Study guide
6.14 Process Capability and Performance Indices
ASQ Certified Six Sigma Black Belt CSSBB
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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
6.14 Process Capability and Performance Indices
Key Takeaways Explain This Topic AI
Process Capability (Cp, Cpk) evaluates short-term potential assuming statistical control using within-subgroup variation, while Process Performance (Pp, Ppk) evaluates overall historical performance using long-term total standard deviation.
Cp measures potential capability regardless of centering, whereas Cpk accounts for process centering relative to specification limits.
Minimum capability benchmarks require Cpk >= 1.33 for existing processes and Cpk >= 1.67 for critical or new processes.
Non-normal data must be transformed using Box-Cox (y^(lambda)) or Johnson transformation families before standard normal capability formulas are applied.
Last updated: August 2026
Evaluating baseline process capability is a central objective of the Measure phase. Process Capability compares the natural voice of the process (process variation, 6 σ 6\sigma 6 σ ) against the voice of the customer (specification limits, USL − LSL \text{USL} - \text{LSL} USL − LSL ). Before conducting capability calculations, a Black Belt must verify two strict prerequisites:
Statistical Stability : The process must be in a state of statistical control, free from special cause variation, as verified by control charts.
Normality : Data must follow a standard Normal distribution (or be transformed to normality).
Capability ( C p , C p k C_p, C_{pk} C p , C p k ) vs. Performance ( P p , P p k P_p, P_{pk} P p , P p k ) Indices
Six Sigma maintains a strict distinction between short-term potential capability and long-term actual performance:
1. Short-Term Process Capability ( C p , C p k C_p, C_{pk} C p , C p k )
Short-term capability represents the inherent technical capability of a stable process operating under controlled conditions over a short timeframe. It utilizes within-subgroup standard deviation ( σ within \sigma_{\text{within}} σ within or σ st \sigma_{\text{st}} σ st ), estimated from control chart ranges or standard deviations: σ ^ st = R ˉ d 2 or σ ^ st = S ˉ c 4 \hat{\sigma}_{\text{st}} = \frac{\bar{R}}{d_2} \quad \text{or} \quad \hat{\sigma}_{\text{st}} = \frac{\bar{S}}{c_4} σ ^ st = d 2 R ˉ or σ ^ st = c 4 S ˉ
2. Long-Term Process Performance ( P p , P p k P_p, P_{pk} P p , P p k )
Long-term performance represents actual historical performance over extended time horizons, incorporating environmental shifts, raw material lot changes, and tool wear. It utilizes total sample standard deviation ( s total s_{\text{total}} s total or s lt s_{\text{lt}} s lt ), calculated across all individual observations: s lt = ∑ i = 1 n ( x i − x ˉ ) 2 n − 1 s_{\text{lt}} = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}} s lt = n − 1 ∑ i = 1 n ( x i − x ˉ ) 2
- Mathematical Formulations of Capability Indices
- Potential Capability Index ( C p C_p C p )
C p C_p C p measures the maximum potential capability of the process assuming the process mean μ \mu μ is perfectly centered between specification limits. It ignores process centering: C p = USL − LSL 6 σ st C_p = \frac{\text{USL} - \text{LSL}}{6\sigma_{\text{st}}} C p = 6 σ st USL − LSL
Centered Capability Index ( C p k C_{pk} C p k )
C p k C_{pk} C p k accounts for actual process centering by evaluating capability relative to both Upper Specification Limit (USL) and Lower Specification Limit (LSL): C p u = USL − μ 3 σ st , C p l = μ − LSL 3 σ st C_{pu} = \frac{\text{USL} - \mu}{3\sigma_{\text{st}}}, \quad C_{pl} = \frac{\mu - \text{LSL}}{3\sigma_{\text{st}}} C p u = 3 σ st USL − μ , C pl = 3 σ st μ − LSL C p k = min ( C p u , C p l ) = min ( USL − μ 3 σ st , μ − LSL 3 σ st ) C_{pk} = \min \left( C_{pu}, C_{pl} \right) = \min \left( \frac{\text{USL} - \mu}{3\sigma_{\text{st}}}, \frac{\mu - \text{LSL}}{3\sigma_{\text{st}}} \right) C p k = min ( C p u , C pl ) = min ( 3 σ st USL − μ , 3 σ st μ − LSL )
Key Relationship : C p k ≤ C p C_{pk} \le C_p C p k ≤ C p always. C p k = C p C_{pk} = C_p C p k = C p only when the process mean is centered perfectly at the specification midpoint USL + LSL 2 \frac{\text{USL} + \text{LSL}}{2} 2 USL + LSL .
Taguchi Capability Index ( C p m C_{pm} C p m )
Developed by Genichi Taguchi, C p m C_{pm} C p m incorporates target centering loss by incorporating the squared deviation of the process mean from the target nominal value T T T : C p m = USL − LSL 6 σ 2 + ( μ − T ) 2 = C p 1 + ( μ − T σ ) 2 C_{pm} = \frac{\text{USL} - \text{LSL}}{6 \sqrt{\sigma^2 + (\mu - T)^2}} = \frac{C_p}{\sqrt{1 + \left( \frac{\mu - T}{\sigma} \right)^2}} C p m = 6 σ 2 + ( μ − T ) 2 USL − LSL = 1 + ( σ μ − T ) 2 C p
Performance Indices ( P p , P p k P_p, P_{pk} P p , P p k )
Calculated using identical formulas to C p C_p C p and C p k C_{pk} C p k , but substituting overall long-term standard deviation s lt s_{\text{lt}} s lt for σ st \sigma_{\text{st}} σ st : P p = USL − LSL 6 s lt , P p k = min ( USL − μ 3 s lt , μ − LSL 3 s lt ) P_p = \frac{\text{USL} - \text{LSL}}{6 s_{\text{lt}}}, \quad P_{pk} = \min \left( \frac{\text{USL} - \mu}{3 s_{\text{lt}}}, \frac{\mu - \text{LSL}}{3 s_{\text{lt}}} \right) P p = 6 s lt USL − LSL , P p k = min ( 3 s lt USL − μ , 3 s lt μ − LSL )
Six Sigma Capability Benchmarks & Worked Numerical Example
Capability Index ( C p k C_{pk} C p k ) Sigma Level ( Z st Z_{\text{st}} Z st ) Short-Term Defect Rate Long-Term DPMO (with 1.5 σ \sigma σ shift) Quality Classification
0.67 0.67 0.67 2.0 σ 2.0\sigma 2.0 σ 4.55 % 4.55\% 4.55% (45,500 PPM) 308,538 DPMO Unacceptable
1.00 1.00 1.00 3.0 σ 3.0\sigma 3.0 σ 0.27 % 0.27\% 0.27% (2,700 PPM) 66,807 DPMO Barely Capable (3-Sigma baseline)
- 1.33 1.33 1.33 4.0 σ 4.0\sigma 4.0 σ 62.1 62.1 62.1 PPM 6,210 DPMO Minimum Benchmark for existing processes
- 1.67 1.67 1.67 5.0 σ 5.0\sigma 5.0 σ 0.57 0.57 0.57 PPM 233 DPMO Required for critical/safety characteristics
- 2.00 2.00 2.00 6.0 σ 6.0\sigma 6.0 σ 0.002 0.002 0.002 PPM 3.4 DPMO World-Class Six Sigma Quality
- Worked Capability Calculation
A machining process produces shaft diameters with specifications LSL = 19.80 mm \text{LSL} = 19.80\text{ mm} LSL = 19.80 mm and USL = 20.20 mm \text{USL} = 20.20\text{ mm} USL = 20.20 mm . Subgroup sample data ( n = 5 n = 5 n = 5 , k = 20 k = 20 k = 20 subgroups) yields a sample mean x ˉ ˉ = 20.08 mm \bar{\bar{x}} = 20.08\text{ mm} x ˉ ˉ = 20.08 mm , an average range R ˉ = 0.092 mm \bar{R} = 0.092\text{ mm} R ˉ = 0.092 mm , and an overall sample standard deviation s lt = 0.045 mm s_{\text{lt}} = 0.045\text{ mm} s lt = 0.045 mm . Using d 2 = 2.326 d_2 = 2.326 d 2 = 2.326 for n = 5 n = 5 n = 5 :
σ ^ st = R ˉ d 2 = 0.092 2.326 = 0.03955 mm \hat{\sigma}_{\text{st}} = \frac{\bar{R}}{d_2} = \frac{0.092}{2.326} = 0.03955\text{ mm} σ ^ st = d 2 R ˉ = 2.326 0.092 = 0.03955 mm .
C p = 20.20 − 19.80 6 × 0.03955 = 0.40 0.2373 = 1.685 C_p = \frac{20.20 - 19.80}{6 \times 0.03955} = \frac{0.40}{0.2373} = 1.685 C p = 6 × 0.03955 20.20 − 19.80 = 0.2373 0.40 = 1.685 .
C p u = 20.20 − 20.08 3 × 0.03955 = 0.12 0.11865 = 1.011 C_{pu} = \frac{20.20 - 20.08}{3 \times 0.03955} = \frac{0.12}{0.11865} = 1.011 C p u = 3 × 0.03955 20.20 − 20.08 = 0.11865 0.12 = 1.011 .
C p l = 20.08 − 19.80 3 × 0.03955 = 0.28 0.11865 = 2.360 C_{pl} = \frac{20.08 - 19.80}{3 \times 0.03955} = \frac{0.28}{0.11865} = 2.360 C pl = 3 × 0.03955 20.08 − 19.80 = 0.11865 0.28 = 2.360 .
C p k = min ( 1.011 , 2.360 ) = 1.011 C_{pk} = \min(1.011, 2.360) = 1.011 C p k = min ( 1.011 , 2.360 ) = 1.011 .
Diagnosis : The potential capability is excellent ( C p = 1.685 > 1.33 C_p = 1.685 > 1.33 C p = 1.685 > 1.33 ), but the process is off-center toward the USL, reducing C p k C_{pk} C p k to 1.011 1.011 1.011 . Re-centering the process mean from 20.08 20.08 20.08 to 20.00 mm 20.00\text{ mm} 20.00 mm will immediately raise C p k C_{pk} C p k to 1.685 1.685 1.685 .
Z Z Z -Score Transformation & The 1.5 σ \sigma σ Shift
Process capability can be expressed on the Standard Normal Distribution Z Z Z -scale: Z USL = USL − μ σ , Z LSL = μ − LSL σ Z_{\text{USL}} = \frac{\text{USL} - \mu}{\sigma}, \quad Z_{\text{LSL}} = \frac{\mu - \text{LSL}}{\sigma} Z USL = σ USL − μ , Z LSL = σ μ − LSL Z min = min ( Z USL , Z LSL ) = 3 ⋅ C p k Z_{\text{min}} = \min(Z_{\text{USL}}, Z_{\text{LSL}}) = 3 \cdot C_{pk} Z min = min ( Z USL , Z LSL ) = 3 ⋅ C p k
The 1.5 σ \sigma σ Shift Assumption : Motorola empirical research established that over long-term operations, process means shift and drift by approximately 1.5 σ 1.5\sigma 1.5 σ . Thus: Z long-term = Z short-term − 1.5 Z_{\text{long-term}} = Z_{\text{short-term}} - 1.5 Z long-term = Z short-term − 1.5 Long-Term Sigma Level = 3 ⋅ C p k − 1.5 \text{Long-Term Sigma Level} = 3 \cdot C_{pk} - 1.5 Long-Term Sigma Level = 3 ⋅ C p k − 1.5
Handling Non-Normal Data: Data Transformations
When data violates normality (e.g., Anderson-Darling test p p p -value < 0.05 < 0.05 < 0.05 ), standard C p k C_{pk} C p k equations produce severe calculation errors. Black Belts apply two primary transformation methods to achieve normality:
1. Box-Cox Power Transformation
The Box-Cox Transformation transforms skewed non-normal data y y y into normal data y ( λ ) y^{(\lambda)} y ( λ ) using a power parameter λ \lambda λ : y ( λ ) = { y λ − 1 λ if λ ≠ 0 ln ( y ) if λ = 0 y^{(\lambda)} = \begin{cases} \frac{y^\lambda - 1}{\lambda} & \text{if } \lambda \neq 0 \\ \ln(y) & \text{if } \lambda = 0 \end{cases} y ( λ ) = { λ y λ − 1 ln ( y ) if λ = 0 if λ = 0
Restriction : Box-Cox can ONLY be applied to strictly positive data ( y > 0 y > 0 y > 0 ).
2. Johnson Transformation System
The Johnson Transformation fits non-normal data into one of three distribution families (Unbounded S U S_U S U , Bounded S B S_B S B , Log-Normal S L S_L S L ), mapping complex non-normal data (including negative values or bounded data) into a Standard Normal distribution.
3. Non-Parametric Percentile Capability Method
If transformations fail, capability is computed directly from empirical percentiles ( P 0.135 P_{0.135} P 0.135 , P 50 P_{50} P 50 , P 99.865 P_{99.865} P 99.865 ): C p k = min ( USL − P 50 P 99.865 − P 50 , P 50 − LSL P 50 − P 0.135 ) C_{pk} = \min \left( \frac{\text{USL} - P_{50}}{P_{99.865} - P_{50}}, \frac{P_{50} - \text{LSL}}{P_{50} - P_{0.135}} \right) C p k = min ( P 99.865 − P 50 USL − P 50 , P 50 − P 0.135 P 50 − LSL )