6. Control Charts for Attributes and Discrete Events

Introduction and context

Statistical process control (SPC) is often introduced through variable control charts, which monitor measurable characteristics such as length, weight, temperature, or cycle time. In many operational settings, however, quality is recorded in a different form. A unit may be classified as conforming or nonconforming; a document may contain several errors; a hospital ward may record a number of medication incidents; or a service centre may count complaints received during each shift. These observations are attributes or discrete events rather than continuous measurements.

Attribute charts provide a disciplined way to distinguish routine variation from unusual variation when data are expressed as counts, proportions, or classifications. Their importance extends beyond manufacturing. Banking, healthcare, logistics, software support, public administration, telecommunications, and hospitality all generate attribute data. The analytical challenge is to select a chart that matches the data-generating process, account for exposure or sample size, and interpret signals without confusing statistical variation with meaningful operational change.

This chapter develops the logic and application of p, np, c, and u charts. It also examines varying sample sizes, rare-event monitoring, overdispersion, underdispersion, service-quality applications, and the managerial actions that should follow a signal.

Learning objectives

After studying this chapter, the learner should be able to:

1. Distinguish defects, defectives, opportunities, and discrete events.

2. Select an appropriate p, np, c, or u chart for a given quality problem.

3. Calculate centre lines, control limits, and standardised statistics.

4. Interpret charts when sample sizes or areas of opportunity vary.

5. Recognise the effects of rare events, overdispersion, and underdispersion.

6. Apply attribute charts to manufacturing and service-quality data.

7. Investigate special causes while avoiding inappropriate tampering and subgrouping.

8. Evaluate the limitations and governance implications of attribute-based monitoring.

The data logic behind attribute charts

The first decision is whether the observation concerns a defective unit or a defect. A defective is a unit that fails to meet one or more requirements. A single defective circuit board is counted once, even if it has five soldering problems. A defect is an individual nonconformity. The same circuit board may therefore contribute five defects. This distinction determines the appropriate chart.

A p chart monitors the proportion of defectives in a sample. An np chart monitors the number of defectives when the sample size is constant. A c chart monitors the number of defects when the inspection area or opportunity is constant. A u chart monitors defects per unit when the number of units or opportunities varies.

The term opportunity refers to a chance for a defect or event to occur. For example, a software module may contain multiple code-review opportunities, or an invoice may contain several fields capable of containing an error. If opportunities vary materially, a raw count can be misleading. Ten errors in one hundred invoices and ten errors in ten invoices do not indicate the same performance.

Attribute charts generally rely on probability models. For p and np charts, the binomial model is appropriate when each unit is classified as defective or nondefective, the sample size is known, and the probability of defectiveness is reasonably stable. For c and u charts, the Poisson model is commonly used when defects are counts of events occurring over a defined and comparable opportunity base. These models are approximations to reality, not substitutes for process understanding.

Choosing among p, np, c and u charts

The following decision logic is useful:

• Use a p chart for the fraction or percentage of defective units, especially when sample sizes vary.
• Use an np chart for the number of defective units when every subgroup has the same sample size.
• Use a c chart for the number of defects when the inspection unit or opportunity area is constant.
• Use a u chart for defects per unit when the number of units, area, time, or opportunities varies.

The distinction between defective and defect is more important than whether the data are collected in a factory or service organisation. A customer account with at least one billing error is a defective account; the total number of billing errors is a defect count. A patient with at least one documentation omission is a defective record; the total omissions are defects.

p charts: monitoring the proportion defective

Suppose subgroup i contains n_i units and x_i units are defective. The observed fraction defective is

p_i = x_i / n_i.

The process average is estimated by

p-bar = total defectives / total units = Σx_i / Σn_i.

For a subgroup of size n_i, the three-sigma limits are

UCL_i = p-bar + 3√[p-bar(1 − p-bar)/n_i]

LCL_i = p-bar − 3√[p-bar(1 − p-bar)/n_i].

If the lower limit is negative, it is set to zero. When sample sizes vary, the centre line remains constant but the limits change. Larger samples produce narrower limits because they provide more information about the underlying proportion.

A p chart is particularly suitable for audit samples, warranty claims, rejected transactions, and service cases classified as acceptable or unacceptable. Percentages are convenient for communication, but calculations should normally use proportions.

Example: variable audit samples

An organisation audits invoices each day. The numbers audited over five days are 100, 150, 80, 120, and 200, with 6, 12, 3, 10, and 18 defective invoices respectively. Total defectives equal 49 and total invoices equal 650, giving p-bar = 0.0754. For the subgroup of 100 invoices, the approximate limits are 0.0754 ± 3√[0.0754(0.9246)/100], or approximately 0.0754 ± 0.077. The lower limit is therefore zero and the upper limit is about 0.152. The limits for the 200-invoice subgroup are narrower. Applying one fixed percentage limit to all days would ignore the different precision of the samples.

When sample sizes differ substantially, a variable-limit p chart is preferable. A standardised p statistic may also be plotted:

Z_i = (p_i − p-bar) / √[p-bar(1 − p-bar)/n_i].

A value beyond approximately ±3 indicates an unusual result under the model.

np charts: monitoring the number defective

The np chart is used when the sample size n is constant and management wishes to interpret the count of defective units directly. Its centre line is

np-bar = n p-bar,

and its limits are

UCL = np-bar + 3√[n p-bar(1 − p-bar)]

LCL = np-bar − 3√[n p-bar(1 − p-bar)],

with a negative lower limit replaced by zero.

The np chart is often easier to explain operationally: “the number of defective claims per batch” may be more meaningful than “the percentage defective.” It should not be used when subgroup sizes vary materially. In that situation, either a p chart or an appropriately standardised approach is required.

A further practical issue is the definition of the subgroup. If each subgroup represents a shift, batch, or transaction period, units should be sampled in a manner that makes the subgroup internally coherent and comparable. Arbitrarily pooling observations can conceal shifts in process behaviour.

c charts: counting defects under constant opportunity

A c chart monitors the number of defects in a constant inspection unit. If c_i is the number of defects in subgroup i, the estimated average count is

c-bar = Σc_i / k,

where k is the number of subgroups. Under a Poisson model,

UCL = c-bar + 3√c-bar,

LCL = c-bar − 3√c-bar.

The lower limit is set to zero if necessary.

Examples include scratches on a fixed area of coated material, errors on forms of identical length, or complaints received during equal one-day periods when the volume and exposure are stable. The “constant opportunity” condition must be examined carefully. Equal calendar periods do not necessarily provide equal opportunity if customer volume, staffing, or transaction counts fluctuate.

A c chart counts all defects, including multiple defects on one unit. If the question is whether a unit is acceptable, a p or np chart is more appropriate.

u charts: defects per unit with varying opportunity

A u chart is used when the number of defects is normalised by the number of inspected units or opportunities. If c_i defects are found in n_i units, then

u_i = c_i / n_i.

The average defects per unit is

u-bar = Σc_i / Σn_i.

The limits for subgroup i are

UCL_i = u-bar + 3√(u-bar/n_i)

LCL_i = u-bar − 3√(u-bar/n_i).

Again, the lower limit cannot be less than zero. If each unit contains a different number of opportunities, the denominator should represent opportunities rather than merely units. For example, software defects per thousand lines of code or billing errors per thousand invoice fields may be more meaningful than defects per file or invoice.

A u chart is valuable in service operations because workload varies. A call centre may monitor complaints per 1,000 calls, a hospital may monitor medication errors per 1,000 administrations, and a transport company may monitor handling incidents per 10,000 parcels. Raw counts would tend to rise with volume even if the underlying rate improved.

Estimation, rational subgrouping and limits

Control limits are estimates of the behaviour expected from common causes. They are not specification limits and do not define customer acceptability. A process can be statistically stable but incapable of meeting a demanding service target; conversely, it can meet an average target while remaining unstable.

Initial limits should be based on a sufficiently representative period. If the data contain a known special cause, blindly incorporating it into the baseline inflates the limits and weakens future detection. However, points should not be deleted merely because they are inconvenient or outside the limits. The cause must be investigated and the decision documented. After a verified process change, a new baseline may be justified.

Rational subgrouping is central. Units grouped together should have been produced under comparable conditions, while differences between subgroups should help reveal changes over time, operator, supplier, machine, shift, location, or case type. If a subgroup combines unrelated populations, the chart may display artificial instability or conceal real changes.

Interpreting signals and investigating causes

A point beyond a control limit is a strong signal of a special cause, but it is not proof of a specific cause. Other useful signals include a sustained run on one side of the centre line, a trend, repeated proximity to a limit, or a nonrandom cycle. The exact supplementary rules should be defined before analysis so that interpretation is not selectively changed after observing the data.

The investigation should move from signal to process evidence. Questions include: Did the input mix change? Was there a new operator, supplier, software release, policy, or machine setting? Did the inspection method change? Was the denominator recorded correctly? Was a case duplicated? Was there a sudden workload or staffing change?

The appropriate response is not automatic adjustment. Tampering occurs when managers react to common-cause fluctuation by making unnecessary process changes. Such reactions can increase variation. A useful discipline is to verify the signal, identify a plausible assignable cause, contain immediate risk, correct the cause, and confirm the result with subsequent data.

Rare events and low-count processes

Attribute data often contain many zeros. A standard Poisson chart may then have a lower limit of zero and a relatively high upper limit, reducing sensitivity to small but important changes. Rare-event charts, such as g charts for the number of opportunities between events or t charts for time between events, may be more informative when the event itself is uncommon. These charts are especially useful for safety incidents, major security breaches, serious clinical events, or catastrophic equipment failures.

A zero count does not necessarily demonstrate perfect quality. It may reflect low exposure, inadequate detection, or a small sample. The analyst should report the opportunity base and the detection system alongside the count.

Overdispersion and underdispersion

The binomial and Poisson models assume a particular relationship between the mean and variance. In real operations, the observed variation may be greater than predicted. This is overdispersion. It can arise from mixtures of populations, changing risk levels, clustering, serial dependence, inconsistent inspection, or omitted explanatory variables. For a Poisson process, if the variance of subgroup counts substantially exceeds the mean, a basic c or u chart may produce too many false alarms.

Possible responses include stratifying the data, redefining rational subgroups, identifying process changes, using an overdispersion or quasi-likelihood adjustment, or applying a model such as negative binomial regression when the objective is explanation or prediction rather than simple monitoring. An inflated dispersion factor may be used cautiously to widen limits, but widening limits without investigating the cause can conceal genuine deterioration.

Underdispersion occurs when variation is lower than the model predicts. It may result from effective prevention controls, quota systems, dependent events, rounding, or a constrained process. Standard limits may then be too wide and insensitive. Underdispersion should not automatically be treated as evidence of superior control; it may indicate that observations are not independent or that the recording system suppresses variation.

Model checking should include plots, variance-to-mean comparisons, stratification, audit of data definitions, and examination of serial dependence. At postgraduate level, analysts should recognise that control-chart rules and probability-model assumptions are related but not identical: a chart can be operationally useful only when its assumptions are credible enough for the decision being made.

Service-quality applications

Attribute charts translate customer experience into monitorable process information. Examples include:

• proportion of calls abandoned, using a p chart when daily call volume varies;
• number of rejected loan applications for documentation errors, using an np chart for fixed-size review samples;
• complaints per equal service period, using a c chart only when exposure is stable;
• delivery exceptions per 1,000 shipments, using a u chart when shipment volumes vary;
• medication administration incidents per 1,000 administrations, using a u chart;
• software incidents per release or per thousand transactions, with careful attention to changing system complexity.

The choice of chart should reflect the managerial question. If leaders want to know how many customers experienced failure, use defective-based monitoring. If they want to understand the number of failures within cases, use defect-based monitoring. A service process may require both: the proportion of customers affected and the average number of errors among affected cases measure different dimensions of quality.

Service data also require attention to case mix. A hospital emergency department, for example, may receive patients of varying severity. A rise in incidents may reflect higher clinical complexity rather than poorer process execution. Stratified charts, risk adjustment, or separate charts for comparable case categories may be necessary.

Illustrative case study: medication administration safety

Background

A regional hospital monitors medication administration incidents across inpatient wards. An incident may be a wrong dose, omitted dose, incorrect timing, or documentation error. The quality team initially reports the number of incidents per ward each week.

Situation and problem

The incident count rises during weeks with higher medication volume. Managers conclude that the process is deteriorating, but ward sizes and patient acuity differ. In addition, one ward reports multiple documentation errors for the same medication round, while another records only whether a medication round had any error.

Analysis

The data definitions are inconsistent, and raw counts do not account for exposure. The team establishes two measures: the proportion of medication rounds with at least one incident, and the number of incidents per 1,000 administrations. The first measure concerns defective rounds and is monitored with a p chart because weekly sample sizes vary. The second concerns defects and is monitored with a u chart because the number of administrations varies.

Initial analysis shows that the u chart has more variation than a Poisson model predicts. Investigation identifies two causes: a new electronic-record interface in one ward and a difference in reporting practice between wards. The team stratifies the data by ward and records the interface change as a process event rather than deleting the observations.

Approach and solution

The hospital standardises incident definitions, trains staff in reporting, and uses separate charts for comparable wards. A special-cause investigation is triggered by a point beyond a limit, a sustained run, or a credible safety pattern. The electronic interface is corrected, and a sample audit is used to assess under-reporting. Management reviews both the statistical signal and the clinical severity of incidents; a low-frequency but high-severity event receives immediate attention even if it does not exceed a statistical limit.

Outcome

The revised charts distinguish workload effects from process changes. The proportion of affected medication rounds becomes stable, while the defect rate falls after the interface correction. The hospital avoids both overreaction to volume-driven counts and false reassurance from a low raw incident count.

Learning points

The case demonstrates that chart selection follows the operational definition of quality. It also shows that data governance, exposure measurement, stratification, and clinical judgement are essential complements to statistical rules. A control chart is a decision aid, not a replacement for safety management.

Technology and contemporary practice

Modern quality systems increasingly generate attribute data automatically from enterprise applications, electronic health records, workflow platforms, sensors, and customer-contact systems. Dashboards can calculate dynamic limits, stratify by process stream, and issue alerts. Automated systems are useful only when event definitions, denominators, timestamps, and revision rules are controlled.

Advanced analytics can supplement charts through anomaly detection, Bayesian updating, hierarchical models, or machine-learning classifiers. These methods may identify complex patterns, but their outputs should remain interpretable and subject to process expertise. Excessive automated alerting creates alarm fatigue, while opaque models can undermine accountability. A sound architecture preserves the original event record, documents rule changes, controls access, and allows an auditor to reconstruct why an alert was generated.

Limitations, risks and professional responsibilities

Attribute charts discard information when a continuous measurement is converted into a pass/fail classification. If measurement data are available and reliable, variable charts may detect shifts earlier. Attribute data may also be affected by subjective inspection, changing definitions, under-reporting, duplicate records, and incentives to classify cases favourably.

Common mistakes include using an np chart with unequal sample sizes, using a c chart when exposure varies, confusing defects with defectives, treating specification limits as control limits, recalculating limits after every signal, ignoring denominator changes, and deleting inconvenient observations without documented justification. Another mistake is to rank departments by raw incident counts when their workloads differ.

Ethically, organisations must avoid using statistical monitoring to punish individuals for system-level variation. Data should support learning, risk reduction, and fair accountability. Sensitive service and healthcare data require confidentiality, access controls, retention rules, and appropriate reporting. If performance measures affect compensation or regulatory reporting, definitions and exception rules should be transparent and independently reviewable.

Best-practice implementation sequence

A robust implementation can follow this sequence:

1. Define the quality event precisely and decide whether it is a defect or defective.

2. Identify the denominator: units, transactions, time, area, or opportunities.

3. Select p, np, c, or u according to the data structure.

4. Establish rational subgroups and verify the measurement or reporting system.

5. Estimate baseline parameters from representative data.

6. Calculate limits using the appropriate sample size or exposure.

7. Display the chart with operational annotations and predetermined rules.

8. Investigate signals using process evidence rather than automatic adjustment.

9. Review dispersion, case mix, and reporting changes.

10. Rebaseline only after a documented, sustained process change.

Chapter Summary

Attribute and discrete-event charts monitor quality when observations are classifications or counts rather than continuous measurements. P charts track proportions defective and accommodate varying sample sizes; np charts track numbers defective when sample size is constant. C charts monitor defect counts under constant opportunity, whereas u charts monitor defects per unit when exposure varies. Correct distinction between defects and defectives, careful denominator selection, rational subgrouping, and credible data definitions are fundamental.

Control limits describe expected process variation and are not specification limits. Signals require investigation, not automatic adjustment. Rare events may require time-between-event or opportunity-between-event charts. Overdispersion and underdispersion reveal departures from simple binomial or Poisson assumptions and should prompt investigation, stratification, or suitable modelling. In service and safety applications, statistical evidence must be combined with case-mix analysis, professional judgement, ethical governance, and risk-based decision-making.

Key Terms

Attribute data: Quality data recorded as classifications, counts, or proportions rather than continuous measurements.

Defective: A unit that fails to meet one or more specified requirements.

Defect: An individual nonconformity or error within a unit.

Opportunity: A defined chance for a defect or event to occur.

p chart: A control chart for the proportion of defective units.

np chart: A control chart for the number of defective units in constant-size samples.

c chart: A control chart for the number of defects under constant opportunity.

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

Common cause: A source of variation inherent in the routine process.

Special cause: An identifiable and nonroutine source of process variation.

Rational subgroup: A logically comparable group of observations formed to reveal process changes.

Control limit: A statistically derived boundary used to assess unusual process behaviour.

Specification limit: A customer, design, or regulatory requirement; it is not estimated from process data.

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

Underdispersion: Variation lower than predicted by the assumed probability model.

Rare event: An event occurring so infrequently that ordinary count charts may have limited sensitivity.

Exposure: The quantity of units, time, area, transactions, or opportunities at risk.

Tampering: Unnecessary process adjustment made in response to common-cause variation.

Standardised statistic: A transformed chart value that accounts for differing sample sizes or exposures.

Review Questions

Short Answer Questions

1. Distinguish a defect from a defective using an operational example.

2. Why is an np chart unsuitable when subgroup sizes vary substantially?

3. State the principal difference between a c chart and a u chart.

4. What is meant by overdispersion in attribute data?

5. Why are control limits different from specification limits?

Long Answer Questions

1. Explain the assumptions, calculation logic, and applications of p and np charts.

2. Compare c and u charts, including the role of exposure and opportunities.

3. Discuss rational subgrouping and its importance in interpreting attribute charts.

4. Explain how rare events, overdispersion, and underdispersion affect chart selection and interpretation.

5. Describe a complete professional procedure for investigating an out-of-control signal.

Analytical / Application Questions

1. A claims department reviews different numbers of claims each week and records the number containing at least one error. Which chart should be used, and how should its limits be calculated?

2. A call centre records complaint counts by day, but daily call volume varies widely. Explain why a c chart may mislead managers and propose a better monitoring approach.

3. A hospital’s incident count rises after patient admissions increase, while the incident rate per 1,000 administrations remains stable. Analyse the result and recommend an appropriate management response.

22. Multiple Choice Questions with Answers

Question 1

A quality team counts the number of customer accounts containing at least one billing error. The number of accounts reviewed varies each week. Which chart is most appropriate?

A. p chart

B. np chart

C. c chart

D. Individuals chart

Correct Answer: A. p chart

Question 2

Which situation best supports use of an np chart?

A. Defects per unit with varying production volume

B. Defective units from samples of constant size

C. Defect counts on areas of varying size

D. Time between rare safety incidents

Correct Answer: B. Defective units from samples of constant size

Question 3

A factory counts scratches on panels, and every inspected subgroup contains the same panel area. Which chart is appropriate?

A. p chart

B. np chart

C. c chart

D. u chart

Correct Answer: C. c chart

Question 4

A u chart is preferred to a c chart primarily because

A. the data are continuous

B. the number of defectives is constant

C. the opportunity or number of units varies

D. the process has no common causes

Correct Answer: C. the opportunity or number of units varies

Question 5

When subgroup sizes vary on a p chart, what changes from subgroup to subgroup?

A. The definition of a defective

B. The centre line necessarily changes

C. The control limits change

D. The specification limits change

Correct Answer: C. The control limits change

Question 6

A process has a stable average defect count, but its observed variance is much larger than its mean. This most directly suggests

A. underdispersion

B. overdispersion

C. perfect control

D. constant sample size

Correct Answer: B. overdispersion

Question 7

Why can raw complaint counts be misleading when daily customer volume changes?

A. Counts are always continuous measurements

B. Higher volume creates more opportunity for complaints

C. Control limits cannot be calculated for counts

D. Complaints are always defectives rather than defects

Correct Answer: B. Higher volume creates more opportunity for complaints

Question 8

Which action is an example of tampering?

A. Investigating a verified supplier change

B. Recalculating a baseline after a documented process redesign

C. Changing process settings after every ordinary fluctuation

D. Checking whether the denominator was recorded correctly

Correct Answer: C. Changing process settings after every ordinary fluctuation

Question 9

For a rare but serious safety event, a time-between-events chart may be useful because

A. it ignores the event occurrence

B. it can provide more information when most count subgroups are zero

C. it requires no exposure data

D. it converts defects into specification limits

Correct Answer: B. it can provide more information when most count subgroups are zero

Question 10

A hospital records multiple documentation errors within some medication rounds but only a yes/no outcome for other rounds. The first priority should be to

A. rank wards by raw error count

B. delete inconsistent observations

C. standardise the event definition and reporting method

D. use an np chart without checking sample size

Correct Answer: C. standardise the event definition and reporting method