How to calculate DPMO and sigma level
You need three counts: units inspected, defects found and defect opportunities per unit. An opportunity is a spot on the unit where a defect can occur and that you actually check: a wire harness with 10 crimps has 10 opportunities.
DPU = defects ÷ units
DPO = defects ÷ (units × opportunities)
DPMO = DPO × 1,000,000
Sigma level = Z(1 − DPO) + 1.5
| Term | Meaning |
|---|---|
| DPU | average defects per unit |
| DPO and DPMO | defects per opportunity, and the same rate per million opportunities |
| Yield | 1 − DPO: the share of opportunities without a defect |
| Z | the standard normal value with the yield to its left (NORM.S.INV in Excel) |
| Defective PPM | bad units per million units: defectives ÷ units × 1,000,000 |
The 1.5 sigma shift
The Z you get from the yield is long-term performance. The sigma level in six sigma tables adds 1.5 to it. The idea came out of Motorola in the 1980s (Bill Smith and Mikel Harry): a process mean drifts by up to 1.5 standard deviations over months, so the process “in the short term” is taken to be 1.5 sigma better than its defect count shows.
It is a convention, not a law. Nobody measured 1.5 on every process, and writers such as Donald Wheeler argue against it. But it is what turns 3.4 DPMO into six sigma, and it is what everyone means when they compare sigma levels. The calculator shows both: the conventional sigma level and the long-term Z without the 1.5.
Sigma level table
With the 1.5 shift:
| Sigma level | DPMO | Yield |
|---|---|---|
| 1σ | 690,000 | 31% |
| 2σ | 308,538 | 69.1% |
| 3σ | 66,807 | 93.32% |
| 4σ | 6,210 | 99.379% |
| 5σ | 233 | 99.9767% |
| 6σ | 3.4 | 99.99966% |
Worked example
The values preloaded in the calculator come from a wire harness inspection:
- 5,000 harnesses inspected, each with 10 crimps (10 opportunities)
- 35 defects found, on 28 harnesses (a few had more than one)
The math:
- DPU = 35 ÷ 5,000 = 0.007 defects per harness
- DPO = 35 ÷ (5,000 × 10) = 35 ÷ 50,000 = 0.0007
- DPMO = 0.0007 × 1,000,000 = 700
- Yield = 1 − 0.0007 = 99.93%
- Long-term Z = Z(0.9993) = 3.19
- Sigma level = 3.19 + 1.5 = 4.69σ, between 4σ and 5σ on the table
With the 28 defective units: defective PPM = 28 ÷ 5,000 × 1,000,000 = 5,600, and harness first pass yield is 1 − 28 ÷ 5,000 = 99.44%. Note the gap: 700 DPMO and 5,600 PPM describe the same lot. DPMO divides by opportunities, PPM counts units.
The Poisson yield, e^(−0.007) = 99.30%, estimates first pass yield when all you have is the defect total, assuming defects land on units at random. Here it comes close to the actual 99.44%.
First pass yield and rolled throughput yield
A step’s first pass yield (FPY) is the share of units that come out right the first time, with no rework or scrap:
FPY = units good the first time ÷ units into the step
RTY = FPY₁ × FPY₂ × … × FPYₙ
The calculator’s example has four steps: cutting 99.2%, crimping 97.5%, assembly 98% and electrical test 99%. Rolled throughput yield = 0.992 × 0.975 × 0.98 × 0.99 = 93.8%. Each step looks fine on its own, yet only 94 harnesses in 100 make it through the whole line untouched. Crimping costs the most, so that is where improvement starts.
RTY is always lower than the worst step’s FPY, and it exposes the “hidden factory”: rework that never shows in final yield because the fixed unit ships as good.
How to read the result
- Compare processes by DPMO, not by defect count. A complex part has more opportunities and will show more defects even when it is well made.
- Below 3σ (over 66,807 DPMO) the process makes defects all the time and inspection becomes routine. 3σ to 4σ is where many plants sit. 4σ and up is a good process for most industries.
- Use sigma level to track a trend on the same process, month by month. To go after the defects, build a Pareto chart by defect type and find the few that make up most of them.
- If the characteristic is measured (a diameter, a torque), use Cp and Cpk instead: they tell spread from centering, which a defect count cannot.
Common mistakes
- Padding the opportunities. Counting 50 opportunities where only 10 get checked cuts DPMO and improves sigma level on paper. Count only what is inspected and can really fail, and keep the definition stable over time.
- Mixing up DPMO and PPM. A customer asking for PPM wants defective units, not defects per opportunity.
- Forgetting the 1.5 when comparing. A long-term Z of 3 is 4.5 on the conventional scale. Always say which one you mean.
- Small samples. With few defects, one more or less moves the result a lot. With zero defects there is no sigma level to compute: inspect more.
- Multiplying final yields. RTY uses each step’s first pass yield, counting a reworked unit as a miss. Using yield after rework hides exactly what RTY is meant to show.
Defects cost money too: see what scrap and rework cost each month with the cost of poor quality calculator.