Six Sigma and quality tools

Why it matters

Six Sigma gives quality improvement a common language (defects per million opportunities and sigma level), a disciplined project route (DMAIC) and trained people (Belts) to lead it. Combined with the basic quality tools, it is how most large Indian manufacturers and service companies run their improvement projects, and Green Belt knowledge is a common expectation in production and quality job interviews.

Key ideas

What "six sigma" means. A process is at six-sigma quality when the nearer specification limit is six process standard deviations from the mean, i.e. the tolerance is ±6σ (Cp = 2). Motorola, which developed the approach in the 1980s, observed that process means drift over time by about 1.5σ. Allowing for this shift, the nearer limit is effectively 4.5σ away, which gives about 3.4 defects per million opportunities (DPMO). Sigma level tables used in industry include this 1.5σ shift:

Sigma level	DPMO (with 1.5σ shift)
2	308 538
3	66 807
4	6 210
5	233
6	3.4

Without the shift, a centred ±6σ process would give only about 0.002 ppm. Always state which convention you are using.

Defect metrics. A unit is the item produced; an opportunity is a place where a defect could occur and that is checked (e.g. each solder joint on a board); a defect is any failure to meet a requirement at an opportunity. DPU counts defects per unit; DPO and DPMO normalise for complexity so that a simple bracket and a complex circuit board can be compared.

Yield measures. First-pass yield (FPY) of a step is the fraction of units that pass that step first time without rework. Rolled throughput yield (RTY) is the product of the FPYs of all steps: the probability that a unit passes the whole process with no defect at any step. It exposes the "hidden factory" of rework that final yield hides.

DMAIC (improving an existing process)

1. Define: problem, customer and critical-to-quality (CTQ) characteristics, project charter, scope (SIPOC).
2. Measure: map the process, check the measurement system (gauge R&R), collect baseline data, compute baseline DPMO or capability.
3. Analyse: find and verify root causes (cause-and-effect diagram, Pareto, scatter plots, hypothesis tests, regression).
4. Improve: generate, test and implement solutions (design of experiments, pilot runs, poka-yoke).
5. Control: hold the gain (control charts, control plan, standard work, mistake-proofing, hand-over to the process owner). For new products or processes, DMADV (Define, Measure, Analyse, Design, Verify), part of Design for Six Sigma, is used instead.

Roles. Champion/Sponsor (senior manager who owns the project and removes obstacles), Master Black Belt (expert coach), Black Belt (full-time project leader), Green Belt (part-time project leader or team member), Yellow Belt (team member with basic training). Lean Six Sigma combines this with lean tools that remove waste and shorten lead time.

The seven basic QC tools (Ishikawa)

1. Check sheet: structured form for collecting data at source.
2. Histogram: shape, centre and spread of a variable.
3. Pareto chart: categories in descending order with a cumulative line; separates the vital few causes (often about 80 % of the effect from about 20 % of the causes).
4. Cause-and-effect (fishbone, Ishikawa) diagram: possible causes grouped under headings such as Man, Machine, Method, Material, Measurement, Environment.
5. Scatter diagram: relationship between two variables.
6. Control chart: stability of a process over time.
7. Stratification (some lists use a flow chart instead): splitting data by source (machine, shift, supplier) to reveal differences. The seven new management tools (affinity diagram, relations diagram, tree diagram, matrix diagram, matrix data analysis or prioritisation matrix, process decision programme chart, arrow diagram) support planning with non-numerical data.

FMEA. Failure mode and effects analysis lists each way a design or process can fail, rates its severity S, occurrence O and detection D (each typically 1–10, with 10 the worst, i.e. hardest to detect), and computes a risk priority number RPN = S × O × D to rank actions. High-severity modes (safety) must be addressed whatever their RPN; newer guidance replaces RPN by action-priority tables for this reason.

Formulas

DPU = D / U

• D = total defects found; U = number of units inspected.

DPO = D / (U × O) and DPMO = DPO × 10⁶

• O = opportunities per unit.

Sigma level ≈ z(1 − DPO) + 1.5

• z( ) = standard normal quantile; the 1.5 is the conventional long-term shift.

FPY ≈ e^(−DPU)

• Poisson estimate of the fraction of units with zero defects.

RTY = FPY₁ × FPY₂ × … × FPYₖ

• k = number of process steps.

RPN = S × O × D

• Severity, occurrence and detection ratings (dimensionless, usually 1–10 each; RPN 1–1000).

Worked examples

Example 1 (standard): DPMO and sigma level Given: 500 circuit boards are inspected; each has 12 defect opportunities; 45 defects are found in total. Find DPU, DPMO, the sigma level (1.5σ shift convention) and the expected first-pass yield.

1. DPU = 45 / 500 = 0.09.
2. DPO = 45 / (500 × 12) = 45 / 6000 = 0.0075; DPMO = 0.0075 × 10⁶ = 7500.
3. z(1 − 0.0075) = z(0.9925) = 2.43; sigma level ≈ 2.43 + 1.5 = 3.93.
4. FPY ≈ e^(−0.09) = 0.914.

DPU = 0.09, DPMO = 7500, about a 3.9-sigma process, FPY ≈ 91.4 %.

Example 2 (GATE level): rolled throughput yield and FMEA priority Given: (a) a four-step process has first-pass yields 0.98, 0.95, 0.99 and 0.97. Find RTY. (b) An FMEA lists four failure modes with (S, O, D) ratings: A (8, 4, 5), B (6, 7, 3), C (9, 2, 6), D (5, 5, 8). Rank them by RPN.

1. (a) RTY = 0.98 × 0.95 × 0.99 × 0.97 = 0.894. So only 89.4 % of units pass all steps first time, although each step is 95 % or better.
2. (b) RPN_A = 8 × 4 × 5 = 160; RPN_B = 6 × 7 × 3 = 126; RPN_C = 9 × 2 × 6 = 108; RPN_D = 5 × 5 × 8 = 200.
3. Ranking by RPN: D (200), A (160), B (126), C (108).

RTY = 0.894 (89.4 %); RPN order D > A > B > C. Mode C has the lowest RPN but severity 9, so it still needs action (for example better detection or a design change) regardless of ranking.

Common mistakes

• Dividing defects by units only (DPU) when the question asks for DPMO; include opportunities.
• Forgetting the 1.5σ shift convention, or applying it twice; 3.4 DPMO corresponds to z = 4.5 on one side.
• Multiplying the final-inspection yield instead of step yields; RTY uses first-pass yields.
• Counting opportunities that are never inspected, which inflates the sigma level.
• Ranking FMEA purely by RPN and ignoring high severity.
• Treating Six Sigma as only statistics; without a Control phase the gains fade.

For GATE PI

Expect DPMO and DPU calculations, the meaning of 3.4 DPMO and Cp = 2, DMAIC phases in order and what happens in each, matching the seven QC tools to their uses (Pareto for prioritising, fishbone for causes, scatter for correlation), RTY products, and RPN calculations. Read carefully whether a sigma-level question uses the 1.5σ shift.

Quick check

1. 2000 units with 5 opportunities each show 30 defects. What is the DPMO?
2. What does the "C" in DMAIC stand for, and what is its purpose?
3. Which tool separates the vital few from the trivial many?
4. Three steps with FPY 0.9 each: what is RTY?

Answers: 1. 3000. 2. Control: to hold the improvement (control charts, control plan, standard work). 3. Pareto chart. 4. 0.729.