Control Charts for Attributes and Events

Introduction and Context

Statistical process control (SPC) is concerned with distinguishing routine process variation from variation that signals a change in the process. Variable control charts use measured quantities such as length, weight, or cycle time. Attribute and event charts are used when observations are classified or counted instead. They are especially valuable when measurements are expensive, when conformity is judged against specifications, or when the phenomenon of interest is inherently a count: defective units, defects, complaints, accidents, failures, or transaction errors.

The principal attribute and event charts are the p chart, np chart, c chart, and u chart. The p and np charts address defective units; the c and u charts address defects or events. This distinction is fundamental. A unit is defective if it fails to conform in at least one respect. A single unit may contain several defects, so the number of defects can exceed the number of defective units.

These charts are based primarily on binomial and Poisson probability models. Their usefulness depends not only on correct calculation but also on appropriate operational definitions, rational subgrouping, stable inspection procedures, and disciplined interpretation. A control chart is not a replacement for process knowledge or causal investigation.

Learning Objectives

By the end of this chapter, the reader should be able to:

• distinguish defective units from defects and events;
• select p, np, c, and u charts appropriately;
• calculate center lines and control limits;
• handle unequal sample sizes and varying areas of opportunity;
• recognize overdispersion and select suitable responses;
• interpret signals without confusing control with capability;
• implement attribute charts in manufacturing and service processes;
• evaluate limitations, governance concerns, and emerging digital applications.

1. The Conceptual Basis of Attribute Charts

1.1 Defectives, defects, and events

A defective unit is a unit that does not meet one or more acceptance criteria. If 7 of 200 invoices contain at least one error, the fraction defective is 7/200. An invoice with three errors is still one defective invoice for a defective-unit chart.

A defect is an individual nonconformity. If the same 200 invoices contain 12 total errors, the defect count is 12. Defect counts require a defined and reasonably constant opportunity structure: the units should be comparable, or the amount inspected should be measured.

An event is an occurrence counted over a defined exposure, such as system outages per month, safety incidents per 10,000 work hours, or customer complaints per 1,000 orders. Events may be desirable or undesirable; the chart concerns their statistical behavior, not their moral status.

1.2 Rational subgrouping

A subgroup should represent output produced under approximately common conditions, while successive subgroups should permit changes to become visible. A subgroup might be a shift's production, a day's transactions, or a defined number of opportunities. Combining observations from unrelated products, machines, or customer segments can create artificial variation and obscure assignable causes.

1.3 Control versus specification

Control limits estimate the range of common-cause variation expected from the process. Specification limits express customer, engineering, legal, or contractual requirements. A stable process may be incapable of meeting specifications, and an unstable process may temporarily meet them. Attribute control charts therefore answer “Is the process behavior stable?” rather than “Does every unit meet the requirement?”

2. Probability Models and General Control-Limit Logic

For a statistic plotted at subgroup i, the usual limits are:

Here W_i is the plotted statistic, E(W_i) is its expected value, and Var(W_i) is its variance under the assumed model. The lower limit is conventionally truncated at zero for count and proportion charts because negative counts or proportions have no practical meaning.

The familiar three-sigma limits are a practical convention, not a guarantee of a particular false-alarm probability. Exact probabilities depend on the distribution, parameter estimation, discreteness, and any autocorrelation or model misspecification.

3. The p Chart: Fraction or Proportion Defective

3.1 Purpose and model

A p chart monitors the fraction defective in each subgroup. Let n_i be the number of units inspected in subgroup i and d_i the number classified as defective. The observed fraction is:

If each unit has a binary outcome, units are independent, and the probability of defectiveness is constant at p, then:

Consequently:

When the process is stable, p is estimated from m preliminary subgroups by the pooled fraction:

The use of total defectives divided by total inspected units is important when subgroup sizes differ. The p-chart limits are:

If a limit is below 0, it is set to 0; if it exceeds 1, it is set to 1. The varying width of the limits is essential when n_i varies.

3.2 Worked example: unequal sample sizes

A service centre records defective applications over five days:

The total is 32 defectives among 650 applications, so p̄ = 32/650 = 0.04923. For day 3, with n_3 = 80:

Thus UCL_3 ≈ 0.04923 + 3(0.02419) = 0.1218 and LCL_3 ≈ 0.04923 − 3(0.02419) = −0.0233, which is truncated to 0. Day 3 is not beyond the limits. Its unusually low fraction should not automatically be treated as a process improvement; special-cause investigation may be appropriate only if additional evidence exists.

For day 5, the standard error is approximately 0.01529, giving limits of approximately 0.0951 and 0.0034. The same underlying process therefore produces narrower limits for the larger sample.

3.3 Interpretation

A point outside a p-chart limit indicates an unlikely result under the fitted binomial model, not proof of a particular cause. Other signals include a sustained run on one side of the center line, a trend, or an unusual pattern of alternation. Rules should be selected in advance and used cautiously; applying many rules increases false signals.

4. The np Chart: Number Defective

The np chart plots the number of defective units, d_i, rather than the fraction defective. It is appropriate when every subgroup has the same sample size n. Under the binomial model:

Using an estimated p̄:

Limits are rounded or otherwise operationalized carefully because the plotted statistic is an integer. The np chart is easy for operators to read: it answers how many defective units occurred in a fixed sample. It should not be used with varying n; a larger sample naturally tends to contain more defectives even when the fraction defective is unchanged.

If n = 100 and p̄ = 0.04, the center line is 4, the standard deviation is √[100(0.04)(0.96)] ≈ 1.96, and the three-sigma limits are approximately −1.88 and 9.88. The lower limit becomes 0; in practice the upper limit is interpreted using the discrete count scale.

5. The c Chart: Count of Defects per Constant Unit

5.1 Model and formula

A c chart monitors the number of defects in a constant inspection unit, such as defects on one standard-length sheet, errors in one fixed-size report, or complaints received during equal time intervals. If defects arise independently, with a constant average rate, and multiple defects can occur in one unit, the Poisson model is appropriate:

For a Poisson random variable, mean and variance are both c. Estimate the average count from m baseline subgroups:

The limits are:

The LCL is set to zero if negative. The c chart requires equal opportunity or equal inspection unit. If one week covers twice as many transactions as another, a c chart on raw counts can mislead.

5.2 Defective units versus defects

Suppose ten software builds contain 0, 1, 2, 0, 3, 1, 4, 2, 1, and 2 coding defects. The total is 16 and c̄ = 1.6. The approximate limits are UCL = 1.6 + 3√1.6 ≈ 5.40 and LCL = 0. No build exceeds the upper limit. A p chart would require the number of builds containing at least one defect, which is 8/10; it would answer a different question.

6. The u Chart: Defects per Unit with Varying Opportunity

6.1 Purpose and equations

A u chart is used when the number of units, area, time, or other exposure varies. Let c_i be the number of defects and n_i the number of equivalent inspection units in subgroup i. The plotted rate is:

Under a Poisson model with a constant defect rate per unit, the pooled estimate is:

Because Var(c_i) = n_iū, the variance of u_i is ū/n_i. Therefore:

Again, LCL_i is truncated at zero. Units must be comparable. If one subgroup contains 500 pages and another contains 500 complex engineering drawings, “per page” may not represent equal opportunity; a more meaningful denominator may be drawing-hours, geometric features, or an exposure measure.

6.2 Worked example

A claims-processing team records errors across three weekly batches:

The pooled rate is ū = 33/1500 = 0.022. For week 2:

Thus UCL_2 ≈ 0.04017 and LCL_2 ≈ 0.00383. The observed rate 0.030 is within the limits. A constant-width c chart would incorrectly ignore the different claim volumes.

7. Selecting the Appropriate Chart

The first decision is whether the observation is a unit classification or a count of nonconformities. The second is whether sample size or exposure is constant. A chart should not be selected merely because its name is familiar.

8. Rare Events and Low Counts

When p is very small, normal approximations can be poor and three-sigma limits may produce a lower limit of zero and a very high upper limit. For rare events, alternatives include exact binomial limits, exact Poisson limits, a g chart for time or units between events, and a t chart for the time between rare events. These approaches can be more sensitive because they model the waiting distance directly rather than aggregating many zero counts.

Exact limits should be derived from the chosen tail probability and the appropriate discrete distribution. They are not automatically superior: estimated parameters, changing exposure, dependence, and delayed reporting still affect validity. Zero counts do not prove zero risk; they may reflect inadequate opportunity or detection.

9. Overdispersion and Model Adequacy

9.1 Meaning of overdispersion

Overdispersion occurs when observed variation exceeds that predicted by the binomial or Poisson model. For a Poisson process, the defining equality is E(C) = Var(C). If the variance is materially larger than the mean, events may be clustered, exposures may differ in unmeasured ways, or several processes may have been mixed. For binomial data, overdispersion may reflect heterogeneous defect probabilities, dependence, or inconsistent inspection.

Simply multiplying limits by an estimated dispersion factor can conceal important causes. First examine stratification, rational subgrouping, measurement consistency, autocorrelation, and changes in opportunity. A mixture of stable processes often creates apparent overdispersion.

9.2 Practical responses

Useful responses include:

1. Stratify by product, machine, shift, supplier, or event type when the strata have distinct mechanisms.
2. Improve the denominator so that exposure genuinely represents opportunity.
3. Use a quasi-binomial or quasi-Poisson model when the mean structure is appropriate but variance is inflated.
4. Use negative-binomial or random-effects models when heterogeneity is substantive and count data are analyzed inferentially.
5. Use Laney p′ or u′ charts when appropriate software and validated methodology are available; these adjust for extra variation in a sequential SPC context.
6. Investigate dependence when events cluster in time or arise from common incidents.

A dispersion estimate should be calculated from adequate, stable baseline data and documented. Overdispersion is a diagnostic clue, not merely a nuisance parameter.

10. Baseline Construction and Operational Use

A sound implementation follows a staged process:

• define the unit, defect, event, and opportunity;
• establish consistent inspection and classification rules;
• select rational subgroups and collect representative baseline data;
• verify data integrity and estimate chart parameters;
• investigate assignable causes before finalizing limits;
• freeze or periodically review limits according to a documented policy;
• monitor prospectively and record responses to signals;
• confirm improvement with new data rather than deleting inconvenient points.

When an assignable cause is verified, the analyst may exclude the affected observation from limits if the cause is documented and the process has genuinely changed. Removing points solely because they look unusual is retrospective bias and weakens the chart.

Attribute data are vulnerable to inspection error. An apparent reduction in defectives may result from fatigue, changed definitions, sampling bias, or pressure not to report. Audits, inter-rater agreement studies, blinded checks, and traceable electronic records support trustworthy charts.

11. Case Application: Hospital Medication-Dispensing Errors

The following is an illustrative case, not a report of a real hospital.

Background

A hospital pharmacy processes prescriptions in daily batches. Management wants to reduce dispensing errors without encouraging under-reporting. Errors may include wrong drug, wrong strength, wrong label, or omitted counseling documentation. A prescription with several errors is counted once for the defective-prescription measure, while every error is counted for the defect measure.

Problem and analysis

Daily volume varies from 700 to 1,100 prescriptions. Therefore an np chart is inappropriate, and a p chart is more suitable for the proportion of prescriptions with at least one error. The pharmacy also wants to monitor total errors per 1,000 prescriptions, for which a u chart is appropriate, with n_i measured in thousands of prescriptions or, equivalently, with a consistent scaling convention.

During the baseline period, 20,000 prescriptions include 180 defective prescriptions, giving p̄ = 0.009. There are 260 total errors, giving a defect rate of 260/20,000 = 0.013 errors per prescription. The quality team checks whether error definitions, reporting channels, and staffing patterns changed across the baseline. One day with a barcode scanner outage is documented as a special cause and is not used to represent routine performance after management confirms that the process was restored.

Approach and solution

The p chart uses daily n_i values and limits based on p̄. The u chart uses total errors divided by daily prescription volume, with limits based on the corresponding exposure. The two charts are reviewed together: a fall in defective prescriptions with a rise in errors per defective prescription may indicate that the process is preventing some events but allowing severity to increase.

A subsequent pilot introduces barcode verification and a standardized interruption policy. The team does not immediately recalculate limits. It first observes whether the post-intervention process is stable, documents the intervention date, and then establishes revised limits if the new process is sustained.

Outcome and learning points

The paired charts distinguish three issues: frequency of affected prescriptions, total error burden, and changing opportunity. The case illustrates why a single chart cannot represent every quality question. It also demonstrates that data governance—consistent definitions, protected reporting, and traceability—is part of statistical control.

12. Technology, Analytics, and Contemporary Practice

Electronic health records, manufacturing execution systems, call-centre platforms, and cloud monitoring tools can calculate attribute charts in near real time. Automation improves timeliness but does not solve conceptual problems. An algorithm may count duplicate events, use an unstable denominator, or silently change a classification rule.

Modern systems may combine SPC with anomaly detection, change-point methods, Bayesian updating, or machine-learning models. Such tools can prioritize investigation, but they should not replace transparent control-chart logic where accountability matters. AI-generated alerts require validation, monitoring for drift, access controls, audit trails, and human review. In regulated settings, the organization should preserve the model version, data definition, parameter history, and decision record.

13. Limitations and Professional Judgment

Attribute charts discard information. Classifying a measurement as pass or fail may hide how close it was to the specification. Defect counts may depend heavily on inspection intensity. Poisson assumptions are often only approximations, and binomial independence can fail when one incident produces multiple related defects. Serial dependence, seasonality, changing product mix, and delayed reporting can create misleading signals.

For these reasons, analysts should combine charts with process maps, Pareto analysis, measurement-system evaluation, stratification, capability analysis where suitable, and subject-matter knowledge. The ethical requirement is especially important: charts should support learning and prevention, not punishment for individual workers or concealment of unfavorable performance.

Chapter Summary

The p and np charts monitor defective units under a binomial framework. The p chart plots a proportion and accommodates unequal sample sizes through subgroup-specific limits; the np chart plots the number defective and requires a constant sample size. The c and u charts monitor counts of defects or events under a Poisson framework. The c chart requires a constant inspection unit, whereas the u chart expresses defects per unit and accommodates varying exposure.

Correct chart selection depends on the question, the unit of observation, and the denominator. Three-sigma limits are model-based operational limits, not specification limits. Rare events may require exact or waiting-time methods. Overdispersion should prompt investigation of heterogeneity, dependence, denominator quality, and inspection systems before statistical adjustment. Effective implementation requires rational subgrouping, documented definitions, stable data collection, disciplined response to signals, and governance of automated analytics.

Key Terms

Defective unit: A unit failing one or more specified conformity requirements.

Defect: An individual nonconformity; one unit may contain multiple defects.

Event: An occurrence counted over a defined exposure or opportunity.

Attribute data: Data based on classifications or counts rather than continuous measurements.

p chart: A chart for the fraction or proportion of defective units.

np chart: A chart for the number of defective units in a fixed sample.

c chart: A chart for the number of defects in a constant inspection unit.

u chart: A chart for defects per unit when exposure varies.

Binomial distribution: A model for the number of defective outcomes among independent fixed trials.

Poisson distribution: A model for counts of events occurring at a stable average rate.

Rational subgroup: A subgroup formed under sufficiently common conditions to make variation interpretable.

Center line: The estimated expected value plotted on a control chart.

Control limit: A statistically derived boundary for unusual process behavior.

Specification limit: A requirement defining acceptable product or service performance.

Overdispersion: Variation greater than predicted by the assumed binomial or Poisson model.

Exposure: The amount of opportunity for an event or defect to occur.

Rare event: An event with a sufficiently low rate that ordinary approximations may be unreliable.

Assignable cause: An identifiable factor producing non-routine variation.

Defect rate: The number of defects divided by a defined opportunity or unit of exposure.

Inspection bias: Systematic distortion caused by inconsistent or non-independent detection and classification.

Review Questions

Short Answer Questions

1. What is the difference between a defective unit and a defect?

1. Why is an np chart unsuitable when subgroup sample sizes vary?

1. State the principal distributional assumptions underlying p and np charts.

1. When is a u chart preferred to a c chart?

1. What does overdispersion indicate in a Poisson-based chart?

Long Answer Questions

1. Explain the selection logic for p, np, c, and u charts, using examples from manufacturing and services.

1. Derive the p-chart limits from the binomial mean and variance, and explain why unequal sample sizes produce variable limits.

1. Discuss the assumptions of the Poisson model and the practical consequences of violating them.

1. Describe a defensible procedure for establishing, revising, and governing attribute-chart limits.

1. Critically evaluate the role of automated and AI-supported alerting in attribute SPC.

Analytical / Application Questions

1. A call centre handles different numbers of calls each day and records multiple complaint categories per call. Recommend charts for defective calls and total complaints, and justify both denominators.

1. A u chart shows repeated high points, but investigation reveals that product mix changes substantially by shift. Diagnose the likely source of the apparent instability and propose an analytical redesign.

1. A hospital reports zero medication errors for several weeks after introducing a new reporting interface. Design a validation plan to determine whether the improvement reflects safer performance, reduced opportunity, or under-reporting.

Multiple Choice Questions with Answers

Question 1

Which chart is most appropriate for the proportion of defective units when subgroup sample sizes vary?

A. p chart

B. np chart

C. c chart

D. u chart

Correct Answer: A. p chart

Question 2

A unit contains three separate nonconformities. How is it represented on a defective-unit chart?

A. As three defective units

B. As one defective unit

C. As zero defective units

D. It cannot be represented

Correct Answer: B. As one defective unit

Question 3

Under the binomial model, the variance of the observed fraction defective p_i is:

A. p(1−p)n_i

B. p(1−p)/n_i

C. n_i/p

D. p/n_i(1−p)

Correct Answer: B. p(1−p)/n_i

Question 4

Which condition is required for an np chart?

A. Variable sample sizes

B. A constant sample size

C. A constant number of defects per unit

D. Normally distributed measurements

Correct Answer: B. A constant sample size

Question 5

A c chart is most appropriate when:

A. The number of defective units is counted in variable samples

B. The fraction defective is monitored with changing sample sizes

C. Defects are counted in equal inspection units

D. Only one defect can occur per unit

Correct Answer: C. Defects are counted in equal inspection units

Question 6

A process records 40 defects across 2,000 equivalent units. What is the pooled u estimate?

A. 0.002 defects per unit

B. 0.02 defects per unit

C. 20 defects per unit

D. 50 defects per unit

Correct Answer: B. 0.02 defects per unit

Question 7

Why should raw counts generally not be compared directly when exposure varies?

A. Counts are always normally distributed

B. A larger exposure naturally provides more opportunity for events

C. Control limits cannot be calculated for counts

D. The Poisson mean is always zero

Correct Answer: B. A larger exposure naturally provides more opportunity for events

Question 8

For a Poisson model, overdispersion means that:

A. The mean is greater than zero

B. The variance is less than the mean

C. The variance is greater than the mean

D. All observations are identical

Correct Answer: C. The variance is greater than the mean

Question 9

If a calculated lower control limit for a count chart is negative, it should normally be:

A. Reported as a negative count

B. Set to zero

C. Replaced by the specification limit

D. Multiplied by the sample size

Correct Answer: B. Set to zero

Question 10

Which statement best distinguishes control limits from specification limits?

A. Control limits describe customer requirements; specifications describe random variation

B. They are two names for the same boundary

C. Control limits describe expected process variation; specifications define acceptability

D. Specification limits are calculated from the Poisson variance

Correct Answer: C. Control limits describe expected process variation; specifications define acceptability