Study guide
9.3 Control Charts for Attributes Data
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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
9.3 Control Charts for Attributes Data
Key Takeaways Explain This Topic AI
The p-chart monitors the proportion (fraction) of nonconforming items in samples of equal or variable sample sizes, based on the binomial distribution with standard error SE(p) = sqrt(p-bar(1-p-bar)/n).
The np-chart tracks the absolute number of nonconforming units per sample and requires a constant sample size n, using center line np-bar and control limits np-bar ± 3*sqrt(np-bar(1-p-bar)).
The c-chart counts the total number of defects per sample unit for a constant area of opportunity based on the Poisson distribution with standard error SE(c) = sqrt(c-bar).
The u-chart monitors defects per unit (u = c/n) across variable sample sizes or varying areas of opportunity, with standard error SE(u) = sqrt(u-bar/ni) creating variable control limits.
Attribute chart selection depends on two binary criteria: nonconforming units (binomial: p, np) vs. count of defects (Poisson: c, u), and constant vs. variable sample size.
Last updated: August 2026
Attributes data consist of discrete qualitative classifications (e.g., pass/fail, go/no-go, conforming/nonconforming) or discrete count data (number of defects, customer complaints, surface scratches). When continuous quantitative variables cannot be economically measured, attributes control charts provide an effective methodology for statistical process control in the Measure and Control phases.
Attribute Data Classification: Defectives vs. Defects
Before selecting an attribute control chart, practitioners must distinguish between two fundamental categories of quality data:
Defective (Nonconforming Unit) : An entire unit or item that fails to meet one or more quality specifications and is classified as unacceptable (binary outcome: pass or fail).
Underlying Probability Model : Binomial Distribution .
Control Charts Used : p p p -chart (fraction defective) and n p np n p -chart (number defective).
Defects (Nonconformities) : Individual quality flaws or occurrences on a single unit that do not necessarily cause the entire unit to be rejected. A single unit may contain multiple defects.
Underlying Probability Model : Poisson Distribution .
Control Charts Used : c c c -chart (count of defects) and u u u -chart (defects per unit).
p p p -Chart (Fraction Nonconforming)
The p p p -chart monitors the proportion (fraction or percentage) of nonconforming items in a subgroup. It is the most versatile attribute chart because it can accommodate both constant and variable subgroup sample sizes ( n i n_i n i ).
Statistical Foundation & Limit Derivations
Under the Binomial distribution with mean parameter p p p :
Subgroup Fraction Nonconforming ( p i p_i p i ) : p i = d i n i p_i = \frac{d_i}{n_i} p i = n i d i (where d i d_i d i is the count of defective units in subgroup i i i ).
Overall Average Fraction Nonconforming ( p ˉ \bar{p} p ˉ ) : p ˉ = ∑ i = 1 k d i ∑ i = 1 k n i \bar{p} = \frac{\sum_{i=1}^{k} d_i}{\sum_{i=1}^{k} n_i} p ˉ = ∑ i = 1 k n i ∑ i = 1 k d i
Standard Error of Proportion : S E ( p i ) = p ˉ ( 1 − p ˉ ) n i SE(p_i) = \sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}} SE ( p i ) = n i p ˉ ( 1 − p ˉ )
Control Limits Formulas U C L p = p ˉ + 3 p ˉ ( 1 − p ˉ ) n i UCL_p = \bar{p} + 3 \sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}} U C L p = p ˉ + 3 n i p ˉ ( 1 − p ˉ ) L C L p = max ( 0 , p ˉ − 3 p ˉ ( 1 − p ˉ ) n i ) LCL_p = \max\left(0, \, \bar{p} - 3 \sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}}\right) L C L p = max 0 , p ˉ − 3 n i p ˉ ( 1 − p ˉ )
Note : If subgroup sample size n i n_i n i varies across days, control limits U C L p UCL_p U C L p and L C L p LCL_p L C L p will vary inversely with n i \sqrt{n_i} n i , resulting in "stepped" control limits. If L C L p LCL_p L C L p calculates to a negative number, it is set to 0.0 0.0 0.0 .
n p np n p -Chart (Number Nonconforming)
The n p np n p -chart tracks the total count of defective items directly rather than a calculated proportion. It is easier for shop-floor operators to interpret because it plots whole numbers ( 0 , 1 , 2 , 3 … 0, 1, 2, 3 \dots 0 , 1 , 2 , 3 … ).
Operational Constraint & Formulas
The n p np n p -chart strictly requires a constant sample size ( n n n ) across all subgroups.
Center Line (CL) : C L n p = n p ˉ = ∑ i = 1 k d i k CL_{np} = n\bar{p} = \frac{\sum_{i=1}^{k} d_i}{k} C L n p = n p ˉ = k ∑ i = 1 k d i
Control Limits : U C L n p = n p ˉ + 3 n p ˉ ( 1 − p ˉ ) UCL_{np} = n\bar{p} + 3\sqrt{n\bar{p}(1-\bar{p})} U C L n p = n p ˉ + 3 n p ˉ ( 1 − p ˉ ) L C L n p = max ( 0 , n p ˉ − 3 n p ˉ ( 1 − p ˉ ) ) LCL_{np} = \max\left(0, \, n\bar{p} - 3\sqrt{n\bar{p}(1-\bar{p})}\right) L C L n p = max ( 0 , n p ˉ − 3 n p ˉ ( 1 − p ˉ ) )
If subgroup sample sizes vary, an n p np n p -chart cannot be used, and the practitioner must default to a p p p -chart.
c c c -Chart (Defects per Unit - Constant Sample Size)
The c c c -chart monitors the total count of individual defects ( c c c ) found across a sample unit with a strictly constant area of opportunity (e.g., defects per 100 m 2 100 \text{ m}^2 100 m 2 of glass sheet, errors per 50 50 50 -page audit ledger, solder joint defects per circuit board).
Statistical Foundation & Limit Derivations
Defect counts follow the Poisson distribution, where the variance equals the mean ( σ 2 = μ = c ˉ \sigma^2 = \mu = \bar{c} σ 2 = μ = c ˉ ).
Average Defect Count ( c ˉ \bar{c} c ˉ ) : c ˉ = ∑ i = 1 k c i k \bar{c} = \frac{\sum_{i=1}^{k} c_i}{k} c ˉ = k ∑ i = 1 k c i
Standard Error : S E ( c ) = c ˉ SE(c) = \sqrt{\bar{c}} SE ( c ) = c ˉ
Control Limits : U C L c = c ˉ + 3 c ˉ UCL_c = \bar{c} + 3\sqrt{\bar{c}} U C L c = c ˉ + 3 c ˉ L C L c = max ( 0 , c ˉ − 3 c ˉ ) LCL_c = \max\left(0, \, \bar{c} - 3\sqrt{\bar{c}}\right) L C L c = max ( 0 , c ˉ − 3 c ˉ )
u u u -Chart (Defects per Unit - Variable Sample Size)
When the area of opportunity or subgroup sample size ( n i n_i n i ) varies from sample to sample, the u u u -chart monitors the rate of defects per unit ( u i = c i / n i u_i = c_i / n_i u i = c i / n i ).
- Statistical Foundation & Limit Derivations
- Subgroup Defect Rate ( u i u_i u i ) : u i = c i n i u_i = \frac{c_i}{n_i} u i = n i c i
Overall Average Defect Rate ( u ˉ \bar{u} u ˉ ) : u ˉ = ∑ i = 1 k c i ∑ i = 1 k n i \bar{u} = \frac{\sum_{i=1}^{k} c_i}{\sum_{i=1}^{k} n_i} u ˉ = ∑ i = 1 k n i ∑ i = 1 k c i
Standard Error : S E ( u i ) = u ˉ n i SE(u_i) = \sqrt{\frac{\bar{u}}{n_i}} SE ( u i ) = n i u ˉ
Control Limits : U C L u = u ˉ + 3 u ˉ n i UCL_u = \bar{u} + 3 \sqrt{\frac{\bar{u}}{n_i}} U C L u = u ˉ + 3 n i u ˉ L C L u = max ( 0 , u ˉ − 3 u ˉ n i ) LCL_u = \max\left(0, \, \bar{u} - 3 \sqrt{\frac{\bar{u}}{n_i}}\right) L C L u = max ( 0 , u ˉ − 3 n i u ˉ )
Attributes Control Chart Selection Matrix
Chart Type Quality Metric Tracked Data Classification Underlying Distribution Subgroup Size ( n n n ) Center Line Control Limits ( U C L / L C L UCL / LCL U C L / L C L )
p p p -Chart Fraction / % Defective Defective Unit (Go/No-Go) Binomial Constant or Variable p ˉ \bar{p} p ˉ p ˉ ± 3 p ˉ ( 1 − p ˉ ) n i \bar{p} \pm 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}} p ˉ ± 3 n i p ˉ ( 1 − p ˉ )
n p np n p -Chart Count of Defectives Defective Unit (Go/No-Go) Binomial Strictly Constant n p ˉ n\bar{p} n p ˉ n p ˉ ± 3 n p ˉ ( 1 − p ˉ ) n\bar{p} \pm 3\sqrt{n\bar{p}(1-\bar{p})} n p ˉ ± 3 n p ˉ ( 1 − p ˉ )
c c c -Chart Count of Defects Individual Defect Flaws Poisson Strictly Constant Area c ˉ \bar{c} c ˉ c ˉ ± 3 c ˉ \bar{c} \pm 3\sqrt{\bar{c}} c ˉ ± 3 c ˉ
u u u -Chart Defects per Unit Rate Individual Defect Flaws Poisson Constant or Variable u ˉ \bar{u} u ˉ u ˉ ± 3 u ˉ n i \bar{u} \pm 3\sqrt{\frac{\bar{u}}{n_i}} u ˉ ± 3 n i u ˉ
Worked Numerical Calculation Example: p p p -Chart
A call center audits k = 20 k = 20 k = 20 daily subgroups of n = 100 n = 100 n = 100 customer service calls each. Over the 20 days, a total of ∑ d i = 120 \sum d_i = 120 ∑ d i = 120 defective calls (compliance errors) are identified.
Overall Fraction Nonconforming: p ˉ = 120 20 × 100 = 120 2000 = 0.060 \bar{p} = \frac{120}{20 \times 100} = \frac{120}{2000} = 0.060 p ˉ = 20 × 100 120 = 2000 120 = 0.060 ( 6.0 % 6.0\% 6.0% ).
Standard Error: S E ( p ) = 0.060 ( 1 − 0.060 ) 100 = 0.060 × 0.940 100 = 0.000564 = 0.02375 SE(p) = \sqrt{\frac{0.060(1 - 0.060)}{100}} = \sqrt{\frac{0.060 \times 0.940}{100}} = \sqrt{0.000564} = 0.02375 SE ( p ) = 100 0.060 ( 1 − 0.060 ) = 100 0.060 × 0.940 = 0.000564 = 0.02375 .
Upper Control Limit: U C L p = 0.060 + 3 ( 0.02375 ) = 0.060 + 0.07125 = 0.13125 UCL_p = 0.060 + 3(0.02375) = 0.060 + 0.07125 = 0.13125 U C L p = 0.060 + 3 ( 0.02375 ) = 0.060 + 0.07125 = 0.13125 ( 13.13 % 13.13\% 13.13% ).
Lower Control Limit: L C L p = max ( 0 , 0.060 − 0.07125 ) = 0.0 LCL_p = \max(0, 0.060 - 0.07125) = 0.0 L C L p = max ( 0 , 0.060 − 0.07125 ) = 0.0 .
Interpretation : The process is in control as long as daily compliance error rates remain between 0.0 % 0.0\% 0.0% and 13.13 % 13.13\% 13.13% . Any day exceeding 13.13 % 13.13\% 13.13% represents a special cause out-of-control signal requiring root cause investigation.
Detailed Mathematical Comparisons of Attribute Charts
Attribute Chart Characteristic Measured Subgroup Size ( n n n ) Center Line Formula Standard Error Formula ( S E SE SE ) Upper Control Limit ( U C L UCL U C L ) Lower Control Limit ( L C L LCL L C L )
p p p -Chart Fraction Defective ( p i p_i p i ) Variable or Constant p ˉ = ∑ d i ∑ n i \bar{p} = \frac{\sum d_i}{\sum n_i} p ˉ = ∑ n i ∑ d i p ˉ ( 1 − p ˉ ) n i \sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}} n i p ˉ ( 1 − p ˉ ) p ˉ + 3 p ˉ ( 1 − p ˉ ) n i \bar{p} + 3 \sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}} p ˉ + 3 n i p ˉ ( 1 − p ˉ ) max ( 0 , p ˉ − 3 p ˉ ( 1 − p ˉ ) n i ) \max\left(0, \bar{p} - 3 \sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}}\right) max ( 0 , p ˉ − 3 n i p ˉ ( 1 − p ˉ ) )
n p np n p -Chart Count of Defectives ( d i d_i d i ) Strictly Constant C L = n p ˉ CL = n\bar{p} C L = n p ˉ n p ˉ ( 1 − p ˉ ) \sqrt{n\bar{p}(1-\bar{p})} n p ˉ ( 1 − p ˉ ) n p ˉ + 3 n p ˉ ( 1 − p ˉ ) n\bar{p} + 3\sqrt{n\bar{p}(1-\bar{p})} n p ˉ + 3 n p ˉ ( 1 − p ˉ ) max ( 0 , n p ˉ − 3 n p ˉ ( 1 − p ˉ ) ) \max\left(0, n\bar{p} - 3\sqrt{n\bar{p}(1-\bar{p})}\right) max ( 0 , n p ˉ − 3 n p ˉ ( 1 − p ˉ ) )
c c c -Chart Count of Defects ( c i c_i c i ) Strictly Constant Area C L = c ˉ CL = \bar{c} C L = c ˉ c ˉ \sqrt{\bar{c}} c ˉ c ˉ + 3 c ˉ \bar{c} + 3\sqrt{\bar{c}} c ˉ + 3 c ˉ max ( 0 , c ˉ − 3 c ˉ ) \max\left(0, \bar{c} - 3\sqrt{\bar{c}}\right) max ( 0 , c ˉ − 3 c ˉ )
u u u -Chart Defects per Unit ( u i u_i u i ) Variable or Constant u ˉ = ∑ c i ∑ n i \bar{u} = \frac{\sum c_i}{\sum n_i} u ˉ = ∑ n i ∑ c i u ˉ n i \sqrt{\frac{\bar{u}}{n_i}} n i u ˉ u ˉ + 3 u ˉ n i \bar{u} + 3 \sqrt{\frac{\bar{u}}{n_i}} u ˉ + 3 n i u ˉ max ( 0 , u ˉ − 3 u ˉ n i ) \max\left(0, \bar{u} - 3 \sqrt{\frac{\bar{u}}{n_i}}\right) max ( 0 , u ˉ − 3 n i u ˉ )
Worked Attribute Chart Calculation: u u u -Chart
A quality team inspects printed circuit boards for solder defects across variable daily production lots.
Day 1: n 1 = 50 n_1 = 50 n 1 = 50 boards, c 1 = 15 c_1 = 15 c 1 = 15 defects ⟹ u 1 = 15 / 50 = 0.30 \implies u_1 = 15/50 = 0.30 ⟹ u 1 = 15/50 = 0.30
Day 2: n 2 = 100 n_2 = 100 n 2 = 100 boards, c 2 = 22 c_2 = 22 c 2 = 22 defects ⟹ u 2 = 22 / 100 = 0.22 \implies u_2 = 22/100 = 0.22 ⟹ u 2 = 22/100 = 0.22
Day 3: n 3 = 80 n_3 = 80 n 3 = 80 boards, c 3 = 18 c_3 = 18 c 3 = 18 defects ⟹ u 3 = 18 / 80 = 0.225 \implies u_3 = 18/80 = 0.225 ⟹ u 3 = 18/80 = 0.225
Overall Average Defect Rate ( u ˉ \bar{u} u ˉ ) : u ˉ = ∑ c i ∑ n i = 15 + 22 + 18 50 + 100 + 80 = 55 230 = 0.2391 defects/board \bar{u} = \frac{\sum c_i}{\sum n_i} = \frac{15 + 22 + 18}{50 + 100 + 80} = \frac{55}{230} = 0.2391\text{ defects/board} u ˉ = ∑ n i ∑ c i = 50 + 100 + 80 15 + 22 + 18 = 230 55 = 0.2391 defects/board
Calculate Control Limits for Day 1 ( n 1 = 50 n_1 = 50 n 1 = 50 ) : S E ( u 1 ) = 0.2391 50 = 0.004782 = 0.06915 SE(u_1) = \sqrt{\frac{0.2391}{50}} = \sqrt{0.004782} = 0.06915 SE ( u 1 ) = 50 0.2391 = 0.004782 = 0.06915 U C L u 1 = 0.2391 + 3 ( 0.06915 ) = 0.2391 + 0.2074 = 0.4465 UCL_{u1} = 0.2391 + 3(0.06915) = 0.2391 + 0.2074 = 0.4465 U C L u 1 = 0.2391 + 3 ( 0.06915 ) = 0.2391 + 0.2074 = 0.4465 L C L u 1 = max ( 0 , 0.2391 − 0.2074 ) = 0.0317 LCL_{u1} = \max(0, 0.2391 - 0.2074) = 0.0317 L C L u 1 = max ( 0 , 0.2391 − 0.2074 ) = 0.0317