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6.15 Capability Studies, Attributes Capability, Yield Metrics, and Long-Term Capability

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1 Chapter 1: Introduction & CSSBB Exam Strategy

1.1 CSSBB Exam Format, Scoring, and Testing Windows 1.2 Eligibility, Project Affidavits, Application, and Recertification 1.3 Open-Book Strategy, Reference Materials, and a 16-Week Study Plan

2 Chapter 2: Organization-Wide Planning and Deployment

2.1 Fundamentals of Six Sigma and Lean Methodologies 2.2 Selecting the Right Methodology: DMAIC, DFSS, and Lean 2.3 Business Systems, Processes, and Stakeholder Relationships 2.4 Strategic Planning and Deployment of Six Sigma Initiatives 2.5 Six Sigma Roles and Responsibilities 2.6 Organizational Barriers and Common Causes of Six Sigma Failure 2.7 Change Management: Stakeholder Analysis, Readiness, and Communication

3 Chapter 3: Organizational Process Management and Measures

3.1 Impact of Six Sigma Projects on Customers, Suppliers, and Stakeholders 3.2 Benchmarking Types and Performance Goal Setting 3.3 Performance Measures: Balanced Scorecard, KPIs, KBIs, OKRs, and Line of Sight 3.4 Leading and Lagging Indicators and Performance Dashboards 3.5 Financial Measures: Revenue Growth, Margin, NPV, ROI, and Cost-Benefit Analysis 3.6 Cost of Poor Quality and Hard versus Soft Savings

4 Chapter 4: Team Management

4.1 Team Types and Constraints 4.2 Team Roles, Responsibilities, and Member Selection 4.3 Team Success Factors and Ground Rules 4.4 Motivational Techniques and Overcoming Demotivators 4.5 Stages of Team Development and Conflict Resolution 4.6 Team Communication Plans 4.7 Team Leadership Models: Direct, Coach, Support, and Delegate 4.8 Group Behaviors, Conflict Resolution, and Meeting Management 4.9 Facilitation Techniques and Team Decision-Making Methods 4.10 Team Training: Needs Assessment, Delivery, and Evaluation

5 Chapter 5: Define Phase

5.1 Customer Identification and Segmentation 5.2 Customer Data Collection: Methods, Validity, and Reliability 5.3 Customer Requirements: CTX, CTQ Trees, and Operational Definitions 5.4 The Kano Model and Quality Function Deployment 5.5 SIPOC and Process Mapping 5.6 Business Case Justification 5.7 The Project Charter and the Problem Statement 5.8 Project Scope and Boundaries 5.9 SMART Goals and Project Performance Measurements 5.10 Project Charter Reviews and Toll-Gate Governance 5.11 Project Management Tools: Gantt Charts, Toll-Gate Reviews, WBS, and RACI 5.12 Analytical Tools Used Throughout the DMAIC Cycle

6 Chapter 6: Measure Phase

6.1 Process Flow Metrics and the Hidden Factory 6.2 Process Analysis Tools and Value Stream Mapping 6.3 Types of Data and Measurement Scales 6.4 Sampling Concepts, Methods, and Sample Size for Estimation 6.5 Data Collection Plans and Common Pitfalls 6.6 Gage R&R for Variable Measurement Systems 6.7 Attribute Agreement Analysis and Audit MSA 6.8 Measurement Systems Across the Organization, and Metrology 6.9 Population Parameters, Sample Statistics, and Valid Statistical Conclusions 6.10 Descriptive Statistics: Central Tendency, Dispersion, and Shape 6.11 Graphical Methods for Exploratory Data Analysis 6.12 Probability Concepts, Common Distributions, and the Central Limit Theorem 6.13 Additional Distributions: Hypergeometric, Exponential, Lognormal, Weibull, and Bivariate 6.14 Process Capability and Performance Indices 6.15 Capability Studies, Attributes Capability, Yield Metrics, and Long-Term Capability

7 Chapter 7: Analyze Phase

7.1 The Correlation Coefficient, Its Confidence Interval, and Causation 7.2 Linear Regression and Residual Analysis 7.3 Multivariate Tools: Factor Analysis, Discriminant Analysis, and MANOVA 7.4 Hypothesis Testing Terminology, Errors, and Single-Sample Tests 7.5 Statistical Versus Practical Significance 7.6 Sample Size for Hypothesis Tests 7.7 Point Estimates, Confidence, Prediction, and Tolerance Intervals 7.8 Tests for Means, Variances, and Proportions 7.9 Analysis of Variance 7.10 Chi-Square Goodness-of-Fit and Tests of Independence 7.11 Types of Risk and Failure Mode and Effects Analysis 7.12 Gap Analysis and Root Cause Analysis 7.13 Waste Analysis: The Seven Classic Wastes and Resource Under-Utilization

8 Chapter 8: Improve Phase

8.1 Design of Experiments: Objectives, Terminology, and Effects 8.2 DOE Design Principles: Randomization, Blocking, Replication, and Resolution 8.3 Planning and Evaluating Designed Experiments 8.4 One-Factor Experiments: Completely Randomized, Randomized Block, and Latin Square Designs 8.5 Full Factorial Experiments and Design Selection 8.6 Two-Level Fractional Factorial Experiments and Confounding 8.7 Response Surface Methodology and Optimization 8.8 Waste Elimination: Pull Systems, Kanban, 5S, and Poka-Yoke 8.9 Cycle-Time Reduction: Continuous Flow, SMED, and Heijunka 8.10 Kaizen, Kaizen Blitz, Theory of Constraints, and OEE 8.11 Implementation Planning, Piloting, and Solution Selection

9 Chapter 9: Control Phase

9.1 SPC Objectives, Variable Selection, and Rational Subgrouping 9.2 Control Charts for Variables Data 9.3 Control Charts for Attributes Data 9.4 Short-Run SPC and Moving Average Charts 9.5 Control Chart Analysis: Common Cause, Special Cause, and Run Rules 9.6 Total Productive Maintenance and Visual Controls 9.7 Measurement System Reanalysis 9.8 Control Plan Development and Process Owner Handoff 9.9 Lessons Learned, Documentation, and Training for Sustainment 9.10 Ongoing Evaluation of the Improved Process

10 Chapter 10: Design for Six Sigma (DFSS)

10.1 DFSS Frameworks: DMADV, DMADOV, and Design Transfer 10.2 Design for X (DFX) 10.3 Robust Design, Parameter Design, and Verification 10.4 Tolerance Design and Statistical Tolerancing

6.15 Capability Studies, Attributes Capability, Yield Metrics, and Long-Term Capability

Key Takeaways Explain This Topic AI

A capability study requires a stable process, a validated measurement system, a verified distribution, and a rational sampling plan that spans the sources of variation.

Natural process limits are set by the process at the mean plus or minus three standard deviations; specification limits are set by the customer and are unrelated.

Rolled throughput yield is the product of the first-pass yields of every step and is always lower than final yield.

For attributes data, process sigma is obtained by converting the defect proportion to a Z value and adding the 1.5 sigma shift when reporting a short-term equivalent.

Short-term capability uses within-subgroup variation and yields Cp and Cpk; long-term performance uses total variation and yields Pp and Ppk.

Last updated: August 2026

  • Designing and conducting a capability study
  • A capability index computed on unsuitable data is worse than no index, because it is believed. Four

prerequisites, in order:

Validated measurement system. If gage R&R consumes 30% of the tolerance, the observed spread is

substantially measurement noise and the capability index is understated.

  • Statistical stability. Capability is a prediction about future output, which is only defensible
  • if the process is in control. Plot a control chart first; if special causes are present, remove them

before computing capability.

Verified distribution. Standard capability formulas assume normality. Test it, and if the data

is non-normal, transform or use a percentile method.

Rational sampling plan. The sample must span the sources of variation the study is meant to

describe.

Sampling plan design

Study purpose Sampling approach Typical size

Short-term (machine) capability Consecutive pieces, single setup, single lot, single operator 30-50 consecutive units

Long-term (process) performance Subgroups spread across shifts, operators, lots, and setups 25+ subgroups of 4-5, over 2-4 weeks

  • Setup verification Small sample immediately after each setup 5 units per setup, several setups
  • The distinction is the whole point. A short-term study deliberately excludes between-setup and

between-lot variation to estimate the process's inherent potential. A long-term study deliberately includes them to estimate what the customer will actually receive.

  • Report with the study: the sample size, the period covered, the stratification included, the normality
  • result, the MSA result, and the control chart evidence. A capability number without that context cannot

be evaluated.

Natural process limits versus specification limits

This distinction is tested repeatedly and confused constantly.

  • Natural process limits Specification limits
  • Source The process itself The customer, the design, or the regulator
  • Formula x ˉ ± 3 σ ^ \bar{x} \pm 3\hat{\sigma} x ˉ ± 3 σ ^ Given; not calculated from data
  • Also called Voice of the process Voice of the customer
  • Appear on Control charts (as control limits for subgroup statistics) Histograms and capability plots
  • Change when The process changes The requirement changes
  • The two are completely independent . A process can be in perfect statistical control and produce
  • 100% defective output, if its natural limits sit entirely outside the specification. Conversely, a
  • process can be wildly out of control and still produce nothing outside specification if the tolerance

is wide.

Two errors follow from confusing them, and both are serious:

  • Putting specification limits on a control chart. Control limits describe what the process does;
  • specification limits describe what is required. Charting subgroup means against specification limits
  • is meaningless, because the distribution of means is narrower than the distribution of individuals by

a factor of n \sqrt{n} n ​ .

Adjusting the process whenever a unit is outside specification. This is tampering: reacting to

common-cause variation increases variation.

Yield and defect metrics

Definitions first. A unit is the item produced; a defect is any single non-conformance; a defective is a unit with one or more defects; an opportunity is a chance for a defect on a unit. D P U = Defects Units D P O = Defects Units × Opportunities per unit D P M O = D P O × 10 6 DPU = \frac{\text{Defects}}{\text{Units}} \qquad DPO = \frac{\text{Defects}}{\text{Units} \times \text{Opportunities per unit}} \qquad DPMO = DPO \times 10^6 D P U = Units Defects ​ D PO = Units × Opportunities per unit Defects ​ D PMO = D PO × 1 0 6 Percent defective = Defective units Total units × 100 P P M = Percent defective × 10 4 \text{Percent defective} = \frac{\text{Defective units}}{\text{Total units}} \times 100 \qquad PPM = \text{Percent defective} \times 10^4 Percent defective = Total units Defective units ​ × 100 PPM = Percent defective × 1 0 4

First pass yield and rolled throughput yield

First pass yield (FPY) for a step is the proportion of units passing that step with no rework. Rolled throughput yield (RTY) is the probability that a unit passes every step with no rework

  • anywhere: R T Y = ∏ i = 1 k F P Y i RTY = \prod_{i=1}^{k} FPY_i RT Y = i = 1 ∏ k ​ FP Y i ​
  • RTY is always lower than final yield, because final yield counts reworked units as good. The gap

between them is the size of the hidden factory.

Worked example, four steps:

  • Step Units in Passed first time FPY
  • 1 Data entry 1,000 950 0.950
  • 2 Underwriting 950 902 0.949
  • 3 Verification 902 875 0.970

4 Issuance 875 866 0.990 R T Y = 0.950 × 0.949 × 0.970 × 0.990 = 0.8658 RTY = 0.950 \times 0.949 \times 0.970 \times 0.990 = 0.8658 RT Y = 0.950 × 0.949 × 0.970 × 0.990 = 0.8658

So 86.6% of applications pass cleanly all the way through, even though final yield after rework may be

99%+. The normalized yield -- the average per-step yield -- is 0.8658 4 = 0.9646 \sqrt[4]{0.8658} = 0.9646 4 0.8658 ​ = 0.9646 , useful for comparing processes with different numbers of steps.

When defects follow a Poisson model, yield relates to DPU as: Y = e − D P U ⟺ D P U = − ln ⁡ ( Y ) Y = e^{-DPU} \qquad \Longleftrightarrow \qquad DPU = -\ln(Y) Y = e − D P U ⟺ D P U = − ln ( Y )

With R T Y = 0.8658 RTY = 0.8658 RT Y = 0.8658 , total D P U = − ln ⁡ ( 0.8658 ) = 0.1441 DPU = -\ln(0.8658) = 0.1441 D P U = − ln ( 0.8658 ) = 0.1441 defects per unit across the whole process.

  • Capability for attributes data
  • There is no Cp or Cpk for attributes data, because there is no continuous distribution and no

specification width. Capability is expressed as a process sigma level derived from the defect rate.

Procedure:

Compute DPMO (or the proportion defective, if working at the unit level).

Convert the corresponding proportion to a Z Z Z value from the standard normal table. This is Z L T Z_{LT} Z L T ​ , the long-term sigma level.

Add 1.5 to report the conventional short-term equivalent: Z S T = Z L T + 1.5 Z_{ST} = Z_{LT} + 1.5 Z ST ​ = Z L T ​ + 1.5 .

Worked example. A process produces 6,210 defects per million opportunities.

D P O = 0.006210 DPO = 0.006210 D PO = 0.006210 , so the yield is 0.993790.

The Z Z Z value with 0.006210 in the upper tail is Z L T = 2.50 Z_{LT} = 2.50 Z L T ​ = 2.50 .

Z S T = 2.50 + 1.5 = 4.0 Z_{ST} = 2.50 + 1.5 = 4.0 Z ST ​ = 2.50 + 1.5 = 4.0 , so the process is reported as a "4 sigma" process.

  • DPMO Long-term Z Reported sigma level Yield
  • 308,538 0.5 2.0 69.15%
  • 66,807 1.5 3.0 93.32%
  • 6,210 2.5 4.0 99.379%
  • 233 3.5 5.0 99.9767%
  • 3.4 4.5 6.0 99.99966%
  • The step from 3 to 4 sigma removes roughly 90% of the defects; the step from 5 to 6 sigma removes
  • another 98.5% of what remains. This is why capability improvement gets progressively harder and why

the business case for the last increment must be made on consequence, not on defect count.

Short-term and long-term capability

Short term Long term

Variation used Within-subgroup, σ ^ w i t h i n = R ˉ / d 2 \hat{\sigma}_{within} = \bar{R}/d_2 σ ^ w i t hin ​ = R ˉ / d 2 ​ or s ˉ / c 4 \bar{s}/c_4 s ˉ / c 4 ​ Total, s s s computed over all individuals

  • Indices C p C_p C p ​ , C p k C_{pk} C p k ​ P p P_p P p ​ , P p k P_{pk} P p k ​
  • Captures Inherent process potential Everything, including drift between subgroups

Data span Short window, homogeneous conditions Weeks or months, all conditions Z s h i f t = Z S T − Z L T ≈ 1.5 Z_{shift} = Z_{ST} - Z_{LT} \approx 1.5 Z s hi f t ​ = Z ST ​ − Z L T ​ ≈ 1.5

  • The 1.5 sigma shift is an empirical convention originating in Motorola's observation that process
  • means drift over time. It is a default, not a law: where enough long-term data exists, compute the

actual shift as Z S T − Z L T Z_{ST} - Z_{LT} Z ST ​ − Z L T ​ rather than assuming 1.5.

Interpreting the gap between the two index sets is diagnostic:

Observation Interpretation Action

C p k ≈ P p k C_{pk} \approx P_{pk} C p k ​ ≈ P p k ​ Process is stable; no meaningful drift between subgroups Reduce inherent variation to improve

C p k ≫ P p k C_{pk} \gg P_{pk} C p k ​ ≫ P p k ​ Good inherent capability, poor control over time Attack drift: setup, tool wear, material lots, shift differences

  • Both low Inherent variation is too large for the tolerance Fundamental process or design change; consider DFSS
  • When only short-term data is available, the convention is to subtract 1.5 from the short-term Z Z Z to
  • estimate long-term performance. When only long-term data is available, add 1.5 to state the short-term
  • equivalent. Say which convention you used; reporting a "6 sigma process" without stating whether it is

short- or long-term is ambiguous by a factor of roughly 20,000 in defect rate.