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

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]

Timeline of the methods and the programme A horizontal timeline from 1920 to 2030. Above the line, the statistical methods: Shewhart's 1924 control chart memo and 1931 book, capability indices and gauge studies in the decades after the war. Below the line, the programme: Motorola's 1981 improvement drive, the 1988 Baldrige award, GE's adoption in 1995, the 2005 AIAG SPC manual and the 2026 AIAG and VDA manual. 1920 1940 1960 1980 2000 2020 1924 Shewhart's control chart memo 1931 Economic Control of Quality 1940s to 1980s: capability indices, gauge studies, DOE, Deming and Juran 1981 Motorola drive 1988 Baldrige 1995 GE adopts 2005 AIAG SPC 2nd ed. 2026 AIAG & VDA SPC Methods (above) Programme and standards (below)
Figure 1. The statistical methods were in place for half a century before the programme that packaged them. Dates from [1], [2], [15], [16] and the AIAG catalogue.

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.

Z = (nearest specification limit − mean) / σ     fraction beyond that limit = P(standard normal > Z) Z is the sigma level in the plain statistical sense. The tail probability is read from the normal distribution; this course computes it by script and the calculator does the same, so no table is needed.

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.

Table 1. Sigma level to parts per million under the two conventions. Centred: both tails, no shift. Shifted: the near tail after the mean moves 1.5σ toward the limit (the convention behind "3.4 PPM at six sigma"); the far tail adds nothing visible above 3σ.
Sigma levelCentred, both tails (PPM)With 1.5σ shift, near tail (PPM)What it means in practice
1317,311691,462Not a process; a lottery
245,500308,538Sorting and rework as a way of life
32,70066,807Cpk = 1.00 (Module 7); the traditional minimum
4636,210Cpk = 1.33; a common customer requirement (Module 7)
4.56.81,350Cpk = 1.50; the shifted "six sigma" process
50.57233Cpk = 1.67; often asked for on critical characteristics (Module 7)
60.0023.4Cpk = 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.

A six-sigma process, centred and after a 1.5 sigma shift Two bell curves against the same pair of specification limits. The solid curve is centred; each limit is six standard deviations from its mean and the tails are invisibly small. The dashed curve is the same width but moved 1.5 standard deviations to the right, so the upper limit is 4.5 standard deviations from its mean and a small tail crosses it. LSLUSL mean, centred mean after a 1.5σ shift 4.5σ Horizontal axis in standard deviations; both curves have the same σ. Tail areas are exaggerated for visibility.
Figure 2. The centred six-sigma process (solid) and the same process after Motorola's assumed 1.5σ drift (dashed). The 3.4 PPM figure is the dashed curve's tail beyond the upper limit, 4.5σ from its mean.

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.

Table A. Hole diameter, mm, 100 consecutive parts in production order, read left to right then down (constructed data).
Values +0+1+2+3+4+5+6+7+8+9
1–108.0358.0067.9888.0308.0368.0277.9627.9848.0298.047
11–207.9858.0368.0408.0167.9888.0038.0088.0318.0357.995
21–308.0148.0158.0128.0368.0108.0157.9858.0197.9968.056
31–408.0198.0208.0187.9918.0078.0188.0358.0217.9888.012
41–508.0137.9818.0278.0327.9978.0628.0238.0048.0578.004
51–608.0197.9787.9938.0537.9957.9988.0117.9997.9907.995
61–708.0128.0488.0328.0647.9988.0208.0218.0278.0317.980
71–808.0478.0168.0068.0218.0047.9908.0087.9757.9928.031
81–908.0267.9927.9698.0208.0248.0157.9958.0178.0088.020
91–1008.0288.0158.0128.0278.0217.9988.0148.0308.0288.009

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.

Zupper = (USL − x̄) / s = (8.050 − 8.0139) / 0.0206 = 0.0361 / 0.0206 = 1.75
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.

Table 2. The same 23 leak-test failures, counted two ways.
QuantityOne opportunity per assemblySix opportunities per assembly (one per joint)
Defects per unit, DPU = 23 / 1,2500.01840.0184
Defects per million opportunities, DPMO18,4003,067
Z from DPMO, no shift2.092.74
"Sigma level" with the 1.5σ shift3.594.24
First-time yield, e−DPU (Poisson)0.98180.9818

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.

Table 3. The five phases, the question each answers, and where this course teaches the tools.
PhaseThe questionTypical deliverablesModules
DefineWhat problem, for whom, how big, and what is out of scope?Problem statement, charter, SIPOC, voice of the customer translated into critical-to-quality characteristics2, 3
MeasureCan we trust the gauge, and how does the process perform now?Measurement system analysis, data collection plan, baseline control chart and capability, defect rate4, 5, 6, 7, 8
AnalyzeWhat causes the variation or the defects?Graphical analysis, root-cause tools, FMEA, hypothesis tests, regression; verified causes, not suspected ones9, 10, 11, 12
ImproveWhat change removes the cause, and did it work?Designed experiments, Lean changes, solution selection, a pilot with a before-and-after test13, 14, 15
ControlHow do we keep it fixed after the team leaves?Control charts, control plan, reaction plan, standard work, handoff to the process owner16, 17, 18
The DMAIC sequence with its gates Five boxes in a row labelled Define, Measure, Analyze, Improve, Control, joined by arrows. Under each arrow is the gate question that must be answered with data before moving on. A return arrow from Control to Define shows that the next project starts from what the control phase revealed. DefineMeasureAnalyzeImproveControl problem sizedgauge trustedcause verifiedchange proven the next project starts from what Control revealed
Figure 3. DMAIC as a gated sequence. Each arrow carries the condition for moving on: the problem is sized with data, the gauge is shown adequate, the cause is verified rather than voted on, the change is proven by a test rather than a bar chart.

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]

Table 4. Roles as they are usually defined in a manufacturing organisation. Definitions are conventions; check your own company's.
RoleWhat they doWhat they need to know
Green BeltLeads 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 resultsEverything in this course: MSA, capability, SPC, the core hypothesis tests, a first designed experiment, DMAIC
Black BeltLeads larger cross-functional projects, usually full-time; coaches Green Belts; owns the statistical workThe Green Belt material in depth plus multiple regression, fractional and response-surface designs, advanced control charts
Master Black BeltTrains and mentors Belts, selects and reviews projects, keeps the method honest across the siteStatistical depth plus programme management
Champion or sponsorA manager who owns the business result, removes obstacles, approves the charter and the closureEnough to read a control chart and a capability report and to ask the questions in Module 8
Process ownerThe person who runs the process every day and inherits the control plan when the project endsThe 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

The other side

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

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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.

Exercise 2 data. Pull strength, N, 60 consecutive bonds in production order, read left to right then down (constructed data).
Values +0+1+2+3+4+5+6+7+8+9
1–109.559.379.227.979.498.089.759.589.719.71
11–209.418.969.0810.088.939.128.858.979.089.39
21–309.099.058.919.308.519.289.978.839.2310.80
31–408.708.369.0210.2410.378.589.569.258.938.18
41–509.818.789.419.3010.289.908.619.119.828.52
51–609.039.388.368.578.319.418.949.769.319.56

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.

1. Motorola's "Six Sigma Quality", as described in the 1988 Baldrige award profile, was
2. Without the 1.5σ shift, how many standard deviations from the mean is a limit whose one-sided tail is 3.4 PPM? (One decimal.)
3. A centred process with both specification limits 6σ from the mean produces about
4. In Worked example 3, counting six joints per assembly instead of one raised the sigma level from 3.59 to 4.24 because
5. Twelve defects are found in 600 units with four opportunities each. What is the DPMO?
6. The Fortune 2006 finding that 91 % of 58 Six Sigma companies trailed the S&P 500 is best described as
7. Swink and Jacobs (2012) found that Six Sigma adoption was associated with
8. According to Schroeder et al. (2008) and Zu et al. (2008), what is new in Six Sigma is
9. The first question of the Measure phase is
10. In Worked example 2 the upper limit was 1.75 standard deviations from the mean. Before that number was read off the histogram, the example first
Answer key
  1. c. A company-wide target.
  2. 4.5.
  3. b. 0.002 PPM.
  4. d. Six times as many opportunities.
  5. 5,000.
  6. a. A correlation from an interested party.
  7. c. A modest positive ROA effect.
  8. b. The organisational structure.
  9. d. Can we trust the gauge?
  10. a. Stability and shape first.

Key takeaways

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.

  1. 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
  2. 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)
  3. 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])
  4. 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])
  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)
  6. 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)
  7. 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)
  8. 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)
  9. 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)
  10. 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
  11. 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)
  12. 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/
  13. 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
  14. 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)
  15. 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/
  16. 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)
  17. 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)
  18. Lean Enterprise Institute. "Seven Wastes." Lean Lexicon. https://www.lean.org/lexicon-terms/seven-wastes/
  19. 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)
  20. 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