12.3 Attribute Control Charts: p, np, c, and u Charts

12.3 Attribute Control Charts: p, np, c, and u Charts

Quick Summary: Attribute control charts monitor discrete, qualitative quality characteristics such as pass/fail classifications, visual defects, clerical errors, or customer complaints. Selecting the correct attribute chart requires answering two sequential questions: (1) Data Classification: Are you evaluating defective units (Binomial distribution: Go/No-Go) or counting defects (Poisson distribution: flaws per unit)? (2) Subgroup Size: Is the sample size or area of opportunity constant or variable? This yields the four standard charts: $p$, $np$, $c$, and $u$.

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Defective Units vs. Defects: The Fundamental Distinction

The most critical conceptual hurdle in attribute SPC is distinguishing between a defective unit and a defect.

                             Attribute Data Type
                                      │
         ┌────────────────────────────┴────────────────────────────┐
         ▼                                                         ▼
     DEFECTIVE                                                  DEFECT
(Nonconforming Unit)                                        (Nonconformity)
  • Whole unit is judged PASS or FAIL                         • Specific flaw, blemish, or error
  • Binary outcome: Yes or No                                 • Countable quantity: 0, 1, 2, 3...
  • Governed by BINOMIAL Distribution                         • Governed by POISSON Distribution
  • Monitored by: p-Chart or np-Chart                         • Monitored by: c-Chart or u-Chart

1. Defective (Nonconforming Unit) — Binomial Distribution

A unit is declared defective when it fails to meet one or more quality specifications, rendering the entire item unacceptable. It is a binary, qualitative assessment: an invoice is either correct or incorrect; a lightbulb either illuminates or does not; a medical claim is either paid or rejected. Regardless of whether an item has 1 flaw or 10 flaws, it counts as exactly one defective unit.

2. Defect (Nonconformity) — Poisson Distribution

A defect is a single, specific failure, blemish, or departure from specification found on a unit. A unit can contain multiple defects and still remain functional. For example, a newly painted luxury car hood might contain 3 paint blisters, 1 scratch, and 2 dust nibs (a total of 6 defects), yet it remains a single unit. Defect counts represent rare events occurring across a continuous area of opportunity, modeled by the Poisson distribution.

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The Attribute Control Chart Selection Decision Tree

To determine which of the four attribute charts to deploy, use the following two-axis selection matrix:

                         Attribute Control Chart Selection
                                         │
                    What type of attribute data is being collected?
                                         │
         ┌───────────────────────────────┴───────────────────────────────┐
         ▼                                                               ▼
  DEFECTIVE UNITS                                                     DEFECTS
 (Binomial: Pass/Fail)                                       (Poisson: Count of Flaws)
         │                                                               │
 Is sample size (n) constant?                                   Is inspection unit constant?
         │                                                               │
    ┌────┴────┐                                                     ┌────┴────┐
    ▼         ▼                                                     ▼         ▼
   YES       NO                                                    YES       NO
 np-Chart  p-Chart                                               c-Chart   u-Chart
 (Number) (Fraction)                                             (Count)   (Rate/Unit)

 *Note: p-charts can also be used for constant n.                 *Note: u-charts can also be used for constant n.

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The $p$-Chart (Proportion Defective)

The $p$-chart monitors the fraction or percentage of defective units in a sample. It is the most versatile attribute chart because it can handle variable subgroup sizes ($n_i$).

Formulas for $p$-Chart:

• Subgroup Proportion ($p_i$): $p_i = \frac{d_i}{n_i}$ (where $d_i$ is number of defectives found in subgroup $i$ of size $n_i$)
• Center Line ($\bar{p}$): $\bar{p} = \frac{\sum_{i=1}^k d_i}{\sum_{i=1}^k n_i}$
• Standard Error of Proportion ($\sigma_{p_i}$): $\sigma_{p_i} = \sqrt{\frac{\bar{p}(1 - \bar{p})}{n_i}}$
• Control Limits: $\text{UCL}_{p_i} = \bar{p} + 3 \sqrt{\frac{\bar{p}(1 - \bar{p})}{n_i}}$ $\text{LCL}_{p_i} = \max\left(0, \bar{p} - 3 \sqrt{\frac{\bar{p}(1 - \bar{p})}{n_i}}\right)$

Stepped (Variable) Control Limits

When sample sizes $n_i$ vary from day to day (e.g., daily call center volumes fluctuating between 800 and 1,500 calls), the denominator inside the square root changes for each subgroup. Consequently, the control limits expand on days with smaller sample sizes and contract on days with larger sample sizes, creating a distinctive "stair-step" appearance.

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The $np$-Chart (Number of Defectives)

The $np$-chart tracks the raw integer count of defective units rather than a fraction. It is highly popular in factory environments because floor operators find integer counts (e.g., "6 bad parts") far easier to interpret than decimals (e.g., "0.012 fraction defective").

Strict Operational Requirement: The sample size $n$ must be identical for every single subgroup. If $n$ varies by even a single unit, an $np$-chart cannot be used; the team must switch to a $p$-chart.

Formulas for $np$-Chart:

• Center Line ($CL_{np}$): $\text{CL}_{np} = n\bar{p} = \frac{\sum d_i}{k}$
• Standard Error of Count ($\sigma_{np}$): $\sigma_{np} = \sqrt{n\bar{p}(1 - \bar{p})}$
• Control Limits: $\text{UCL}_{np} = n\bar{p} + 3 \sqrt{n\bar{p}(1 - \bar{p})}$ $\text{LCL}_{np} = \max\left(0, n\bar{p} - 3 \sqrt{n\bar{p}(1 - \bar{p})}\right)$

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The $c$-Chart (Defect Counts for Constant Area)

The $c$-chart tracks the total count of nonconformities (defects) across a constant area of opportunity (such as one airplane wing, 100 printed pages, or 10 square meters of glass). Because it models Poisson events, variance equals the mean ($\sigma^2 = \lambda = \bar{c}$).

Formulas for $c$-Chart:

• Center Line ($\bar{c}$): $\bar{c} = \frac{\sum_{i=1}^k c_i}{k}$
• Standard Error ($\sigma_c$): $\sigma_c = \sqrt{\bar{c}}$
• Control Limits: $\text{UCL}_c = \bar{c} + 3\sqrt{\bar{c}}$ $\text{LCL}_c = \max\left(0, \bar{c} - 3\sqrt{\bar{c}}\right)$

Notice: No sample size $n$ appears in the formula because the area of opportunity is standardized as a single unit ($n = 1$).

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The $u$-Chart (Defects per Unit for Variable Area)

When the area of opportunity varies across inspection periods (e.g., inspecting rolls of fabric of varying lengths, hospital wards with different patient-day counts, or varying batch sizes), the $c$-chart cannot be used. Instead, the $u$-chart monitors the average defect rate per unit: $u_i = \frac{c_i}{n_i}$ (where $c_i$ is total defects observed across $n_i$ inspection units).

Formulas for $u$-Chart:

• Center Line ($\bar{u}$): $\bar{u} = \frac{\sum_{i=1}^k c_i}{\sum_{i=1}^k n_i}$
• Standard Error ($\sigma_{u_i}$): $\sigma_{u_i} = \sqrt{\frac{\bar{u}}{n_i}}$
• Control Limits: $\text{UCL}_{u_i} = \bar{u} + 3 \sqrt{\frac{\bar{u}}{n_i}}$ $\text{LCL}_{u_i} = \max\left(0, \bar{u} - 3 \sqrt{\frac{\bar{u}}{n_i}}\right)$

Like the $p$-chart, the $u$-chart displays stepped control limits when subgroup sizes $n_i$ vary.

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Step-by-Step Worked Calculation Examples

Example 1: $np$-Chart Calculation in Manufacturing

Scenario: An injection molding line inspects a fixed sample of $n = 400$ medical syringe barrels every hour. Over $k = 25$ hours, inspectors identify a total of $300$ defective barrels.

1. Calculate the overall proportion defective ($\bar{p}$): $\bar{p} = \frac{300}{25 \times 400} = \frac{300}{10,000} = 0.030 \quad (3.0\%)$
2. Calculate the Center Line for the $np$-chart: $\text{CL}_{np} = n\bar{p} = 400 \times 0.030 = 12.0 \text{ defectives}$
3. Calculate the standard error ($\sigma_{np}$): $\sigma_{np} = \sqrt{n\bar{p}(1 - \bar{p})} = \sqrt{400 \times 0.030 \times 0.970} = \sqrt{11.64} \approx 3.4117$
4. Compute Control Limits: $\text{UCL}_{np} = 12.0 + 3(3.4117) = 12.0 + 10.235 = 22.235 \approx 22.24 \text{ defectives}$ $\text{LCL}_{np} = \max(0, 12.0 - 10.235) = 1.765 \approx 1.76 \text{ defectives}$

Example 2: $c$-Chart Calculation in Automotive Quality

Scenario: A luxury automotive assembly plant inspects finished vehicle paintwork for surface blemishes (clearcoat craters, dust inclusions, orange peel). Exactly $1$ vehicle body ($n = 1$) is inspected per audit period. Over $k = 30$ vehicles, auditors count a total of $270$ defects.

1. Calculate the Center Line ($\bar{c}$): $\bar{c} = \frac{270}{30} = 9.0 \text{ defects per car}$
2. Calculate the standard error ($\sigma_c$): $\sigma_c = \sqrt{\bar{c}} = \sqrt{9.0} = 3.0$
3. Compute Control Limits: $\text{UCL}_c = \bar{c} + 3\sqrt{\bar{c}} = 9.0 + 3(3.0) = 9.0 + 9.0 = 18.0 \text{ defects}$ $\text{LCL}_c = \max(0, 9.0 - 3(3.0)) = \max(0, 0) = 0 \text{ defects}$

Example 3: $u$-Chart Calculation with Variable Subgroup Sizes

Scenario: A software testing firm logs functional bugs per module. Because software modules vary in complexity, the size is measured in thousands of lines of code (KLOC). Across $k = 20$ modules totaling $100$ KLOC, testers discover $400$ bugs.

1. Calculate the average defect rate per unit ($\bar{u}$): $\bar{u} = \frac{400}{100} = 4.0 \text{ bugs per KLOC}$
2. Calculate limits for a specific module measuring $n_i = 4$ KLOC: $\sigma_{u_i} = \sqrt{\frac{\bar{u}}{n_i}} = \sqrt{\frac{4.0}{4}} = \sqrt{1.0} = 1.0$ $\text{UCL}_{u} = 4.0 + 3(1.0) = 7.0 \text{ bugs/KLOC}$ $\text{LCL}_{u} = 4.0 - 3(1.0) = 1.0 \text{ bugs/KLOC}$
3. Calculate limits for a smaller module measuring $n_i = 1$ KLOC: $\sigma_{u_i} = \sqrt{\frac{4.0}{1}} = 2.0$ $\text{UCL}_{u} = 4.0 + 3(2.0) = 10.0 \text{ bugs/KLOC}$ $\text{LCL}_{u} = \max(0, 4.0 - 3(2.0)) = \max(0, -2.0) = 0 \text{ bugs/KLOC}$

Example 4: $p$-Chart Calculation with Variable Subgroup Sizes

Scenario: A commercial bank processes incoming mortgage applications. Over $k = 20$ business days, total applications received are $10,000$ (averaging 500 per day), and $400$ applications are rejected due to compliance defects.

1. Compute the process average proportion defective ($\bar{p}$): $\bar{p} = \frac{400}{10,000} = 0.040 \quad (4.0\%$
2. Control limits for a high-volume day ($n_i = 1,000$ applications): $\sigma_{p} = \sqrt{\frac{0.040(1 - 0.040)}{1,000}} = \sqrt{\frac{0.0384}{1,000}} = \sqrt{0.0000384} \approx 0.006197$ $\text{UCL}_p = 0.040 + 3(0.006197) = 0.040 + 0.01859 = 0.0586 \quad (5.86\%)$ $\text{LCL}_p = \max(0, 0.040 - 0.01859) = 0.0214 \quad (2.14\%)$
3. Control limits for a low-volume day ($n_i = 400$ applications): $\sigma_{p} = \sqrt{\frac{0.040 \times 0.960}{400}} = \sqrt{0.000096} \approx 0.009798$ $\text{UCL}_p = 0.040 + 3(0.009798) = 0.040 + 0.02939 = 0.0694 \quad (6.94\%)$ $\text{LCL}_p = \max(0, 0.040 - 0.02939) = 0.0106 \quad (1.06\%)$

Notice that on the lower-volume day ($n=400$), the control limits widen considerably ($[1.06%, 6.94%]$ vs $[2.14%, 5.86%]$), demonstrating why stepped control limits naturally expand and contract based on sample size.

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Critical CSSC Exam Traps

• Trap 1: Choosing an $np$-Chart with Variable Subgroup Sizes — This is one of the most common trick questions on the exam. If the scenario mentions that sample size varies from day to day (e.g., "inspecting all incoming invoices each day"), you cannot use an $np$-chart. You must choose a $p$-chart.
• Trap 2: Choosing a $c$-Chart for Variable Inspection Areas — If a quality auditor counts surface scratches on steel coils, but the square footage of the inspected coils varies, a $c$-chart is mathematically invalid. The $u$-chart must be used to normalize defects per unit of area.
• Trap 3: Confusing Binomial and Poisson Formulas — Memorize that Binomial charts ($p, np$) always include the term $(1 - \bar{p})$ under the square root. Poisson charts ($c, u$) never contain $(1 - \bar{p})$; their variance is based strictly on $\bar{c}$ or $\bar{u} / n$.
• Trap 4: Negative Lower Control Limits — When calculating LCL for attribute charts where the average is small, the mathematical formula often results in a negative number (e.g., $1.5 - 3(1.0) = -1.5$). In physical reality, you cannot have negative defectives or negative defects. On the exam, you must report $\text{LCL} = 0$.