units at random, the number of defects per unit follows a Poisson distribution. The chance a unit has none is then \(e^{-DPU}\). If the observed first-pass yield is clearly higher than the Poisson estimate, defects are clustering on a few bad units, which points to a specific cause. The rolled throughput yield of a multi-step process is the product of each step's first-pass yield, \(RTY = \prod_i FPY_i\). It is the probability that a unit passes every step without rework, which is the yield the factory floor actually experiences.
Worked example
An inspection finds 38 defects on 1200 units, with 12 defect opportunities per unit. That gives DPU = 0.0317, DPO = 0.00264, and DPMO = 2639. This corresponds to Z = 2.79 long-term, or a 4.29σ short-term sigma level. The Poisson first-pass yield, \(e^{-0.0317} = 96.88\,\%\), matches the observed 1163/1200 = 96.92 %, so defects are spread randomly rather than clustered. Five steps with first-pass yields of 98.5, 99.2, 97.8, 99.6, and 98.9 % give RTY = 94.13 %. About one unit in seventeen needs rework somewhere, even though no single step looks worse than 97.8 %.
Assumptions and limits
- Opportunity counts are a judgment call. Counting more opportunities per unit lowers DPMO and raises the sigma level without any change to the product. Define opportunities once, document the definition, and keep it fixed when comparing over time.
- Normal tails. Converting a defect rate to a z-value assumes normally distributed variation. For attribute defects (missing features, contamination) the sigma level is only a common scale, not a physical distance to a limit.
- One-sided convention. Every defect is assigned to one tail. For a centered process with two-sided limits, use the two-sided column in the table or the capability calculator.
- Independent steps. RTY assumes step yields are independent. Upstream defects that make downstream failures more likely make the true RTY lower.
References
- D. C. Montgomery, Introduction to Statistical Quality Control, 8th ed., Wiley (chapter on Six Sigma and process capability).
- AIAG, Statistical Process Control (SPC) Reference Manual, 2nd ed. (2005).