# Quality Management Concept Card: [Your process or course] > A concept card on quality for the operations core of an MBA: the cost of quality, variation and statistical process > control with Shewhart control charts, process capability (Cp and Cpk), six sigma and the DMAIC cycle, lean waste, > Deming's 14 points and the basic root-cause tools. It is for MBA students in the operations course, applicants > preparing for operations and consulting interviews, and managers who own a service or production process. The > examples use two fictional settings: the laboratory and emergency department of Tallow Creek Medical Center and the > distribution centre of Northfield Outdoor Co.; all data are invented. > Source basis: Shewhart (1931), Deming (1986), Juran (1951), Feigenbaum (1983), Crosby (1979), Ishikawa (1976), Ohno > (1988), Womack and Jones (1996), Nelson (1984) and the NIST/SEMATECH e-Handbook of Statistical Methods, written up in > original words and examples. Every number was computed in Python and is shown with the Excel formula. > Replace every [bracketed] value and every row marked *Example*. ## Document control | Field | Value | |---|---| | Course or project | [Operations management core / field project / improvement project] | | Learner or analyst | [Your name] | | Process and quality measure | [Lab turnaround time at Tallow Creek Medical Center (fictional) / your process and measure] | | Data source and period | [System extract, YYYY-MM-DD to YYYY-MM-DD, subgroup size and frequency] | | Specification or target source | [Clinical protocol, customer contract, engineering drawing] | | Tools | Excel (AVERAGE, MAX, MIN, STDEV.S, NORM.S.DIST); Python 3 with numpy and scipy for the checks | | Reviewed by | [Tutor, study group or process owner] | | Last updated | YYYY-MM-DD | | Version | [0.1] | ## When to use it | Question you face | Tool in this card | |---|---| | Is the process stable, or did something change? | Control chart (X-bar and R for measurements, p-chart for proportions defective) | | Can a stable process meet the specification? | Capability indices Cp, Cpk (or Cpu, Cpl for one-sided limits) | | How much does poor quality cost us, and where should we spend? | Cost of quality (prevention, appraisal, internal and external failure) | | How do we run an improvement project? | Six sigma DMAIC cycle, with the root-cause tools inside Analyse | | Where is effort spent without adding value? | Lean waste walk and value stream map | | Why does the problem happen? | Pareto chart, cause-and-effect (fishbone) diagram, five whys, stratification | | How should management behave to sustain quality? | Deming's 14 points as a self-assessment | Do not use control-chart limits as specification limits, and do not compute capability for a process that is not in statistical control: the index then describes no stable process at all. ## The idea in plain language Every process varies. Lab results take 38 minutes one time and 49 the next; one tent pole is 0.3 mm long, the next 0.2 mm short. Shewhart (1931) separated two kinds of variation. **Common-cause** variation (Shewhart's chance causes) is the background noise built into the system: many small causes that are always present. **Special-cause** variation (Shewhart's assignable causes) comes from something identifiable that is not normally part of the process: a miscalibrated analyser, a new supplier lot, a missing staff member. The two need opposite management responses. A special cause should be found and removed by the people close to the process, quickly. Common-cause variation can only be reduced by changing the system (equipment, method, training, design), which is management's job. Deming (1986) argued that most problems belong to the system, and that blaming individuals for common-cause variation (tampering) makes performance worse. A **control chart** is the tool that tells the two apart: it plots a statistic over time against limits set at three standard errors from the centre line. A point outside the limits, or a non-random pattern, signals a special cause; points inside the limits with no pattern are treated as common-cause noise. **Capability** is a different question: given that the process is stable, does its natural spread fit inside what the customer requires? A process can be perfectly in control and still produce many defects, because its spread is too wide or its centre is off target. **Cost of quality** turns quality into money. Spending on prevention and appraisal is visible in budgets; failure costs (scrap, rework, returns, warranty, lost customers) are spread across many accounts and are usually larger. Feigenbaum's prevention, appraisal and failure categories and Crosby's (1979) argument that it is nonconformance, not quality, that costs money lead to the same management conclusion: move spending upstream to prevention. **Six sigma** and **lean** are two improvement programmes built on these ideas. Six sigma, developed at Motorola in the 1980s and spread by General Electric in the 1990s, uses a structured project cycle (DMAIC) and statistics to reduce variation. Lean, drawn from the Toyota Production System (Ohno 1988) and popularised in the West by Womack, Jones and Roos (1990), removes waste and improves flow. Many organisations combine them. ## Formulas and notation | Symbol | Meaning | |---|---| | $n$ | Subgroup size (measurements taken together, for example 4 tests per hour) | | $k$ | Number of subgroups used to set the limits (20 to 25 is common; 15 is a minimum for a first look) | | $\bar{x}_i$, $R_i$ | Mean and range of subgroup $i$ | | $\bar{\bar{x}}$, $\bar{R}$ | Grand mean and average range over the $k$ subgroups | | $A_2$, $D_3$, $D_4$, $d_2$ | Control-chart constants that depend on $n$ | | $\hat{\sigma}$ | Estimate of the within-subgroup (short-term) standard deviation, $\bar{R}/d_2$ | | $\bar{p}$ | Average proportion defective across subgroups | | USL, LSL | Upper and lower specification limits (set by the customer or design, not by the data) | | $\mu$ | Process mean | **X-bar and R charts:** $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2\bar{R}, \qquad LCL_{\bar{x}} = \bar{\bar{x}} - A_2\bar{R}, \qquad UCL_R = D_4\bar{R}, \qquad LCL_R = D_3\bar{R}$$ **p-chart** (subgroups of constant size $n$): $$UCL_p = \bar{p} + 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}, \qquad LCL_p = \max\left(0,\ \bar{p} - 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}\right)$$ **Capability** (stable process, roughly normal data): $$C_p = \frac{USL - LSL}{6\hat{\sigma}}, \qquad C_{pk} = \min\left(\frac{USL - \mu}{3\hat{\sigma}},\ \frac{\mu - LSL}{3\hat{\sigma}}\right), \qquad C_{pu} = \frac{USL - \mu}{3\hat{\sigma}}$$ **Defects per million opportunities** and the six sigma convention, which assumes the mean drifts by 1.5 standard deviations in the long run: $$DPMO = \frac{\text{defects}}{\text{units} \times \text{opportunities per unit}} \times 10^6, \qquad DPMO(z) = \left(1 - \Phi(z - 1.5)\right) \times 10^6$$ Control-chart constants (computed from the normal distribution of the range): | $n$ | $d_2$ | $A_2$ | $D_3$ | $D_4$ | |---|---|---|---|---| | 4 | 2.059 | 0.729 | 0 | 2.282 | | 5 | 2.326 | 0.577 | 0 | 2.114 | Sigma level and defects (one-sided, with the 1.5 sigma shift convention). Excel: `=(1-NORM.S.DIST(z-1.5,TRUE))*10^6`. | Sigma level $z$ | DPMO with 1.5 sigma shift | |---|---| | 2 | 308,537.5 | | 3 | 66,807.2 | | 4 | 6,209.7 | | 5 | 232.6 | | 6 | 3.4 | The 1.5 sigma shift is a convention adopted in six sigma practice, not a law of nature; many statisticians criticise it. A process with a measured long-term spread needs no assumed shift. **Quick capability example (fictional).** A supplier makes tent-pole sections for Northfield Outdoor Co. to a specification of 500.0 mm plus or minus 1.5 mm (LSL 498.5, USL 501.5). The stable process has mean 500.4 mm and $\hat{\sigma} = 0.40$ mm. $C_p = 3.0/(6 \times 0.40) = 1.250$; $C_{pk} = \min(1.1, 1.9)/(3 \times 0.40) = 0.917$. The spread would fit, but the process is off centre: about 2,981 parts per million fall outside (2,980 above USL, 1.0 below LSL). Excel: `=(1-NORM.S.DIST((501.5-500.4)/0.4,TRUE)+NORM.S.DIST((498.5-500.4)/0.4,TRUE))*10^6`. Centring the mean at 500.0 raises $C_{pk}$ to 1.250 and cuts the defect rate to 176.8 ppm, with no new equipment. Reaching $C_{pk} = 1.33$ (a common customer requirement) when centred needs $\hat{\sigma} \le 0.376$ mm. ## Cost of quality Cost of quality groups spending into four categories. Prevention and appraisal are the costs of good quality; internal and external failure are the costs of poor quality. | Category | Meaning | Northfield Outdoor Co. items *(Example, USD thousand per year)* | Total | Share of revenue (USD 240.0m) | Share of COQ | |---|---|---|---|---|---| | Prevention | Stops defects from occurring | Supplier quality audits 420; design reviews for new products 380; picking-standards training 260 | 1,060 | 0.44% | 11.0% | | Appraisal | Finds defects before customers do | Incoming inspection 910; fabric and coating lab tests 340; order audits before dispatch 450 | 1,700 | 0.71% | 17.7% | | Internal failure | Defects found before shipment | Rework and relabelling 620; scrap of damaged stock 780; re-picking mis-picks 300 | 1,700 | 0.71% | 17.7% | | External failure | Defects found by customers | Defect returns processing 2,150; warranty repairs 1,640; reshipping wrong items 870; quality-related service contacts 480 | 5,140 | 2.14% | 53.5% | | **Total** | | | **9,600** | **4.00%** | **100.0%** | Excel: share of revenue `=E2/240000`; share of COQ `=E2/SUM($E$2:$E$5)`. Reading the table: more than half of Northfield's cost of quality is external failure, the most expensive kind because it also costs reputation and future sales, which the table does not even include. Only 11 percent goes to prevention. The classic management move is to fund prevention (supplier development, design for quality, error-proofing in picking) and expect failure and later appraisal costs to fall by more than the extra spend. Test that expectation with a pilot and a before-and-after cost of quality, not with a slogan. ## Statistical process control in practice Steps for a variables chart (measurements such as time, length, weight): 1. Choose the quality characteristic that matters to the customer and can be measured reliably (check the measurement system first: a gauge that varies as much as the process makes the chart useless). 2. Choose rational subgroups: for example, 4 consecutive lab tests each hour. 3. Collect at least 20 to 25 subgroups (15 for a first look), compute $\bar{x}_i$ and $R_i$. 4. Compute trial limits. Check the R chart first: if the ranges are out of control, the X-bar limits (which use $\bar{R}$) are not valid. 5. Investigate points outside the limits and non-random patterns. If you find and remove an assignable cause, drop those subgroups and recompute the limits. Do not drop points without a documented cause. 6. Freeze the limits and use them to monitor new data. Recompute only when the process has been changed on purpose. 7. Once in control, assess capability against the specification. Signals of a special cause, beyond a point outside the three-sigma limits, come from rule sets such as the Western Electric rules (1956) and Nelson (1984): for example, a run of eight or nine points on one side of the centre line (rule sets differ in the count), six points steadily rising or falling, or two of three points beyond two sigma on the same side. More rules catch smaller shifts faster but raise false alarms; choose a rule set before you look at the data. ## Six sigma, DMAIC and the root-cause tools ```mermaid flowchart LR D["Define: problem, customer, CTQ, scope, charter"] --> M["Measure: baseline, measurement system check, capability"] M --> A["Analyse: Pareto, fishbone, five whys, data tests of causes"] A --> I["Improve: pilot solutions, error-proofing, standard work"] I --> C["Control: control chart, control plan, handover to owner"] C -->|"New problem or new target"| D ``` - **Define.** State the problem in numbers, the customer and the critical-to-quality characteristic (CTQ), the scope, the team and the expected benefit in a project charter. - **Measure.** Establish the baseline, check the measurement system, and compute current capability or DPMO. - **Analyse.** Find the vital few causes. Use a **Pareto chart** (Juran's name for the observation that a few causes explain most defects) to choose where to look, a **cause-and-effect diagram** (Ishikawa) to list possible causes by category (people, methods, machines, materials, measurement, environment), **five whys** (from Toyota practice) to push past symptoms, and **stratification** and **scatter plots** to test causes with data. Ishikawa (1976) grouped these with the check sheet, histogram and control chart as the basic quality tools. - **Improve.** Pilot solutions, error-proof steps (poka-yoke), standardise the new method and measure the effect. - **Control.** Hand the improved process to its owner with a control chart, a control plan and a response plan. Six sigma projects are led by trained practitioners (often titled green belts and black belts) and sponsored by a manager who owns the benefit. Critics point to bureaucracy and to projects chosen for certification rather than for value; treat the belts as a means, not the goal. ## Lean and the seven wastes Lean thinking starts from value as the customer defines it, then maps the value stream, makes the work flow, lets the customer pull it and pursues perfection (Womack and Jones 1996). Ohno (1988) described the wastes that the Toyota Production System attacks; a commonly added eighth is unused talent. | Waste | Meaning | *Example* at Tallow Creek or Northfield | |---|---|---| | Overproduction | Making more or earlier than needed | Running full lab panels when one test was ordered | | Waiting | People or items idle | Patients waiting for a bed after the admission decision | | Transport | Moving items without adding value | Samples walked to a lab on another floor | | Over-processing | More work than the customer values | Triple data entry of the same patient details | | Inventory | More stock than needed | Distribution centre holding 40 weeks of slow-moving tents | | Motion | Unnecessary movement of people | Pickers walking the full aisle for fast-moving items placed at the back | | Defects | Errors, rework, returns | Mis-picked orders reshipped to customers | | Unused talent (added later) | Ideas and skills not used | Nurses' improvement ideas never collected | ## Deming's 14 points Deming set out 14 points for management in *Out of the Crisis* (1986). The paraphrases below are original summaries, not his wording; read the book for his full argument. | # | Point (short name) | Plain paraphrase | Self-check question | |---|---|---|---| | 1 | Constancy of purpose | Commit to long-term improvement of products and services over quarterly results | Does our plan fund improvement beyond this year? | | 2 | The new philosophy | Accept that delays, errors and defects are not normal | Do we treat defects as a cost of doing business? | | 3 | Stop relying on mass inspection | Build quality into the process instead of sorting it at the end | What share of our COQ is appraisal? | | 4 | Do not buy on price alone | Work with fewer suppliers on total cost and long-term quality | Do we compare suppliers on total cost? | | 5 | Improve the system constantly | Keep reducing waste and variation in every process | Who owns improvement of each process? | | 6 | Train on the job | Teach everyone the job and the method properly | Do new staff learn the standard or a colleague's habit? | | 7 | Lead | Supervisors should help people and remove obstacles, not just police | What did I remove from my team's way this month? | | 8 | Drive out fear | People must feel safe to report problems and ask questions | Are errors reported or hidden? | | 9 | Break down barriers between departments | Functions should solve problems together | Do handovers between units have a shared owner? | | 10 | Drop slogans and exhortations | Posters asking for zero defects blame people for system problems | Are we asking people to try harder at a broken process? | | 11 | Drop numerical quotas and management by numbers alone | Targets without a method distort behaviour | Do our targets come with a method to reach them? | | 12 | Remove barriers to pride in work | Let people do good work and know they did | What stops our people from doing the job well? | | 13 | Education and self-improvement | Invest in everyone's learning beyond the current job | What is our learning budget per person? | | 14 | Everyone transforms | Make the transformation everyone's job, led from the top | Who on the leadership team owns quality? | ## Worked example A (fully solved): X-bar and R charts for lab turnaround time **Situation (fictional).** Tallow Creek Medical Center's laboratory measures turnaround time (sample received to result available) for urgent blood tests. Each hour for 15 hours, the quality lead records 4 consecutive urgent tests ($n = 4$). The clinical protocol requires results within 60 minutes. | Subgroup *(Example)* | Times (min) | $\bar{x}_i$ | $R_i$ | |---|---|---|---| | 1 | 38, 43, 33, 49 | 40.75 | 16 | | 2 | 45, 41, 40, 44 | 42.50 | 5 | | 3 | 41, 41, 46, 45 | 43.25 | 5 | | 4 | 42, 42, 43, 39 | 41.50 | 4 | | 5 | 40, 45, 41, 35 | 40.25 | 10 | | 6 | 40, 45, 41, 41 | 41.75 | 5 | | 7 | 45, 51, 38, 49 | 45.75 | 13 | | 8 | 36, 43, 36, 49 | 41.00 | 13 | | 9 | 46, 48, 38, 45 | 44.25 | 10 | | 10 | 39, 40, 45, 46 | 42.50 | 7 | | 11 | 43, 39, 38, 49 | 42.25 | 11 | | 12 | 57, 58, 65, 50 | 57.50 | 15 | | 13 | 55, 58, 50, 46 | 52.25 | 12 | | 14 | 39, 47, 40, 42 | 42.00 | 8 | | 15 | 41, 43, 48, 39 | 42.75 | 9 | Excel, with the four times in B2:E2: subgroup mean `=AVERAGE(B2:E2)`, range `=MAX(B2:E2)-MIN(B2:E2)`. **Step 1: trial limits from all 15 subgroups.** $\bar{\bar{x}} = 44.017$ minutes, $\bar{R} = 9.533$ minutes. R chart: $UCL_R = 2.282 \times 9.533 = 21.755$, $LCL_R = 0$. All ranges are inside, so the spread is stable and the X-bar limits can be used. X-bar chart: $UCL = 44.017 + 0.729 \times 9.533 = 50.966$, $LCL = 44.017 - 0.729 \times 9.533 = 37.067$. Excel: `=AVERAGE(F2:F16)+0.729*AVERAGE(G2:G16)`. **Step 2: signals.** Subgroups 12 (57.50) and 13 (52.25) are above the upper limit. **Step 3: investigate.** The quality lead finds that an analyser calibration drifted during hour 12 and was corrected at the end of hour 13. This is a documented assignable cause, so subgroups 12 and 13 are removed from the limit calculation (they stay on the chart, marked). **Step 4: revised limits from 13 subgroups.** $\bar{\bar{x}} = 42.346$, $\bar{R} = 8.923$; $UCL_{\bar{x}} = 48.851$, $LCL_{\bar{x}} = 35.841$; $UCL_R = 20.362$. All 13 remaining subgroups are inside the revised limits with no run or trend, so the process is treated as in control. **Step 5: capability against the 60-minute protocol (one-sided).** $\hat{\sigma} = \bar{R}/d_2 = 8.923/2.059 = 4.334$ minutes. $C_{pu} = (60 - 42.346)/(3 \times 4.334) = 1.358$. Under a normal model, about 23 tests per million would exceed 60 minutes. Excel: `=(60-42.346)/(3*8.923/2.059)` and `=(1-NORM.S.DIST((60-42.346)/4.334,TRUE))*10^6`. **Step 6: decision.** In normal operation the laboratory is stable and capable against the protocol. The real risk is special causes such as the calibration drift, which pushed five of eight results in hours 12 and 13 to 55 minutes or more. The management response is a calibration check in the control plan and an alert when a subgroup mean crosses 48.9 minutes, not a general speed-up campaign. Turnaround times are often right-skewed; check the normal model with a histogram before you rely on a parts-per-million estimate. ```vega-lite { "$schema": "https://vega.github.io/schema/vega-lite/v5.json", "description": "X-bar chart of urgent lab turnaround time at Tallow Creek Medical Center (fictional) with revised limits", "width": 420, "height": 220, "data": {"values": [ {"subgroup": 1, "mean": 40.75}, {"subgroup": 2, "mean": 42.5}, {"subgroup": 3, "mean": 43.25}, {"subgroup": 4, "mean": 41.5}, {"subgroup": 5, "mean": 40.25}, {"subgroup": 6, "mean": 41.75}, {"subgroup": 7, "mean": 45.75}, {"subgroup": 8, "mean": 41.0}, {"subgroup": 9, "mean": 44.25}, {"subgroup": 10, "mean": 42.5}, {"subgroup": 11, "mean": 42.25}, {"subgroup": 12, "mean": 57.5}, {"subgroup": 13, "mean": 52.25}, {"subgroup": 14, "mean": 42.0}, {"subgroup": 15, "mean": 42.75} ]}, "layer": [ { "mark": {"type": "line", "point": true}, "encoding": { "x": {"field": "subgroup", "type": "quantitative", "title": "Subgroup (hour)", "axis": {"tickMinStep": 1}}, "y": {"field": "mean", "type": "quantitative", "title": "Mean turnaround (minutes)", "scale": {"domain": [30, 60]}} } }, {"mark": {"type": "rule", "strokeDash": [6, 4], "color": "firebrick"}, "encoding": {"y": {"datum": 48.851}}}, {"mark": {"type": "rule", "color": "gray"}, "encoding": {"y": {"datum": 42.346}}}, {"mark": {"type": "rule", "strokeDash": [6, 4], "color": "firebrick"}, "encoding": {"y": {"datum": 35.841}}} ] } ``` The dashed lines are the revised control limits (35.841 and 48.851) and the solid grey line the revised centre line (42.346). Subgroups 12 and 13 sit above the upper limit; they were excluded from the revised limits because their cause was found. ## Worked example B (half solved): p-chart for order picking errors **Situation (fictional).** Northfield Outdoor Co.'s distribution centre audits 500 dispatched orders each day and counts orders with at least one picking error. Over 20 days the counts were: 5, 8, 7, 7, 5, 14, 14, 10, 4, 7, 7, 6, 6, 7, 4, 6, 17, 8, 2, 2 (146 errors in 10,000 audited orders). **Solved part.** $\bar{p} = 146/10{,}000 = 0.01460$. Standard error $\sqrt{0.0146 \times 0.9854/500} = 0.00536$. $UCL_p = 0.0146 + 3 \times 0.00536 = 0.03069$, which is 15.35 errors in 500 orders; $LCL_p = \max(0, 0.0146 - 0.01608) = 0$. Excel: `=0.0146+3*SQRT(0.0146*(1-0.0146)/500)`. **To complete.** 1. Which day or days signal a special cause? [ ] 2. The supervisor finds that on that day most picks were made by temporary staff on their first shift. Recompute $\bar{p}$ without that day: [ ] errors in [ ] orders, $\bar{p}$ = [ ]. 3. Revised $UCL_p$ = [ ], which is [ ] errors per 500 orders. 4. Days 6 and 7 had 14 errors each. Are they signals under the revised limit? What would you still check? [ ] 5. State one prevention action and one appraisal action for the temporary-staff cause, and say which cost of quality category each belongs to. [ ] ## Exercise C (unsolved): returns inspection time **Situation (fictional).** Northfield's returns centre times the inspection of returned jackets. Twenty-five subgroups of 5 consecutive jackets give $\bar{\bar{x}} = 12.0$ minutes and $\bar{R} = 3.2$ minutes. The service standard is between 5 minutes (shorter suggests skipped checks) and 20 minutes (longer delays refunds). 1. Compute the X-bar chart limits and the R chart upper limit. 2. Estimate $\hat{\sigma}$, then $C_p$ and $C_{pk}$. Write the Excel formulas. 3. Is the process capable? What would you check before you report the answer to the operations director? 4. The director proposes a target of "every inspection under 12 minutes". Use the chart and Deming's point 11 to explain the likely effect. ## Connections to other cards - `mba/mba-process-analysis-card`: yield and rework loops change capacity and flow time; defects at the bottleneck are the most expensive. - `mba/mba-queueing-card`: rework raises utilisation and therefore waiting; variability reduction is a queueing lever and a quality lever at the same time. - `mba/mba-inventory-card`: lean's view of inventory as waste and the safety stock logic of the inventory card meet in the question of how much buffer variability really requires. - `mba/mba-operations-guide`: the course map and the mini-case that uses this card. - `academic/stats-sampling-distributions` and `academic/stats-distributions-sheet`: why limits sit at three standard errors, and the normal and binomial models behind X-bar and p-charts. - `academic/stats-proportion-tests`, `academic/stats-hypothesis-testing` and `academic/stats-experiment-design`: testing whether an improvement worked. - `compliance/capa-nonconformity`, `ops/fmea-software`, `dev/postmortem` and `healthcare/clinical-audit`: the documentation formats for corrective action, risk analysis and learning from incidents. - `quality/sqa-plan` and `quality/test-strategy`: the same quality ideas in software delivery. ## Common mistakes and misconceptions - **Using specification limits as control limits.** Control limits come from the process data; specification limits come from the customer. A process can be in control and out of specification, or the reverse. - **Reacting to every point (tampering).** Adjusting a stable process after each common-cause movement adds variation. Act on signals, not on noise. - **Removing out-of-control points without a cause.** Dropping points to make the chart look clean hides the problem and narrows the limits falsely. - **Computing capability for an unstable process.** Cp and Cpk assume a stable, predictable process; first bring it into control. - **Reporting Cp without Cpk.** Cp ignores centring; the tent-pole example has Cp 1.250 but Cpk 0.917. - **Using the overall standard deviation for short-term limits.** Limits use within-subgroup variation ($\bar{R}/d_2$); the overall standard deviation includes special causes and inflates the limits. - **Treating sigma level as exact.** The DPMO table assumes normality and a 1.5 sigma shift; real tails differ. - **Cost of quality without failure costs.** Teams often count inspection budgets but miss returns, reshipping and service time in other accounts, which understates the case for prevention. - **Lean as headcount reduction.** Using lean to cut jobs destroys the trust that improvement depends on (Deming's point 8). - **Five whys stopping at a person.** "The picker made a mistake" is not a root cause; ask why the system allowed it. ## Self-test questions 1. Define common-cause and special-cause variation and give the right management response to each. 2. Why do you check the R chart before you read the X-bar chart? 3. A process has Cp = 1.6 and Cpk = 0.8. What does that tell you, and what is the cheapest fix? 4. Using the sigma-level table, how many defects per million does a 4-sigma process produce under the six sigma convention? Give the Excel formula. 5. Why does external failure usually cost more than the same defect caught internally? Give two reasons. 6. Name the five DMAIC phases and one tool used in each. 7. Name five of the seven wastes and give a service-industry example for two of them. 8. A manager says: "The chart shows the process is in control, so our customers must be satisfied." What is wrong? 9. Which two of Deming's 14 points are most at odds with a sales team paid on individual quarterly quotas, and why? 10. A p-chart's subgroup size changes from 500 to 200 orders a day. What happens to the control limits? 11. The five whys analysis of a mis-picked order ends with "the picker was careless". Continue it two more levels. ## My notes and next steps | Item | My entry | |---|---| | Confidence with quality tools (1 = lost, 5 = could teach it) | [ ] | | A process I know and its key quality measure | [ ] | | My estimate of its cost of quality split, and where the data would come from | [ ] | | Questions for my professor or study group | [ ] | | Mistakes logged in `mba/mba-error-log` | [Date and short description] | | Next review date (spacing: +2 days, +1 week, +1 month) | YYYY-MM-DD | | Next card to study | [mba/mba-queueing-card / mba/mba-process-analysis-card] | ## Sources and further reading - Shewhart, W. A. (1931). *Economic Control of Quality of Manufactured Product*. Van Nostrand. - Juran, J. M. (Ed.) (1951). *Quality Control Handbook*. McGraw-Hill. - Western Electric Company (1956). *Statistical Quality Control Handbook*. Western Electric. - Ishikawa, K. (1976). *Guide to Quality Control*. Asian Productivity Organization. - Crosby, P. B. (1979). *Quality Is Free*. McGraw-Hill. - Feigenbaum, A. V. (1983). *Total Quality Control* (3rd ed.). McGraw-Hill. - Nelson, L. S. (1984). The Shewhart control chart: tests for special causes. *Journal of Quality Technology*, 16(4), 237-239. - Deming, W. E. (1986). *Out of the Crisis*. MIT Center for Advanced Engineering Study. - Ohno, T. (1988). *Toyota Production System: Beyond Large-Scale Production*. Productivity Press. - Womack, J. P., Jones, D. T., and Roos, D. (1990). *The Machine That Changed the World*. Rawson Associates. - Womack, J. P., and Jones, D. T. (1996). *Lean Thinking*. Simon and Schuster. - NIST/SEMATECH e-Handbook of Statistical Methods, chapter 6, Process or Product Monitoring and Control, https://www.itl.nist.gov/div898/handbook/ (accessed 2026-10-28). Open resource, US government work. - Further reading (by name only): Montgomery, *Introduction to Statistical Quality Control*; Cachon and Terwiesch, *Matching Supply with Demand*; Wheeler, *Understanding Variation*. ## Answer key **Worked example B.** (1) Day 17 (17 errors, $p = 0.034$) is above the UCL of 0.03069 (15.35 errors). (2) 129 errors in 9,500 orders, $\bar{p} = 0.01358$. (3) $UCL_p = 0.01358 + 3 \times \sqrt{0.01358 \times 0.98642/500} = 0.01358 + 3 \times 0.00518 = 0.02911$, which is 14.55 errors per 500 orders. (4) Fourteen errors (0.028) are below 0.02911, so days 6 and 7 are not signals under a three-sigma rule. They are, however, two consecutive high days; check run rules, and check what happened on those days (product launch, staffing) before you dismiss them. (5) Prevention: a structured first-shift induction with a buddy and simplified pick paths (prevention cost). Appraisal: a 100 percent check of orders picked by staff in their first week (appraisal cost). Prevention is preferred in the long run; appraisal is a sensible temporary containment. **Exercise C.** (1) $UCL_{\bar{x}} = 12.0 + 0.577 \times 3.2 = 13.846$; $LCL_{\bar{x}} = 12.0 - 1.846 = 10.154$; $UCL_R = 2.114 \times 3.2 = 6.765$, $LCL_R = 0$. (2) $\hat{\sigma} = 3.2/2.326 = 1.376$ minutes; $C_p = 15/(6 \times 1.376) = 1.817$; $C_{pk} = \min(8, 7)/(3 \times 1.376) = 1.696$. Excel: `=(20-5)/(6*3.2/2.326)` and `=MIN(20-12,12-5)/(3*3.2/2.326)`. (3) Yes, comfortably (above 1.33), provided the process is in control: check that all 25 subgroups are inside the limits with no runs, that the data are roughly normal, and that the timing method is reliable. (4) The process mean is 12.0, so about half of inspections take longer than 12 minutes from common causes. A target without a method will either be missed or met by skipping checks (and times near the 5-minute lower bound would rise), exactly the distortion Deming's point 11 warns about. **Self-test.** 1. Common cause: built-in noise of the system; respond by changing the system (management). Special cause: an identifiable, unusual event; find and remove it locally and quickly. 2. The X-bar limits are computed from $\bar{R}$. If the ranges are unstable, $\bar{R}$ does not describe a single short-term spread and the X-bar limits are meaningless. 3. The natural spread ($6\hat{\sigma}$) uses only about 62 percent of the tolerance (1/1.6), but the mean sits only 2.4 $\hat{\sigma}$ from the nearer limit, half the distance it would be if centred. Centring the process (an adjustment, not new equipment) is usually the cheapest fix. 4. 6,209.7 DPMO. Excel: `=(1-NORM.S.DIST(4-1.5,TRUE))*10^6`. 5. Examples: it adds handling, reshipping and service costs on top of the original cost; it damages reputation and future sales; it may carry warranty, liability or regulatory costs. 6. Define (project charter, SIPOC), Measure (measurement system check, capability study), Analyse (Pareto, fishbone, hypothesis test), Improve (pilot, error-proofing), Control (control chart, control plan). 7. Overproduction, waiting, transport, over-processing, inventory, motion, defects. Service examples: waiting (customers on hold), over-processing (approvals that add no value). 8. In control means stable, not good. A stable process can be off target or too wide for the customer's needs; capability and customer measures answer that question. 9. Examples: point 11 (quotas and numerical targets without a method) and point 9 (barriers between departments), or point 8 (fear). Individual quotas encourage short-term behaviour and competition instead of cooperation. 10. The standard error grows by $\sqrt{500/200} = 1.58$, so the limits widen; with varying sizes, compute limits for each day. 11. Example: Why was the picker careless? Because two similar products sit in adjacent bins with similar labels. Why? Because slotting is by supplier, not by pick risk, and nobody owns slotting rules. Root cause: the slotting rule, which management can change. ## Revision history | Version | Date | Author | Change summary | |---|---|---|---| | 0.1 | YYYY-MM-DD | [Your name] | First draft from the template | | 0.2 | YYYY-MM-DD | [Your name] | *Example:* Replaced lab data with own process, froze revised limits | ## Standard and licence note Based on the published methods of Shewhart (1931), the Western Electric rules (1956), Nelson (1984), Deming (1986) and the NIST/SEMATECH e-Handbook of Statistical Methods, and on the cost of quality, lean and six sigma literature cited above. The text, the paraphrase of Deming's 14 points, the examples and the data are original; Tallow Creek Medical Center and Northfield Outdoor Co. are fictional. This card is not affiliated with or endorsed by any business school, standards body, company or author, and it is not a certification body of knowledge. Check your course's constant tables and rule sets, which may differ slightly. Template licence: MIT.