Module 0 · Foundations
Introduction: what Six Sigma is and is not
Six Sigma is three things wearing one name: a statistical target that Motorola set in the 1980s, a project method with five phases, and a management programme with belts, champions and savings claims. The statistics are old, sound, and worth your career. The programme has a mixed record, and the record is published. This module separates the three so that you know which one you are being asked to believe in.
Learning objectives
- Describe where Six Sigma came from, using the primary sources rather than training-course folklore, and say which of its claims are the companies' own.
- Convert a defect rate into a Z-value and a "sigma level" under both conventions (with and without the 1.5σ shift), and explain why the two are always labelled.
- Read a histogram against specification limits and estimate the fraction out of tolerance from the mean and standard deviation.
- Name the five DMAIC phases, the question each one answers, and the roles of a Green Belt, Black Belt, Master Black Belt, champion and process owner.
- Summarise the published critiques of Six Sigma programmes and the peer-reviewed evidence on the other side, and say what quality of evidence each rests on.
Why this matters
You will meet Six Sigma in one of three forms. A drawing note or a supplier quality manual will ask for a capability index, and someone will call the process "four sigma" or "six sigma" without saying what they mean. A manager will ask you to run a DMAIC project on a scrap problem and expect a charter, a gauge study and a control plan at the end of it. Or a programme with belts and savings targets will arrive at your plant, and you will be sent on a course. In each case the useful part is the same: a set of statistical methods for measuring variation, finding its causes and holding the gains. Those methods are the subject of this course. The wrapping is the subject of this module.
The name has a specific origin, and it is worth knowing because the number in it is quoted everywhere and understood almost nowhere. In 1981 Motorola launched a drive for a tenfold improvement in the quality of its products and services; by the time it received the Malcolm Baldrige National Quality Award in 1988 it employed 99,000 people, had 1987 sales of $6.7 billion, and put "Six Sigma Quality" at the top of its list of goals, described as a statistical measure of variation from a desired result that translated, in concrete terms, into a target of no more than 3.4 defects per million products, customer services included.[1] That figure, 3.4 defects per million, is the whole of what most people know about Six Sigma. Section 2 of this module shows where it comes from, and why a statistician will tell you it does not mean what it appears to mean.
The second reason this matters is that the programme's business record is contested, and a junior engineer who repeats the savings figures from a training slide will eventually meet someone who has read the critiques. Section 6 gives both sides with their sources. The short version: the methods are sound; the evidence that a company-wide programme improves financial results is mixed and depends on how it was run; and the value to you personally is in learning to measure, chart and test properly, which no critique disputes.
1. Where Six Sigma came from
Motorola, 1981 to 1988
The primary source for the early history is not a textbook but the profile that the US National Institute of Standards and Technology published when Motorola received the Baldrige award. It records the 1981 tenfold-improvement goal, the corporate objective of "total customer satisfaction" printed on a card that managers carried, a stated quality goal of zero defects in everything the company did, a spend of more than $170 million on employee education between 1983 and 1987, and Six Sigma Quality as the headline metric, with employees recording the defects found in every function of the business.[1] Note the last point: Six Sigma at Motorola was a measurement discipline applied to every function, not only the factory, and the 3.4 figure was a target, not a report of results.
The engineer usually credited with proposing the approach is Bill Smith, who put the idea to Motorola's chairman in the mid-1980s; the account of how the term was coined and later registered as a trademark comes from secondary sources and is not independently verified here.[3] Mikel Harry's 1988 monograph The Nature of Six Sigma Quality, published by Motorola University Press, is the earliest formal statement of the method that the course has been able to identify by catalogue, and is the usual attribution for the 1.5σ shift convention discussed below.[4]
General Electric, 1995 onward
Six Sigma became a household name in management when General Electric adopted it. GE's 1998 annual report letter to share owners is the primary source for the company's own account, and it should be read as exactly that: the company's account of itself. It describes Six Sigma as one of three company-wide growth initiatives, says GE "plunged into" it just over three years before, in 1995, claims an investment of more than a billion dollars, claims savings of more than three quarters of a billion dollars beyond that investment in 1998 with a billion and a half in sight for 1999, and reports 5,000 full-time Master Black Belts and Black Belts driving tens of thousands of projects, with virtually every professional in the company trained as a Green Belt with a project completed.[2] None of those numbers has been audited by anyone outside GE, and this course does not repeat them as facts about Six Sigma; it repeats them as facts about what GE said, which is a different thing and the reason the programme spread.
Two features of the GE version shaped everything that followed. The belt hierarchy became the standard organisational structure (Section 4), and the Define phase was added to Motorola's original Measure, Analyze, Improve, Control sequence to give the DMAIC acronym now in universal use. The account of that evolution comes from GE insiders and from secondary summaries, and is cited as such.[3][17]
The methods are older than the name
Nothing statistical in Six Sigma was invented at Motorola or GE. Walter Shewhart's memo of 16 May 1924 at Western Electric contained the first control chart and the distinction between chance causes and assignable causes; his 1931 book set out the economic argument for three-sigma limits that Module 16 still uses.[15][16] Capability indices, gauge studies, designed experiments, hypothesis tests and regression all predate the programme by decades. The academic literature is explicit about this: Schroeder and colleagues, in the first attempt at a formal definition, found the tools and techniques of Six Sigma "strikingly similar" to earlier approaches to quality management and located the novelty in the organisational structure, the parallel meso-structure of belts, projects and metrics that sits beside the normal management hierarchy.[8] Zu and colleagues reached a similar conclusion the same year: three practices are new, the role structure, the structured improvement procedure and the focus on metrics; the rest is inherited.[9] The ASQ Green Belt body of knowledge itself asks candidates to know the evolution of Six Sigma from quality leaders such as Juran, Deming, Shewhart and Ishikawa.[10]
2. What "six sigma" means as a number
Sigma, σ, is the standard deviation of a process characteristic: a measure of its spread, defined properly in Module 1. If the characteristic follows a normal distribution, 99.73 % of parts fall within three standard deviations either side of the mean, and 0.27 %, or 2,700 parts per million, fall outside. The "sigma level" of a process is the distance from the process mean to the nearest specification limit, expressed in standard deviations. A three-sigma process has its nearest limit 3σ away; a six-sigma process has it 6σ away.
That definition is enough to compute a defect rate, and the computation is the first place where the arithmetic and the convention part company.
For a process centred between two limits each 6σ away, the fraction beyond both limits together is 2 × P(Z > 6) = 0.002 parts per million. That is not 3.4. The 3.4 figure comes from Motorola's convention that a process mean will drift over the long run by up to 1.5σ from where it was set, so that a limit set 6σ away in the short term is effectively only 4.5σ away in the long term, and the one-sided tail beyond 4.5σ is 3.4 parts per million.[1][4] Table 1 gives both conventions for the usual levels. All values are computed from the normal distribution by the course's verification scripts.
| Sigma level | Centred, both tails (PPM) | With 1.5σ shift, near tail (PPM) | What it means in practice |
|---|---|---|---|
| 1 | 317,311 | 691,462 | Not a process; a lottery |
| 2 | 45,500 | 308,538 | Sorting and rework as a way of life |
| 3 | 2,700 | 66,807 | Cpk = 1.00 (Module 7); the traditional minimum |
| 4 | 63 | 6,210 | Cpk = 1.33; a common customer requirement (Module 7) |
| 4.5 | 6.8 | 1,350 | Cpk = 1.50; the shifted "six sigma" process |
| 5 | 0.57 | 233 | Cpk = 1.67; often asked for on critical characteristics (Module 7) |
| 6 | 0.002 | 3.4 | Cpk = 2.00; Motorola's target |
Read down the shifted column and you have the table printed in every Six Sigma training deck. Read across any row and you see that "sigma level" is ambiguous by a factor of 1.5σ, which at the three-sigma line is the difference between 2,700 and 66,807 parts per million. Whenever anyone quotes a sigma level, ask which column. This course, and every calculator in it, prints both and labels them.
The 1.5σ shift: a convention, its rationale and its critics
The shift is contested and the course treats it as such. Its published statistical rationale is Bothe's 2002 paper, which argues that with subgroups of four an X̄ chart tends not to detect shifts of up to about 1.5σ quickly, so a process under routine control can wander that far without anyone acting.[6] Against it, Tadikamalla pointed out in 1994 that sigma quality levels translate into defect rates only under a centring assumption, that Motorola's 3.4 PPM comes from placing the mean 1.5σ off centre, and that the terminology confuses statisticians who take "six sigma" to mean what it says.[5] Burns's 2018 critique goes further: a process whose mean is permanently 1.5σ off where it was set is not in statistical control and is therefore not predictable, so attaching a long-run defect rate to it is a contradiction in terms.[7] Module 6 goes through the argument in full. For now the rule is: never report a sigma level without saying whether 1.5 was added, and prefer to report the defect rate itself, which needs no convention.
Worked example 1: from a defect rate to a sigma level, both ways
This is arithmetic on the normal distribution, not data. Every value is computed by the verification scripts and reproduced by the calculator below.
Start from Motorola's target of 3.4 defects per million. The Z-value whose upper tail is 3.4 × 10⁻⁶ is 4.50. Add the 1.5σ convention and the sigma level is 6.00. So "six sigma" and "3.4 PPM" are the same statement only with the shift included; without it, 3.4 PPM is a 4.5σ process.
Now the other direction, for three defect rates you will actually meet. A supplier delivering 1,000 PPM has Z = 3.09, a "sigma level" of 4.59 with the shift. At 100 PPM: Z = 3.72, sigma level 5.22. At 10 % defective (100,000 PPM): Z = 1.28, sigma level 2.78. Notice that a line scrapping one part in ten is called a "2.8 sigma process" in the shifted convention, which sounds respectable and is not. The defect rate is the honest number.
Worked example 2: a histogram against a specification
The data below is a constructed example generated by a seeded script, not a real production run. Setting: a drilled hole of Ø8.000 ± 0.050 mm in a steel bracket, measured with a bore gauge reading to 0.001 mm, 100 consecutive parts in production order.
This is the sigma-level idea applied to something you can hold. The full capability treatment, with the two sigma estimates and the indices, is Module 7; here the point is only that a defect rate is a tail area, and a tail area is set by where the mean sits and how wide the spread is.
| Values | +0 | +1 | +2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1–10 | 8.035 | 8.006 | 7.988 | 8.030 | 8.036 | 8.027 | 7.962 | 7.984 | 8.029 | 8.047 |
| 11–20 | 7.985 | 8.036 | 8.040 | 8.016 | 7.988 | 8.003 | 8.008 | 8.031 | 8.035 | 7.995 |
| 21–30 | 8.014 | 8.015 | 8.012 | 8.036 | 8.010 | 8.015 | 7.985 | 8.019 | 7.996 | 8.056 |
| 31–40 | 8.019 | 8.020 | 8.018 | 7.991 | 8.007 | 8.018 | 8.035 | 8.021 | 7.988 | 8.012 |
| 41–50 | 8.013 | 7.981 | 8.027 | 8.032 | 7.997 | 8.062 | 8.023 | 8.004 | 8.057 | 8.004 |
| 51–60 | 8.019 | 7.978 | 7.993 | 8.053 | 7.995 | 7.998 | 8.011 | 7.999 | 7.990 | 7.995 |
| 61–70 | 8.012 | 8.048 | 8.032 | 8.064 | 7.998 | 8.020 | 8.021 | 8.027 | 8.031 | 7.980 |
| 71–80 | 8.047 | 8.016 | 8.006 | 8.021 | 8.004 | 7.990 | 8.008 | 7.975 | 7.992 | 8.031 |
| 81–90 | 8.026 | 7.992 | 7.969 | 8.020 | 8.024 | 8.015 | 7.995 | 8.017 | 8.008 | 8.020 |
| 91–100 | 8.028 | 8.015 | 8.012 | 8.027 | 8.021 | 7.998 | 8.014 | 8.030 | 8.028 | 8.009 |
Before any number is read off a histogram, the data must be shown to come from one stable process, because a mean and a standard deviation of a process that shifted halfway through describe nothing. The individuals and moving range chart of these 100 values (Module 16 explains the chart; the calculator below draws it for the capability case) has no point beyond its limits and no run rule firing, and the Anderson-Darling test gives p = 0.61, no evidence against a normal shape. So the summary is meaningful: mean x̄ = 8.0139 mm, sample standard deviation s = 0.0206 mm, and 5 of the 100 holes are outside the tolerance, all of them oversize.
Zlower = (x̄ − LSL) / s = (8.0139 − 7.950) / 0.0206 = 3.10
The upper limit is 1.75 standard deviations from the mean. The normal tail beyond Z = 1.75 is 4.0 %, or 40,096 PPM; the lower tail adds 978 PPM, for a predicted 41,074 PPM, about 4.1 %. The sample shows 5 %, which is consistent: at 4 % you expect four in a hundred and five is unremarkable. In sigma-level language this is a 1.75σ process without the shift, or "3.25σ" with it. To become a centred six-sigma process on this tolerance the spread would have to fall from 0.0206 mm to 0.0083 mm, and even a shift of the mean back to 8.000 would only move the nearest limit out to 2.4σ. The histogram below, with the specification limits drawn on it, is the picture behind the arithmetic.
Descriptive statistics and histogram (pre-loaded with the hole data)
Paste your own values to get the summary statistics, the count outside specification and a histogram with the specification limits and a fitted normal curve. Nothing leaves your browser.
Worked example 3: counting defects, and how "opportunities" move the sigma level
Constructed counts, not a real production record. Setting: a line brazing aluminium heat-exchanger assemblies leak-tests every unit; in one week 1,250 assemblies were tested and 23 failed. Each assembly has six braze joints.
When the characteristic is a count of defects rather than a measurement, the sigma level is reached through defects per million opportunities, DPMO: the number of defects divided by the number of chances to make one, times a million. Module 6 covers the definitions properly. The trap is the word "opportunities", and this example shows it.
| Quantity | One opportunity per assembly | Six opportunities per assembly (one per joint) |
|---|---|---|
| Defects per unit, DPU = 23 / 1,250 | 0.0184 | 0.0184 |
| Defects per million opportunities, DPMO | 18,400 | 3,067 |
| Z from DPMO, no shift | 2.09 | 2.74 |
| "Sigma level" with the 1.5σ shift | 3.59 | 4.24 |
| First-time yield, e−DPU (Poisson) | 0.9818 | 0.9818 |
Nothing on the line changed between the two columns. The same 23 assemblies leaked. Counting each joint as an opportunity divides the defect count by six times as many chances, and the "sigma level" rises from 3.59 to 4.24. Defects per unit and first-time yield, which do not involve opportunities, are identical in both columns, which is the reason this course prefers them. When a report quotes a sigma level from a DPMO, ask how the opportunities were counted, and ask what DPU was. The calculator below does the conversion in both directions and shows both Z conventions.
Sigma level and DPMO converter (pre-loaded with the leak-test counts)
Part A converts a defect count into DPU, DPMO, yield and both Z conventions; Part B converts a DPMO or a sigma level directly; Part C multiplies step yields into a rolled throughput yield (Module 6).
3. DMAIC in one page
DMAIC is the project method. It is a sequence of five phases, each of which answers one question and produces a small number of deliverables, and the discipline of the method is refusing to move to the next phase before the current question has been answered with data. Most engineers who have run a good root-cause investigation have already done DMAIC without the letters; the letters add a checklist and a vocabulary that a team and its sponsor can share.
| Phase | The question | Typical deliverables | Modules |
|---|---|---|---|
| Define | What problem, for whom, how big, and what is out of scope? | Problem statement, charter, SIPOC, voice of the customer translated into critical-to-quality characteristics | 2, 3 |
| Measure | Can we trust the gauge, and how does the process perform now? | Measurement system analysis, data collection plan, baseline control chart and capability, defect rate | 4, 5, 6, 7, 8 |
| Analyze | What causes the variation or the defects? | Graphical analysis, root-cause tools, FMEA, hypothesis tests, regression; verified causes, not suspected ones | 9, 10, 11, 12 |
| Improve | What change removes the cause, and did it work? | Designed experiments, Lean changes, solution selection, a pilot with a before-and-after test | 13, 14, 15 |
| Control | How do we keep it fixed after the team leaves? | Control charts, control plan, reaction plan, standard work, handoff to the process owner | 16, 17, 18 |
The Capstone (Module 19) runs one project through all five phases on a fully specified dataset, and uses every tool in the course at least once.
4. Belts and roles
The belt structure is the part of Six Sigma that the academic literature agrees is new.[8][9] It is an organisational device: a way of deciding who leads projects, who supports them, who signs off on them and who owns the process afterwards. The names vary between companies; the ASQ Green Belt body of knowledge lists black belt, master black belt, green belt, champion, executive, coach, facilitator, team member, sponsor and process owner as the roles a candidate should be able to describe.[10]
| Role | What they do | What they need to know |
|---|---|---|
| Green Belt | Leads a bounded project in their own area part-time, or supports a Black Belt's project; collects the data, runs the basic analyses, presents the results | Everything in this course: MSA, capability, SPC, the core hypothesis tests, a first designed experiment, DMAIC |
| Black Belt | Leads larger cross-functional projects, usually full-time; coaches Green Belts; owns the statistical work | The Green Belt material in depth plus multiple regression, fractional and response-surface designs, advanced control charts |
| Master Black Belt | Trains and mentors Belts, selects and reviews projects, keeps the method honest across the site | Statistical depth plus programme management |
| Champion or sponsor | A manager who owns the business result, removes obstacles, approves the charter and the closure | Enough to read a control chart and a capability report and to ask the questions in Module 8 |
| Process owner | The person who runs the process every day and inherits the control plan when the project ends | The control plan, the reaction plan, and how to read the chart on the wall |
This course is written at the Green Belt level and covers the topics in the ASQ body of knowledge that a manufacturing engineer meets: measurement systems, capability studies with stability and normality verified, the Cp/Cpk and Pp/Ppk distinction, sigma level and the shift, multi-vari and graphical analysis, hypothesis tests, correlation and regression, designed experiments, and SPC.[10] It is not a certification course and does not claim to prepare you for any examination. Certification bodies test specific vocabulary and formats; this course tests whether you can run and read the study.
5. Lean and Six Sigma
Lean is the other improvement tradition you will meet, and most programmes today are called Lean Six Sigma. Lean descends from the Toyota Production System described by Ohno, and the term and its five principles were set out by Womack and Jones in Lean Thinking; its focus is flow, and its central idea is the removal of waste from the value stream, the sequence of activities that turns raw material into a delivered product.[20][19] Ohno's seven wastes, in the Lean Enterprise Institute's wording, are overproduction, waiting, conveyance, processing, inventory, motion and correction, and the Institute calls overproduction the worst form of waste.[18] Six Sigma's focus is variation and defects. The two are complementary rather than competing: a process can flow beautifully and make scrap, or make perfect parts one batch a month. Antony, Snee and Hoerl review the history of how the two were integrated into the combined programme now usual, and where it is heading.[11]
In this course, Lean appears where a Green Belt needs it: process mapping and the wastes in Module 2, and 5S, standard work, poka-yoke, SMED and kaizen in Module 14. The statistical core is Six Sigma's, because that is the part that a one-semester statistics course did not leave you with.
6. What the critiques say, and what the evidence is
Six Sigma programmes have been criticised in the business press and defended in the academic literature, and an engineer should know both, with a sense of how good each piece of evidence is.
The critiques
- Fortune, 2006. In an article arguing that Jack Welch's management rules no longer worked, Betsy Morris reported that of 58 large companies that had announced Six Sigma programmes, 91 % had trailed the S&P 500 since, according to an analysis by Charles Holland of the consulting firm Qualpro, which the article itself notes sells a competing improvement method. The article's complaint is that the method is designed to fix existing processes and leaves little room for new ideas.[12] Read it for what it is: a share-price correlation from an interested party, not a peer-reviewed study, with no control for what those companies would have done without the programme. It is nevertheless the most quoted critique.
- BusinessWeek on 3M, 2007. Brian Hindo described how James McNerney, arriving from GE at the end of 2000, brought GE's Six Sigma programme to 3M, had thousands of staff trained as Black Belts, cut 8,000 jobs, and raised operating margins from 17 % to 23 % between 2001 and 2005, and how the share of 3M's sales from products introduced in the previous five years fell from the traditional one-third to about one-quarter. His successor George Buckley, quoted as saying that invention is by its nature a disorderly process, removed the obligation on research scientists to follow Six Sigma objectives.[13] This is journalism with named sources, and its point is narrow and probably right: a method built to reduce variation is the wrong tool for a laboratory whose job is to produce variation.
- The academic view that the tools are not new. Schroeder and colleagues and Zu and colleagues, cited above, both conclude that the statistics were inherited and that the organisational structure is the contribution.[8][9] That is not a critique of the tools; it is a warning against paying for a brand.
The other side
- Swink and Jacobs, 2012. The peer-reviewed counterweight to the Fortune figure is an event study in the Journal of Operations Management comparing about 200 Six Sigma adopters with matched firms. It found a positive effect on return on assets, arising mostly from reductions in indirect costs, with the size of the effect depending on the firm's context.[14] This is a properly controlled study, and its conclusion is modest: the programme tends to help, in ways that depend on how it is run.
- GE's own account. The 1998 letter quoted in Section 1 is the strongest statement of the positive case and the weakest as evidence, because it is self-reported and unaudited.[2]
What to take from it
Three things. First, the statistical methods are sound, are older than the programme, and are not what any critic is attacking. Second, whether a company-wide programme pays is an empirical question with a mixed answer, and the honest sources on both sides agree that management, project selection and context decide it. Third, the method has a domain: it is built to reduce variation in a process that should be repeatable, and it is the wrong tool for work whose value comes from novelty. When you are asked to believe a savings figure, ask who counted it and whether the counting was audited; when you are asked to apply DMAIC to something, ask whether the something is a process.
7. How to use this course
The course has twenty modules in seven parts, following the DMAIC sequence after three foundation modules. Process capability (Modules 7 and 8) and statistical process control (Modules 16 and 17) are the core and get the most depth, because they are what a manufacturing engineer is asked for most often and gets wrong most expensively. Each module has the same shape: objectives, why it matters, the content with every formula explained in words and then applied to numbers, at least three worked examples with their full datasets printed, a calculator pre-loaded with the module's data, a list of common mistakes, exercises with collapsible solutions, a quiz, takeaways and references.
Two rules govern every number on every page. Real-world claims are cited to a source that is listed at the end of the module and in the course's source register, with its verification status; a source that could not be read in the original is marked as secondary. Constructed examples are labelled as such, generated by a seeded script, and every statistic derived from them has been computed by one route, recomputed independently by a second, checked for consistency by a third, and reproduced by the in-page calculator before it was allowed on the page. If you copy any dataset in this course into Excel, Minitab or Python you will get the same numbers, and you should try it.
The quiz at the end of each module is scored in your browser and a score of 70 % or more marks the module complete on this device; nothing is sent anywhere. The Calculators page collects every calculator for use after the course. Where practice varies, as it does over the capability index naming conventions in Module 7 and the 1.5σ shift here, the course teaches both usages and tells you which one a given document is likely to be using.
Common mistakes
- Quoting a sigma level without saying whether 1.5 was added. Consequence: a 3σ process is reported at 66,807 PPM by one reader and 2,700 PPM by another, from the same words. Fix: report the defect rate, and if a sigma level is required, label the convention.
- Treating 3.4 PPM as a property of the normal distribution. Consequence: an engineer computes 6σ tails correctly, gets 0.002 PPM, and concludes that the training material or their own arithmetic is wrong. Fix: 3.4 is 4.5σ one-sided; the extra 1.5σ is Motorola's convention.
- Inflating the opportunity count. Consequence: the sigma level rises with no change on the line, as in Worked example 3. Fix: report DPU and first-time yield, which have no opportunities in them, and define opportunities once, in writing, before the study.
- Repeating programme savings figures as facts. Consequence: the first person who has read the Fortune article or the 3M story stops trusting anything else you say. Fix: attribute claims to the company that made them and know the peer-reviewed evidence.
- Applying DMAIC to work that is not a process. Consequence: 3M's laboratories. Fix: the method reduces variation in something that should be repeatable; if novelty is the value, use a different method.
- Starting a project without a gauge study. Consequence: months of analysis on data that mostly measured the measurement. Fix: the Measure phase begins with Module 4, every time.
- Reading a mean and standard deviation off a histogram of an unstable process. Consequence: a predicted defect rate for a process that no longer exists. Fix: chart the data in time order first, as Worked example 2 does; Modules 7 and 16 make this the rule.
Exercises
Exercise 1: DPMO and sigma level from an inspection count
Constructed counts, not a real production record. Setting: an automated optical inspection station checked 4,800 assembled circuit boards in a month. Each board has 120 solder joints. The station logged 331 joint defects in total.
Tasks. (a) Compute defects per unit, defects per opportunity and DPMO, taking each joint as one opportunity. (b) Compute the first-time yield from DPU using the Poisson relation and compare it with 1 − DPMO/10⁶. (c) Convert the DPMO into a Z-value and into a sigma level with the 1.5σ shift, and check the Z against Table 1. (d) A manager says the line is "nearly five sigma". What should the report say instead?
Show the worked solution
(a) DPU = 331 / 4,800 = 0.0690 defects per board. Opportunities = 4,800 × 120 = 576,000 joints. DPO = 331 / 576,000 = 0.000575, so DPMO = 575.
(b) First-time yield from DPU: e−0.0690 = 0.9334, so about 93.3 % of boards leave the station with no defect at all. From DPMO: 1 − 575/10⁶ = 0.9994, which is the yield per joint, not per board. The two numbers answer different questions, and a report that quotes 99.94 % without saying "per joint" is misleading: one board in fifteen needs rework.
(c) Z from DPMO, no shift: 3.25. With the shift: 4.75. Table 1 check: 575 PPM lies between the shifted rows for 4σ (6,210 PPM) and 5σ (233 PPM), closer to 5, so 4.75 is consistent.
(d) "Nearly five sigma" is the shifted figure for the joint-level DPMO. The report should say: 4,800 boards, 331 joint defects, DPU 0.069, first-time yield 93.3 % per board, 575 DPMO per joint, Z = 3.25 (sigma level 4.75 with the 1.5σ convention). Every one of those numbers is defensible; the phrase on its own is not. The calculator on this page reproduces all of them from the three inputs.
Exercise 1 in the calculator
Exercise 2: a one-sided specification from measured data
Constructed data, not a real production run. Setting: wire-bond pull strength on a power module, with a minimum of 8.0 N and no maximum, measured on a pull tester reading to 0.01 N, 60 consecutive bonds in production order.
| Values | +0 | +1 | +2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1–10 | 9.55 | 9.37 | 9.22 | 7.97 | 9.49 | 8.08 | 9.75 | 9.58 | 9.71 | 9.71 |
| 11–20 | 9.41 | 8.96 | 9.08 | 10.08 | 8.93 | 9.12 | 8.85 | 8.97 | 9.08 | 9.39 |
| 21–30 | 9.09 | 9.05 | 8.91 | 9.30 | 8.51 | 9.28 | 9.97 | 8.83 | 9.23 | 10.80 |
| 31–40 | 8.70 | 8.36 | 9.02 | 10.24 | 10.37 | 8.58 | 9.56 | 9.25 | 8.93 | 8.18 |
| 41–50 | 9.81 | 8.78 | 9.41 | 9.30 | 10.28 | 9.90 | 8.61 | 9.11 | 9.82 | 8.52 |
| 51–60 | 9.03 | 9.38 | 8.36 | 8.57 | 8.31 | 9.41 | 8.94 | 9.76 | 9.31 | 9.56 |
Tasks. (a) Compute the mean and sample standard deviation, and count the bonds below 8.0 N. (b) State why the time order matters before you use those two numbers, and what you would check. (c) Compute Z for the lower limit and the predicted fraction below 8.0 N; compare with the count. (d) Express the result as a sigma level under both conventions and say which one you would put in a report.
Show the worked solution
(a) x̄ = 9.2105 N, s = 0.5839 N. 1 of the 60 bonds (7.97 N, bond 4) is below 8.0 N.
(b) The mean and standard deviation summarise one distribution. If the pull strength drifted during the 60 bonds, say because the bonding tool wore, there is no single distribution and the prediction in part (c) is fiction. The check is an individuals and moving range chart of the 60 values in order (Module 16): here no point falls outside the limits of 7.44 to 10.98 N and no run rule fires, and the Anderson-Darling p-value of 0.95 gives no reason to doubt a normal shape. Proceed.
(c) Z = (x̄ − LSL) / s = (9.2105 − 8.0) / 0.5839 = 1.2105 / 0.5839 = 2.07. The normal tail below Z = 2.07 is 19,078 PPM, about 1.9 %. Observed: 1 in 60, 1.7 %. Consistent. (Module 7 would write this as Cpl = Z/3 = 0.69 from the overall s, with a 95 % interval of 0.54 to 0.84 from only 60 values.)
(d) Without the shift the process is at 2.07σ; with it, "3.57σ". Put the defect rate in the report: about 2 % of bonds are predicted below the minimum, one was observed in sixty, and the process needs its mean raised or its spread reduced before it is acceptable. If a sigma level is demanded, write both figures with their labels.
Exercise 2 in the calculator
Quiz
Ten questions. Score 70 % or more to mark the module complete on this device.
Answer key
- c. A company-wide target.
- 4.5.
- b. 0.002 PPM.
- d. Six times as many opportunities.
- 5,000.
- a. A correlation from an interested party.
- c. A modest positive ROA effect.
- b. The organisational structure.
- d. Can we trust the gauge?
- a. Stability and shape first.
Key takeaways
- Six Sigma began as a Motorola target in the 1980s, spread through GE in the late 1990s, and packages statistical methods that are decades older. The packaging is what varies; the statistics are where the value is.
- A sigma level is a distance from the mean to the nearest limit in standard deviations, and therefore a defect rate. Centred, 6σ means 0.002 PPM; Motorola's 3.4 PPM includes a 1.5σ shift, which is a contested convention. Always say which.
- A defect rate from a count depends on how opportunities are defined; DPU and first-time yield do not. Prefer them, or define opportunities in writing first.
- A mean and standard deviation, and any defect prediction from them, are meaningful only for a process shown to be stable over time. Chart first, every time.
- DMAIC is five questions answered in order with data: what problem, can we trust the gauge and where are we now, what causes it, what fixes it and did it work, how do we keep it fixed.
- The belt structure is the genuinely new part of Six Sigma. This course covers the Green Belt body of knowledge but is not a certification course.
- The business case is contested: a consultancy's correlation and the 3M story on one side, a peer-reviewed event study on the other. Attribute savings claims to whoever made them, and remember that the method has a domain.
References
All web sources accessed 2026-09-09. Sources marked "secondary" were not read in the original by the course author; the claim is taken from the source shown. Standards and books cited by catalogue record are cited only for their existence, edition and general subject.
- National Institute of Standards and Technology. "Malcolm Baldrige National Quality Award 1988 Recipient: Motorola Inc." Baldrige Performance Excellence Program profile. https://www.nist.gov/system/files/documents/2017/10/11/1988_Motorola_Inc.pdf
- General Electric Company. 1998 Annual Report, letter to share owners. 1999. https://www.annualreports.com/HostedData/AnnualReportArchive/g/NYSE_GE_1998.pdf (the company's own claims, unaudited)
- Wikipedia. "Bill Smith (Motorola engineer)" and "Six Sigma". https://en.wikipedia.org/wiki/Bill_Smith_(Motorola_engineer) (secondary: used only for the attribution to Smith and the MAIC to DMAIC evolution; dates cross-checked against [1])
- Harry, M. J. The Nature of Six Sigma Quality. Motorola University Press, 1988. https://openlibrary.org/books/OL9828527M/The_Nature_of_Six_Sigma_Quality (catalogue record; secondary for the 1.5σ convention, which is attributed to it by [5])
- Tadikamalla, P. R. "The Confusion over Six-Sigma Quality." Quality Progress 27(11):83 to 85, 1994. https://www.proquest.com/openview/df184b19ba7003517881a00d6f967c6d/1 (abstract read)
- Bothe, D. R. "Statistical Reason for the 1.5σ Shift." Quality Engineering 14(3):479 to 487, 2002. https://www.tandfonline.com/doi/abs/10.1081/qen-120001884 (abstract read)
- Burns, T. "Predictable." Quality Digest, 13 June 2018. https://www.qualitydigest.com/inside/six-sigma-article/predictable-061318.html (opinion piece, cited as one side of the controversy)
- Schroeder, R. G., Linderman, K., Liedtke, C., and Choo, A. S. "Six Sigma: Definition and Underlying Theory." Journal of Operations Management 26(4):536 to 554, 2008. https://strategicimprovementsystems.com/wp-content/uploads/2011/07/PAPER_SIX_SIGMA.pdf (abstract read)
- Zu, X., Fredendall, L. D., and Douglas, T. J. "The Evolving Theory of Quality Management: The Role of Six Sigma." Journal of Operations Management 26(5):630 to 650, 2008. https://onlinelibrary.wiley.com/doi/10.1016/j.jom.2008.02.001 (abstract read)
- ASQ. Certified Six Sigma Green Belt (CSSGB) Body of Knowledge Map 2014 to 2022. American Society for Quality, 2022. https://www.asq.org/cert/resource/pdf/certification/2022-CSSGB-BoK-Map.pdf
- Antony, J., Snee, R., and Hoerl, R. "Lean Six Sigma: Yesterday, Today and Tomorrow." International Journal of Quality & Reliability Management 34(7):1073 to 1093, 2017. https://www.emerald.com/insight/content/doi/10.1108/ijqrm-03-2016-0035/full/html (abstract read)
- Morris, B. "Jack Welch's Rules for Winning Don't Work Anymore (But We've Got 7 New Ones That Do)." Fortune, 24 July 2006. https://fortune.com/article/fortune-archives-jack-welch/
- Hindo, B. "At 3M, a Struggle Between Efficiency and Creativity." BusinessWeek, 11 June 2007. https://effectuation.org/hubfs/Journal%20Articles/2016/06/3m-struggle-between-efficiency-and-creativity.pdf
- Swink, M., and Jacobs, B. W. "Six Sigma Adoption: Operating Performance Impacts and Contextual Drivers of Success." Journal of Operations Management 30(6):437 to 453, 2012. https://onlinelibrary.wiley.com/doi/10.1016/j.jom.2012.05.001 (abstract read)
- Best, M., and Neuhauser, D. "Walter A Shewhart, 1924, and the Hawthorne Factory." Quality and Safety in Health Care 15(2):142 to 143, 2006. https://pmc.ncbi.nlm.nih.gov/articles/PMC2464836/
- Shewhart, W. A. Economic Control of Quality of Manufactured Product. Van Nostrand, 1931 (reissued ASQ, 1980). https://archive.org/details/in.ernet.dli.2015.150272 (catalogue record)
- Hahn, G. J., Doganaksoy, N., and Hoerl, R. "The Evolution of Six Sigma." Quality Engineering 12(3):317 to 326, 2000. https://www.tandfonline.com/doi/abs/10.1080/08982110008962595 (secondary: citation only; GE insiders' account of the programme's evolution)
- Lean Enterprise Institute. "Seven Wastes." Lean Lexicon. https://www.lean.org/lexicon-terms/seven-wastes/
- Womack, J. P., and Jones, D. T. Lean Thinking. Simon & Schuster, 1996. https://www.lean.org/the-lean-post/articles/lean-thinking-a-look-back-and-a-look-forward/ (catalogue record and the Lean Enterprise Institute's retrospective)
- Ohno, T. Toyota Production System: Beyond Large-Scale Production. Productivity Press, 1988. https://books.google.com/books/about/Toyota_Production_System.html?id=QebEDwAAQBAJ (catalogue record; the waste list itself is cited from [18])
Further reading
- Harry, M., and Schroeder, R. Six Sigma: The Breakthrough Management Strategy Revolutionizing the World's Top Corporations. Currency/Doubleday, 2000. The popular account from inside the programme; its savings figures are the companies' own and are not repeated in this course.
- Pyzdek, T., and Keller, P. A. The Six Sigma Handbook, 5th ed. McGraw-Hill, 2018. A general reference for the whole body of knowledge.