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Module 6 · Measure

Sigma level, DPMO, DPU, RTY

Module 0 introduced "sigma level" as a preview: a defect rate expressed as a Z-value. This module is where that preview becomes a toolkit. Defects versus defectives, how an "opportunity" is defined and how that definition can be gamed, first-time yield versus the rolled throughput yield of an entire process, the arithmetic that converts a defect rate to a sigma level, and the 1.5σ shift — where it comes from, the statistical argument for it, and the standing objection to it. None of this is difficult arithmetic. All of it is easy to state sloppily in a way that quietly changes what a reported number means.

Learning objectives

Why this matters

Every sigma-level number this course computes traces back to one sentence in a NIST profile of Motorola's 1988 Baldrige Award: the company's Six Sigma Quality programme set "a target of no more than 3.4 defects per million products, customer services included."[1] That number is real, it is Motorola's own stated target, and it is also, by itself, incomplete: 3.4 parts per million is not what a process centred at six standard deviations from its nearest specification limit produces (that figure is roughly two thousandths of a PPM, six orders of magnitude smaller). The 3.4 comes from a specific, named convention — the 1.5σ shift — that this module explains in full, because reporting a sigma level without saying which convention produced it is one of the most common ways a Six Sigma number misleads an audience that assumes it means something more literal than it does.

The scenario below is a constructed illustration, not a real incident.

A second, quieter way the same numbers mislead: two lines can report identical defect counts on identical products and publish two different sigma levels, purely from how "opportunity" was defined. A connector with 5 solder joints inspected as 5 separate opportunities per unit produces a very different DPMO than the same connector's identical defects counted as 1 opportunity (the unit either has a defect or it does not) — and a line under pressure to show improvement has an obvious incentive to redefine opportunities upward until the number looks better, with nothing about the actual process having changed. Worked example 1 below runs this exact comparison on one dataset.

Defects versus defectives, and how opportunities get gamed

A defective unit is one that fails, however many things are wrong with it: one defective is one defective, whether it has one flaw or five. A defect is one specific nonconformance; a single defective unit can carry several defects. Counting defectives answers "how many units failed"; counting defects answers "how many things went wrong," and the two questions need different denominators and can move independently — scrap rate (defectives / units) can hold steady while defects per unit quietly rises, if failures are increasingly multi-cause.

An opportunity is a defined chance for a defect to occur on a unit: a solder joint, a dimension, a required field on a form. The count is a judgement call, and Six Sigma's critics have long pointed out that it is judgement with an obvious incentive attached: define more opportunities per unit for the same defect count, and DPMO falls, and the resulting sigma level rises, with nothing about the process itself different. There is no universal rule for what counts as an opportunity; the discipline that prevents gaming is entirely procedural — define opportunities once, in writing, before comparing periods or lines, and never let the definition change just because the number needs to look better.

DPU, DPO, DPMO

DPU = defects / units
DPO = defects / (units × opportunities per unit)
DPMO = DPO × 1,000,000 DPU does not need an opportunity count and is the most defensible number to compare over time on one process, because there is nothing to redefine. DPO and DPMO normalise by opportunities, which makes them comparable only when the opportunity definition is held fixed.

First-time yield versus rolled throughput yield

First-time yield (FTY) at one step is the probability a unit passes that step without rework, scrap, or a defect found; under a Poisson defect model, FTY = e−DPU. Rolled throughput yield (RTY) is the probability a unit passes every step in a multi-step process the first time, with no rework anywhere along the way.

RTY = FTY1 × FTY2 × … × FTYk
Normalised yield per step = RTY1/k
Total DPU implied = −ln(RTY) RTY is always at or below the worst individual step's yield, and it falls faster than intuition expects: five steps that each individually look "pretty good" at 95 to 99 % can roll up to well under 90 %. A final-inspection pass rate reported after rework can look far better than RTY, because rework lets a unit pass on a second or third try; RTY, computed from first-time step yields, is what reveals the cost of that rework — sometimes called the hidden factory, because the rework loop consumes real capacity that never shows up on a yield chart built only from final results.

Converting a defect rate to a sigma level

A "sigma level" is a Z-value: how many standard deviations the nearest specification limit sits from the process mean, under a normal model, read backward from an observed defect rate rather than forward from a measured mean and sigma the way Module 7 computes Cpk. Two numbers are commonly reported, and this course always labels which one is in use.

Z (long-term, no shift) = Φ−1(1 − DPMO / 1,000,000)
Sigma level (Motorola convention) = Z + 1.5 Φ−1 is the inverse standard normal CDF (Module 1). The plain Z is a direct reading of the observed defect rate as a normal tail probability, nothing more.[5] Adding 1.5 is the shift convention explained below; skipping the label on which one a report is quoting is how "we're at four sigma" ends up meaning two different defect rates depending on who is asked.

The 1.5 sigma shift: history and controversy

Motorola's own articulation of Six Sigma treats a process as capable of drifting up to 1.5σ off-centre over the long run without being considered out of control, and reports the sigma level for the resulting worst-case, one-sided tail rather than for a perfectly centred process.[6], [7], [8] Arithmetically: a process truly centred 6σ from both specification limits has a predicted defect rate of about 0.002 PPM (Table 1 below computes this exactly); a process whose mean has drifted 1.5σ toward one limit, leaving only 4.5σ of margin on the near side, has a predicted one-sided defect rate of about 3.4 PPM — Motorola's published target. The two numbers describe different things: the first is "how good is this process if it never moves," the second is "how bad could this process get if it drifts as far as we're willing to tolerate before calling it a problem."

Tadikamalla's 1994 critique is that the shift is a definitional convention baked into the standard sigma-to-PPM lookup table, not a property measured on any particular process, and that presenting "6σ = 3.4 PPM" without the shift explained lets readers assume a Z of 6 when the honest claim is closer to a Z of 4.5.[2] Bothe's 2002 paper supplies the strongest statistical argument for allowing the margin: with the small subgroup sizes (commonly n = 4 or 5) typical of an X̄-R chart, a mean shift smaller than roughly 1.5σ has a real chance of going undetected for some time by the chart's own limits, so a process that has drifted that far without tripping an alarm is not a hypothetical, it is a realistic blind spot of the monitoring method itself.[3] Burns' 2018 practitioner column takes the opposite position: if a process really does drift 1.5σ routinely and this is built into the standard conversion table as normal, expected behaviour, that process is, by Shewhart's own definition, not in a state of statistical control at all, and the shift convention quietly normalises an out-of-control condition instead of flagging it as one.[4] This course takes no side: every sigma-level figure it reports states, in words and in the calculator output, whether the 1.5σ shift is included, so the reader is never left to guess which claim is being made.

Which metric to report, and to whom

Four numbers have now been defined for the same process, and a Green Belt is regularly asked for all of them by different people in the same week. They are not interchangeable, and the honest answer to "what is our sigma level?" often begins by asking what the number is for.

DPU is the number to track over time on one process. It has no opportunity count in it, so nobody can move it by redefining anything, and it is the only one of the four whose trend line means exactly what it appears to mean. If a project baseline and its after-improvement result are reported in one metric, this is the one that will survive a sceptical review.

RTY is the number to report for a line. It answers the question a plant manager is actually asking, which is what fraction of what we start comes out the far end without being touched again. Its value is that it exposes rework: a line whose final yield is 99 % and whose RTY is 78 % is running a second factory inside the first one, and the gap between the two numbers is roughly the size of it.

DPMO and the sigma level are the numbers for talking to someone outside the process. They compress a defect rate onto a scale that a customer or a corporate scorecard already understands, and that compression is exactly what makes them easy to abuse. Two rules keep them defensible. First, write down the opportunity definition next to the number, in words, once, and do not change it during the project; if it has to change, report both. Second, state whether the 1.5 sigma shift is included. A sigma level quoted without either of those is not a measurement, it is a claim.

What none of the four will tell you is whether the process is stable, which is the question that decides whether any of them predicts anything about next week. A defect rate computed over a period that contained a tooling change and a material lot change is an average of two processes, and the sigma level derived from it describes neither. That is why a p chart (Module 17) belongs beside a DPMO figure, and why the capability indices of Module 7, which come from measured values rather than counts, carry more information per part measured than any count-based metric can.

One practical consequence for project work: prefer a measured characteristic to a count wherever the process gives you a choice. Fifty leak-test results tell you the reject rate and nothing else; fifty measured joint gaps tell you the reject rate, the margin you have, which direction you are drifting, and how much variation reduction would be needed to close the gap. Counting is what you do when the characteristic genuinely is an attribute, not a first resort.

Worked examples

Worked example 1: DPMO, and what happens when "opportunity" changes

The data below is a constructed example, not a real production run.

A wire-to-board connector has 5 solder joints. Over 2,400 units, 54 joint defects were logged — a specific, unambiguous count, independent of any opportunity definition.

DPU = 54 / 2,400 = 0.0225    DPO = 54 / (2,400 × 5) = 0.0045    DPMO = 4500.0 Z (no shift) = 2.61; sigma level (with the 1.5σ shift) = 4.11.

Now recount the identical 54 defects on the identical 2,400 units, changing only how many opportunities are declared per unit.

Table 1. The same 54 defects in 2,400 units, three opportunity definitions.
Opportunities per unitDPMOSigma level (shifted)
1 (unit pass/fail only)22500.03.50
5 (one per solder joint)4500.04.11
10 (each joint split into 2 sub-checks)2250.04.34

Ten times more declared opportunities produces exactly one-tenth the DPMO (22,500 falls to 2,250, precisely proportional, since DPU and defect count never changed) and a reported sigma level that climbs by nearly a full point, from 3.50 to 4.34. Nothing about the connector, the solder process, or the 54 real defects changed at all. This is the entire mechanism behind opportunity-count gaming, and it is why DPU — which has no opportunity count to redefine — is the more defensible number to track over time on a single process.

Sigma level and DPMO converter

Pre-loaded with the connector's 5-opportunity count. Change "Opportunities per unit" to 1 or 10 to reproduce Table 1, or edit the RTY section's step yields to reproduce Worked example 2.

Worked example 2: rolled throughput yield of a 5-step line

Constructed example, not a real production line.

A wire-harness line runs five steps — crimp, insulate, sub-assemble, function test, final inspection — each with its own first-time yield.

Table 2. Step yields and cumulative RTY, 5-step wire-harness line (constructed data).
StepFirst-time yieldCumulative RTY through this step
1. Crimp98.0 %0.9800
2. Insulate95.0 %0.9310
3. Sub-assemble99.0 %0.9217
4. Function test97.0 %0.8940
5. Final inspection98.5 %0.8806
RTY = 0.98 × 0.95 × 0.99 × 0.97 × 0.985 = 0.8806, i.e. 88.06 %
Normalised yield per step = RTY1/5 = 0.9749    Total DPU implied = 0.1271 Every individual step looks solid, 95 % or better. Rolled up, only about 88 % of units pass all five steps with no rework anywhere. If this line's shipped-quality report only counts units after rework, it could plausibly show 99 %+ final yield — correct, and telling a very different story than the 88.06 % first-time number about how much rework capacity the line is actually consuming.

Worked example 3: a sigma-level table

Arithmetic on the normal distribution, not measured data.

Table 3. Sigma level to PPM, half-sigma steps, both conventions (generated by scipy in verification/tables.py).
Sigma level (k)Centred, two-sided PPMShifted 1.5σ, one-sided PPM
1.0317311691462
2.045500308538
3.02699.866807
4.063.36209.7
5.00.57232.6
6.00.00203.4

Reading straight across the k = 6.0 row is the whole controversy in one line: a process genuinely centred at six standard deviations from both specification limits predicts about 0.002 PPM; the number everyone actually quotes, 3.4 PPM, is the one-sided tail after subtracting the 1.5σ shift, i.e. what a true 4.5σ process produces on its worst side. Both numbers are in this table. Only one of them is what "six sigma" literally means.

Common mistakes

  1. Reporting a sigma level without saying whether the 1.5σ shift is included. Consequence: "four sigma" can mean a Z of 4 (PPM about 32) or a Z of 2.5 after subtracting the shift (PPM about 6,210) depending on which convention is meant, a more than 190-fold difference. Fix: always state the convention next to the number.
  2. Redefining "opportunities" to make a trend look better. Consequence: Worked example 1 — the identical defects and units produce a DPMO that swings by a factor of ten purely from the opportunity count. Fix: fix the opportunity definition in writing before comparing any two periods or lines.
  3. Confusing defectives with defects. Consequence: a scrap rate (defectives) can look stable while the average number of things wrong per unit (DPU) is quietly rising, and nobody notices because the wrong metric is being watched. Fix: track DPU alongside the defective rate, not instead of it.
  4. Computing RTY as if it were an average of the step yields. Consequence: five steps at 95 to 99 % "feel" like a process running at roughly 97 %; the actual RTY (Worked example 2) is meaningfully lower, because yields multiply, they do not average. Fix: always compute the product, never eyeball an average.
  5. Reporting a final-inspection pass rate as if it were RTY. Consequence: rework lets units pass on a second try, so a final pass rate can look good while masking a large, costly hidden factory of rework loops. Fix: compute RTY from first-time step yields specifically, not from a final good/bad tally.
  6. Treating the 1.5σ shift as settled science in either direction. Consequence: presenting Motorola's convention as an arbitrary marketing trick, or presenting it as a rigorously derived universal constant, both overstate what the sources actually support. Fix: state Bothe's statistical rationale and Burns' and Tadikamalla's objections together, as this module does.
  7. Using DPMO to compare processes with genuinely different, non-comparable opportunity structures. Consequence: a simple one-step attribute inspection and a 40-step electronics assembly are not meaningfully compared by DPMO alone, because the opportunity-counting judgement calls are not equivalent. Fix: compare DPU or RTY trends within a process over time before comparing sigma levels across different processes.

Exercises

Exercise 1: RTY of a 4-step board assembly line

Constructed example, not a real production line.

A board assembly line runs place, reflow, test, and conformal coat, with first-time yields 99.5 %, 96.0 %, 98.0 % and 99.0 %. Tasks. (a) Compute RTY. (b) Compute the normalised yield per step. (c) A colleague says "average yield is about 98 %, so RTY should be about 98 %." Explain what is wrong with that reasoning, using your answer to (a).

Show the worked solution

(a) RTY = 0.995 × 0.96 × 0.98 × 0.99 = 0.9267, i.e. 92.67 %.

(b) Normalised yield = RTY1/4 = 0.9812, i.e. 98.12 % per step on average.

(c) The arithmetic mean of 99.5, 96.0, 98.0 and 99.0 is indeed about 98.1 %, close to the normalised yield in (b) — but RTY itself is 92.67 %, over 5 points lower, because yields compound multiplicatively, not additively. The weakest step (96.0 %, reflow) drags the product down by more than a simple average would suggest, and the gap between "average-looking" and "actual" widens as more steps are added.

Exercise 2: leak-test DPMO

Constructed example, not a real production run.

A brazed assembly has 2 leak-test points per unit (inlet and outlet joints). Over 3,000 units, 33 point failures were logged. Tasks. (a) Compute DPU. (b) Compute DPMO. (c) Compute the sigma level with the 1.5σ shift. (d) If a second plant reports the identical 33 failures on 3,000 units but counts only 1 opportunity per unit (a unit either leaks or it does not), will its DPMO be higher or lower than yours, and by what factor?

Show the worked solution

(a) DPU = 33 / 3,000 = 0.011.

(b) DPMO = 33 / (3,000 × 2) × 1,000,000 = 5500.0.

(c) Sigma level (shifted) = 4.04.

(d) Higher, by a factor of exactly 2: with 1 opportunity instead of 2, the same defect count is divided by half as many total opportunities, so DPMO doubles to 11,000. The two plants' processes could be identical; only the opportunity definition differs.

Quiz

Ten questions. Score 70 % or more to mark the module complete on this device.

1. A single unit with three separate nonconformances on it counts as
2. Redefining "opportunities per unit" upward, with the same defect count and units, causes DPMO to
3. Which metric is least sensitive to how "opportunity" is defined?
4. Rolled throughput yield of a multi-step process is computed as
5. A final-inspection pass rate computed after units that failed have been reworked and re-tested
6. Motorola's stated Six Sigma target, in defects per million, is
7. A process truly centred at six standard deviations from both specification limits, with no shift applied, has a predicted defect rate of approximately
8. Bothe's (2002) statistical rationale for the 1.5 sigma shift is that
9. Burns' (2018) objection to the 1.5 sigma shift is that
10. This course's policy on reporting a sigma level is to
Answer key
  1. b. One defective, three defects.
  2. c. DPMO decreases, sigma level rises, nothing real changed.
  3. d. DPU.
  4. b. Product of the step yields.
  5. a. Can hide the hidden factory of rework.
  6. 3.4 defects per million.
  7. c. About 0.002 PPM.
  8. b. Small-subgroup charts can miss a shift that size.
  9. d. Routine drift of that size means the process is not in control.
  10. a. Always state the convention.

Key takeaways

References

All web sources accessed 2026-09-09 unless noted.

  1. NIST. "Malcolm Baldrige National Quality Award 1988 Recipient: Motorola Inc." https://www.nist.gov/system/files/documents/2017/10/11/1988_Motorola_Inc.pdf
  2. Tadikamalla, P. R. (1994). The confusion over six-sigma quality. Quality Progress, 27(11), 83–85. https://www.proquest.com/openview/df184b19ba7003517881a00d6f967c6d/1
  3. Bothe, D. R. (2002). Statistical reason for the 1.5σ shift. Quality Engineering, 14(3), 479–487. https://www.tandfonline.com/doi/abs/10.1081/qen-120001884
  4. Burns, T. "Predictable." Quality Digest, 13 June 2018. https://www.qualitydigest.com/inside/six-sigma-article/predictable-061318.html
  5. NIST/SEMATECH. e-Handbook of Statistical Methods, 1.3.6.6.1 "Normal Distribution". NIST. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm
  6. ASQ. Certified Six Sigma Green Belt (CSSGB) Body of Knowledge Map 2014–2022. ASQ, 2022. https://www.asq.org/cert/resource/pdf/certification/2022-CSSGB-BoK-Map.pdf
  7. Harry, M. J. (1988). The Nature of Six Sigma Quality. Motorola University Press. https://openlibrary.org/books/OL9828527M/The_Nature_of_Six_Sigma_Quality (secondary: catalogue-level; Motorola's own articulation of the shift convention, cited via [1] and [2] for the specifics used here)
  8. Harry, M., & Schroeder, R. (2000). Six Sigma: The Breakthrough Management Strategy Revolutionizing the World's Top Corporations. Currency/Doubleday. https://books.google.com/books/about/Six_Sigma.html?id=RY0rAAAAYAAJ (secondary: catalogue-level; RTY and belts background)