A thousand units with five ways each to go wrong, and seventeen defects found. That is 3,400 defects per million opportunities and a yield of 99.66% — which sounds excellent and is 4.21 sigma with the conventional shift, or 2.71 sigma without it. The rolled throughput yield, the chance a unit passes every one of its five opportunities untouched, is 98.31%.
1,000 units × 5 opportunities = 5,000
3,400.00 DPMO — 4.21 sigma
That sigma level includes the conventional 1.5 sigma allowance for long-term drift. Without it the same defect rate is 2.71 sigma — the two differ by exactly 1.5 by construction, and almost every published figure uses the first without saying so. The yield per opportunity is 99.6600%, and the rolled throughput yield — the chance a unit clears all 5 opportunities — is 98.3144%.
DPMO
3,400.00
0.01700 defects per unit
Sigma level
4.206
with the 1.5 shift
Without the shift
2.706
the short-term figure
Rolled throughput yield
98.314%
per opportunity 99.6600%
Yield per opportunity is 99.6600% and rolled throughput yield is 98.3144%. The gap is compounding: a unit has to clear 5 opportunities, and 99.66% raised to that power is a smaller number than it looks.
What a sigma level actually means
Defects per million at each sigma level, with and without the 1.5 shift
Sigma
DPMO with the shift
DPMO without it
Ratio
3
66,807.2
1,349.90
49.5×
3.5
22,750.1
232.63
97.8×
4 ←
6,209.7
31.67
196.1×
4.5
1,349.9
3.40
397.3×
5
232.6
2.8665e-1
811.5×
5.5
31.7
1.8990e-2
1,667.8×
6
3.4
9.8659e-4
3,443.9×
The familiar “3.4 defects per million” is the shifted column at six sigma. The unshifted figure is 0.00099 — a factor of 3,444 — and the gap widens as the sigma level rises. Neither number is wrong; quoting one as though it were the only one is.
Both sigma conventions Rolled throughput yield The shift made explicit Opportunity count is a judgement
What this tool shows
“Six sigma is 3.4 defects per million” already contains a 1.5 sigma allowance for long-term drift. Without it, six sigma is 0.00098659 defects per million — a factor of 3,444. Both are computed here rather than one being quoted as though it were the only one. The gap widens as the level rises: 49.5× at three sigma, 196× at four, 3,444× at six.
Defects per million opportunities from units, opportunities and defect count
Sigma level computed both ways — with the conventional 1.5 shift and without it
Rolled throughput yield, which compounds across opportunities and is always below the per-opportunity yield
Defects per unit alongside defects per opportunity
The full sigma-to-DPMO table under both conventions, with the ratio between them
Why the opportunity count is the softest number in the calculation
Both conventions Rolled yield Shift made explicit Conversion table
Raising the opportunity count lowers DPMO without changing the process.
Updated 13 September 2026 · Works in any browser, no installation
DPMO is defects divided by units times opportunities, scaled to a million. It exists to make processes of different complexity comparable: a unit with fifty chances to go wrong and a unit with two are not comparable on defect count alone. The sigma level is that rate read back through a normal distribution — and it is read back with a 1.5 sigma allowance added, by a convention that is almost never stated alongside the number.
At a glance
Formula shown
DPO = defects / (units × opportunities), and DPMO = DPO × 10⁶. The sigma level is Φ⁻¹(1 − DPO) + 1.5, where the 1.5 is the conventional allowance for long-term drift; without it the level is simply Φ⁻¹(1 − DPO). Rolled throughput yield is e^(−DPU) with DPU = defects/units, which is the probability a unit passes every opportunity untouched and is always below the per-opportunity yield.
Scenario support
Six Sigma project baselines and after-measures, comparing processes with different numbers of failure modes, setting improvement targets in DPMO rather than defect counts, supplier quality scorecards, and translating a defect rate into the sigma language a steering committee expects.
Educational estimate
Planning support from the values you enter — not professional advice.
The 1.5 sigma nobody mentions
The most quoted number in quality management contains an adjustment that is rarely stated alongside it.
Six sigma from a normal distribution is 0.00098659 defects per million. Two parts per billion.
The familiar figure is 3.4 per million, which is the tail at 4.5 sigma. The difference is a 1.5 sigma allowance for the process mean drifting over the long run.
That is a factor of 3,444, and it grows with the level: 49.5× at three sigma, 196× at four, 812× at five.
Neither number is wrong and the convention is defensible. Quoting one as though it were the only one is the problem, which is why both are printed here and the table shows them side by side.
The opportunity count is a judgement
DPMO looks like a measurement and contains a number that is decided rather than observed — and that number moves the result directly.
An opportunity is a distinct way one unit can be defective. How many a unit has is a choice about how finely to subdivide it.
The shipped transactional preset shows the leverage: three errors in five hundred cases is 500 DPMO at twelve opportunities each and 6,000 DPMO at one. The same three errors, and 4.79 sigma against 4.01.
So a DPMO is only comparable against another DPMO counted the same way. Across organisations, and often across departments, it is not.
The discipline is to fix the opportunity definition before measuring anything and to publish it with the number. Where that is impossible, defects per unit is the honest alternative: it has no opportunity count in it at all.
Per-opportunity yield flatters, rolled yield does not
A yield of 99.3% per opportunity sounds close to perfect. It is not what a customer experiences.
Rolled throughput yield is the chance a unit clears every opportunity untouched, which is the per-opportunity yield compounded.
On the struggling preset that is 99.30% per opportunity and 94.55% rolled, across eight opportunities. One unit in eighteen leaves with something wrong.
The gap widens with complexity. At 99.9% per opportunity, a unit with 100 of them has a rolled yield of 90.5%; with 500, it is 60.6%.
Which is why DPMO alone is a poor project metric on complex products. It can fall while rolled yield stays flat, if the opportunity count grew.
Defects, not defective units
The commonest arithmetic error here is counting the wrong thing, and it always moves the answer in the same direction.
A defect is one thing wrong. A defective unit is a unit with at least one. A unit with three defects is one defective unit and three defects.
DPMO counts defects. Counting defective units instead understates it whenever anything carries more than one problem.
The related figure is DPU — defects per unit — and it can exceed 1. A proportion cannot, which is the quickest way to tell which number you are holding.
If units are simply good or bad, a p chart is the right tool and DPMO is the wrong frame; it is built for processes where one unit can fail in several independent ways.
A sigma level is a normal-distribution translation
The conversion from a defect rate to a sigma level is not a measurement of anything. It is a lookup through a distribution the process may not have.
It assumes the characteristic is normally distributed and that defects come from the tails of that distribution beyond a specification.
For a count of transactional errors that story does not apply at all. There is no underlying measurement, no specification and no tail — only a rate.
The translation is still useful as a common language, which is what it is for: it puts a service process and a machining process on one scale.
It is not useful as a claim about the distribution. A process at 4.2 sigma has a defect rate; whether anything about it is normally distributed is a separate question and usually unasked.
What DPMO cannot see
DPMO is a rate over a period, and rates hide everything about when things went wrong.
It has no time dimension. A process at a steady 3,400 DPMO and one that was perfect for six months and then broke produce the same figure.
It weights every defect equally. A cosmetic blemish and a safety failure both count as one, which is rarely what anyone means.
It says nothing about capability against a specification. That is a Cpk question, and a process can have a low DPMO and no margin at all.
And it cannot separate common cause from special cause. A control chart over the same period answers that, and is the natural companion to a DPMO baseline rather than a competitor to it.
Reporting a DPMO
Four items, and the first two are what make two DPMO figures comparable at all.
Give the opportunity definition. Not the count — the definition. It is the number a reader cannot reconstruct and the one that moves the answer most.
Say whether the sigma level includes the 1.5 shift. The two differ by exactly 1.5, and almost every published figure uses the shifted one silently.
Give units, opportunities and defects, not just the DPMO. Three numbers instead of one, and every derived figure follows from them.
And give rolled throughput yield on anything with more than a few opportunities. It is the figure that corresponds to what a customer actually receives.
Sources and methodology
References for DPMO, the sigma scale and the shift.
Method. The sigma level is computed from the inverse normal of the defect rate, both with and without the conventional 1.5 allowance, and the suite asserts on 200 generated cases that the two differ by exactly 1.5 and that converting the sigma level back through the same table returns the DPMO it came from. Rolled throughput yield uses e^(−DPU) rather than the per-opportunity yield raised to a power — the two agree closely and the exponential form is the one that holds when defects are Poisson rather than independent per opportunity. A defect count larger than the number of opportunities returns no result rather than a negative yield, and a flawless process reports an infinite sigma level rather than a large finite one, since the inverse normal has no value there. The conversion table is generated from the same functions the calculator uses, so the two cannot disagree, and the suite checks the six-sigma entries against their published values under both conventions. That engine is verified on every change against 139 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
CpkCp, Cpk, Pp and Ppk with the defect rates they predict and the rate actually observed — including the built-in case where Cp is 2.05, Cpk is 0.57 and a quarter of the sample is already out of spec.
p Chartp and np charts with limits computed per point when the sample sizes differ, and the cost of flat limits measured on your own chart.
c Chartc and u charts for defect counts, with the exact Poisson probability of exceeding the three-sigma limit computed rather than assumed.
X-bar and R ChartX-bar with R or S charts, every Shewhart constant computed from its definition rather than looked up, and all eight Nelson rules.
Percent ErrorPercent error against an accepted value, rounded to your significant figures, for one measurement, repeated trials, or a whole column of pairs.
Gauge R&RANOVA gauge repeatability and reproducibility with both acceptance criteria, the operator-by-part interaction tested, and the full variance decomposition.
An educational tool. DPMO depends on an opportunity count that is decided rather than measured, so two figures are only comparable when counted the same way. The sigma level is a translation through a normal distribution the process may not have, and it conventionally includes a 1.5 sigma allowance for long-term drift — a factor of 3,444 at six sigma — which this page reports both ways rather than silently.