What this calculator does
You have an inspection record: so many units checked, so many defects found. This calculator turns that into the four numbers a quality report actually needs — DPU, DPO, DPMO and a process sigma level — plus the yield figures that tell you what the defect rate costs you in rework.
It reports the sigma level both ways: with the conventional 1.5σ long-term shift applied, and as measured. The difference is a full 1.5 sigma, so anyone quoting a sigma level without saying which convention they used is quoting a number that cannot be checked.
The one input that needs thought is opportunities per unit. Everything downstream of DPO depends on it, and it is the only input nobody else can supply for you.
The formula
Three layers. The first is arithmetic, the second is a Poisson model, the third is a normal-distribution lookup.
DPU = D / U defects per unit
DPO = D / (U × O) defects per opportunity
DPMO = DPO × 1 000 000
Y = 1 − DPO opportunity-level yield
RTY = e^(−DPU) rolled throughput yield
Y_nrm = RTY^(1/k) = e^(−DPU/k) normalised yield of an average step
Z_lt = Φ⁻¹(1 − DPO) long-term sigma, measured as-is
Z_st = Z_lt + 1.5 process sigma, conventional scale
Cpk ≈ Z / 3
D is total defects, U units inspected, O opportunities per unit, k process steps, and Φ⁻¹ the standard normal quantile function (this tool evaluates it with Acklam’s rational approximation, accurate to about nine significant figures — no table lookup, so the answer is continuous rather than stepped).
Two of these deserve explanation.
RTY is not 1 − DPU. A unit can carry two defects. If your line runs at DPU 0.035, the fraction of units needing rework is not 3.5% but 1 − e^(−0.035) = 3.44% — close here, but at DPU 1.0 the difference is the gap between “100% defective” (nonsense) and 63.2% (correct). The Poisson form is also the only one that composes: the RTY of a chain of steps is the product of the step yields, which is exactly e^(−ΣDPU).
The 1.5σ shift is a convention, not a physical law. It comes from Motorola in the 1980s, as an allowance for a process mean that wanders over the long run by up to 1.5 standard deviations relative to what a short capability study would show. There is no derivation of it. It is an empirical rule of thumb that got frozen into the definition of “six sigma” so that 6σ would correspond to 3.4 DPMO rather than the 0.002 DPMO a genuinely centred normal process would give. Use it, because everyone else does, but say so, and never apply it to a process whose stability you have actually measured — measure the drift instead.
Reading the result
DPMO is the comparable number, and only comparable between processes that count opportunities the same way. Sigma level is DPMO restated on a scale people find intuitive. RTY is the one that maps to money: it is the proportion of units that get through the entire process with no rework at all, and 1 − RTY sized against your throughput is the hidden factory.
The normalised yield per step is the sanity check on RTY. A process with 20 steps each running at 99% has an RTY of 81.8% — each step looks excellent and the line as a whole is poor. If your step owners are all reporting good yields and the plant is still reworking one unit in five, this is why.
The equivalent Cpk is the bridge to a variables capability study. Cpk = Z/3, so 6σ ↔ Cpk 2.0, 4.5σ ↔ 1.5, 3σ ↔ 1.0. Note that if you selected “no shift”, the number this calculator labels Cpk is strictly Ppk — a long-term index — because it was derived from a long-term defect count.
The single largest source of dishonest sigma levels is the opportunity count. Credit each unit with 50 opportunities instead of 5 and DPMO falls by a factor of ten and the sigma level rises by roughly 0.6, without a single defect being prevented. That is why this tool warns above 6σ: at that level the opportunity definition is far more likely to be the cause than the process.
Typical values
Rough anchors on the conventional shifted scale, for orientation only:
| Sigma (shifted) | DPMO | Yield per opportunity |
|---|---|---|
| 3.0σ | 66 807 | 93.32% |
| 3.5σ | 22 750 | 97.73% |
| 4.0σ | 6 210 | 99.379% |
| 4.5σ | 1 350 | 99.865% |
| 5.0σ | 233 | 99.9767% |
| 6.0σ | 3.4 | 99.99966% |
Most established manufacturing sits between 3.5σ and 4.5σ. Transactional and administrative processes are often below 3σ. Safety-critical aviation and medical-device processes are the realistic home of 5σ and above. Anything reported above 6σ from a sample of a few thousand units is measuring the sample size, not the process — you cannot demonstrate 3.4 DPMO without on the order of a million opportunities.
Worked example
An assembly line inspects 1200 units, each carrying 8 defect opportunities, over a process of 5 steps, and finds 42 defects. The 1.5σ shift is applied.
Total opportunities: 1200 × 8 = 9600.
DPU = 42 / 1200 = 0.035
DPO = 42 / 9600 = 0.004375
DPMO = 0.004375 × 1 000 000 = 4375
Y = 1 − 0.004375 = 0.995625 → 99.5625%
Rolled throughput yield:
RTY = e^(−0.035) = 0.9656054 → 96.5605%
Y_nrm = e^(−0.035 / 5) = e^(−0.007) = 0.9930244 → 99.3024%
So each of the five steps averages 99.30% yield, and yet 3.44 units in every 100 leave the line having needed rework somewhere.
Sigma level. We need the z with an upper-tail area of 0.004375:
Z_lt = Φ⁻¹(1 − 0.004375) = 2.6217
Z_st = 2.6217 + 1.5 = 4.1217 → 4.12σ
Cpk = 4.1217 / 3 = 1.374
4.12σ, 4375 DPMO, Cpk 1.37. A respectable, unremarkable manufacturing process — and one where the useful lever is the 3.44% rework rate, not the sigma number.
FAQ
Should I count defects or defective units? Defects. A unit with three defects contributes three. If your data only records defective units you can still use the tool — enter defective units as defects — but be aware DPU is then a lower bound and RTY will come out optimistic.
How do I decide the number of opportunities? Count the distinct ways the unit can fail that you actually inspect for, independently. A solder joint is an opportunity; a “solder quality” attribute covering 200 joints is one opportunity, not 200, unless you inspect all 200 separately. Whatever rule you choose, write it down and apply it consistently — a sigma level without its opportunity definition is uninterpretable.
Why does 3.4 DPMO equal six sigma?
Because of the shift. Φ⁻¹(1 − 3.4e-6) = 4.4998, and 4.4998 + 1.5 = 6.0. Without the shift, a centred normal process with specification limits at ±6σ would produce 0.00197 DPMO — about two defects per billion.
Can I compare a sigma level to a Cpk from a capability study?
Yes, via Cpk = Z/3, but only loosely. Cpk comes from measured variation against specification limits and assumes normality; a sigma level from DPMO comes from an attribute count and assumes nothing about the distribution. They agree when the process is normal and the defect definition is exactly “outside specification”. Use the equivalence to communicate, not to substitute one study for the other.
What if I found zero defects? Then the tool reports a bound, not a value, using DPO = 0.5 / (total opportunities). Zero defects in 9600 opportunities is consistent with anything from a perfect process to about 300 DPMO. Inspect more, or move to a variables measurement, which extracts far more information per unit.
This tool implements the standard DPMO and sigma-level definitions for analysis and reporting. It is not a conformity assessment; capability and acceptance decisions must follow the standard applicable in your jurisdiction and the requirements agreed with your customer.