11.2 Statistical Process Control for Attributes (p, np, c, and u Charts)

Statistical Process Control for Attributes ($p$, $np$, $c$, and $u$ Charts)

Core Principle: Attribute control charts monitor qualitative, count-based, or pass/fail quality characteristics. Attribute SPC is organized according to whether the inspection records nonconforming units (defective products modeled by the binomial distribution) or nonconformities (defects per inspection unit modeled by the Poisson distribution).

In many manufacturing, assembly, and service systems, measuring continuous numerical dimensions is neither practical nor cost-effective. Instead, quality inspectors evaluate parts using "go/no-go" functional gauges, visual optical inspection, or defect counts per assembly. Statistical Process Control for attributes provides the quantitative framework for tracking these discrete metrics. Industrial engineers must select the correct attribute chart based on the mathematical nature of the data and whether the sample inspection size remains constant or varies.

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1. Fundamentals of Attribute Data: Defectives vs. Defects

A critical distinction on the PE exam is between nonconforming units (defective items) and nonconformities (defects):

Attribute Data Classification
├── Nonconforming Units (Defective Items)
│   ├── Binary outcome: Conforming (Pass) vs Nonconforming (Fail)
│   ├── Underlying distribution: Binomial Distribution B(n, p)
│   └── Appropriate Control Charts:
│       ├── p-chart: Fraction nonconforming (constant or variable n)
│       └── np-chart: Number nonconforming (strictly constant n)
└── Nonconformities (Defects per Unit)
    ├── Count of discrete imperfections, flaws, or blemishes per unit
    ├── Underlying distribution: Poisson Distribution Poisson(lambda)
    └── Appropriate Control Charts:
        ├── c-chart: Total defects across constant area of opportunity (n = 1)
        └── u-chart: Defects per inspection unit (constant or variable n_i)

The Binomial Model for Nonconforming Units

When an inspection station evaluates whether a manufactured part meets functional standards, each part represents a Bernoulli trial with two mutually exclusive outcomes: conforming (probability $1 - p$) or nonconforming (probability $p$). In a random sample of $n$ independent units, the probability of observing exactly $d$ nonconforming units is governed by the Binomial Distribution:

$P(D = d) = \binom{n}{d} p^d (1 - p)^{n - d}$ $\text{Expected Count: } E(D) = np \qquad \text{Variance: } \operatorname{Var}(D) = np(1 - p)$

For the fraction nonconforming $\hat{p} = D / n$:

$E(\hat{p}) = p \qquad \operatorname{Var}(\hat{p}) = \frac{\operatorname{Var}(D)}{n^2} = \frac{p(1 - p)}{n}$

The Poisson Model for Nonconformities

A single complex product (e.g., an automobile door panel, a printed circuit board, or an aircraft turbine blade) can satisfy overall functional requirements while exhibiting several discrete blemishes (scratches, pinholes, solder bridges). When the opportunities for defects across a continuous area of opportunity are vast, but the probability of a defect at any specific point is extremely small, defect occurrences follow the Poisson Distribution:

$P(C = c) = \frac{e^{-\lambda} \lambda^c}{c!} \quad \text{for } c = 0, 1, 2, \dots$ $\text{Expected Count: } E(C) = \lambda \qquad \text{Variance: } \operatorname{Var}(C) = \lambda$

The defining characteristic of the Poisson distribution is that its variance equals its mean ($\sigma_c^2 = \mu_c = \lambda$).

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2. Fraction Nonconforming ($p$) and Number Nonconforming ($np$) Charts

The $p$-Chart (Fraction Nonconforming)

The $p$-chart monitors the proportion of nonconforming items in a subgroup. It is the most versatile attribute chart because it accommodates both constant sample sizes and variable sample sizes ($n_i$).

Given $m$ subgroups where subgroup $i$ contains $n_i$ inspected units and $d_i$ nonconforming units, the subgroup fraction nonconforming is $p_i = d_i / n_i$. The overall weighted process fraction nonconforming is:

$\bar{p} = \frac{\sum_{i=1}^m d_i}{\sum_{i=1}^m n_i}$

Case A: Constant Sample Size ($n_i = n$)

When every subgroup contains exactly $n$ units, the control limits remain fixed across all intervals:

$CL_p = \bar{p}$ $UCL_p = \bar{p} + 3 \sqrt{\frac{\bar{p}(1 - \bar{p})}{n}}$ $LCL_p = \max\left(0, \bar{p} - 3 \sqrt{\frac{\bar{p}(1 - \bar{p})}{n}}\right)$

Case B: Variable Sample Size ($n_i$ varies)

In high-mix manufacturing or shipping dock receiving inspection, the number of inspected units varies per lot. Control limits must be calculated individually for each subgroup $i$:

$UCL_{p,i} = \bar{p} + 3 \sqrt{\frac{\bar{p}(1 - \bar{p})}{n_i}}$ $LCL_{p,i} = \max\left(0, \bar{p} - 3 \sqrt{\frac{\bar{p}(1 - \bar{p})}{n_i}}\right)$

Notice that as $n_i$ increases, the standard error decreases, causing the control limits to narrow. Conversely, smaller sample sizes cause the limits to expand. Alternatively, engineers construct a Standardized $p$-Chart where the plotted statistic is the standard normal variate $Z_i$:

$Z_i = \frac{p_i - \bar{p}}{\sqrt{\frac{\bar{p}(1 - \bar{p})}{n_i}}}$ $CL_Z = 0 \qquad UCL_Z = +3 \qquad LCL_Z = -3$

The $np$-Chart (Number Nonconforming)

When operators on a shop floor inspect a fixed number of components each hour (e.g., $n = 50$), plotting fractional percentages (such as $0.04$ or $0.06$) can introduce calculation errors. The $np$-chart plots the integer count of nonconforming items ($d_i$) directly.

Formulation for the $np$-chart:

$CL_{np} = n\bar{p}$ $UCL_{np} = n\bar{p} + 3 \sqrt{n\bar{p}(1 - \bar{p})}$ $LCL_{np} = \max\left(0, n\bar{p} - 3 \sqrt{n\bar{p}(1 - \bar{p})}\right)$

Where $\bar{p} = \frac{\sum d_i}{m \cdot n}$. Note the mathematical relationship: the $np$ limits are simply the $p$-chart limits multiplied through by $n$.

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3. Count of Nonconformities ($c$) and Nonconformities per Unit ($u$) Charts

The $c$-Chart (Total Defects per Constant Unit)

The $c$-chart monitors the total count of nonconformities ($c$) observed in a single, strictly constant area of opportunity ($n = 1$). Typical industrial units include:

• Number of weave defects in exactly $100\text{ m}^2$ of composite cloth.
• Number of solder bridge defects on a single complex avionics circuit board.
• Number of surface paint imperfections on an automobile hood.

Given $m$ inspection intervals with defect counts $c_1, c_2, \dots, c_m$:

$\bar{c} = \frac{1}{m} \sum_{i=1}^m c_i$

Because the variance of a Poisson random variable equals its mean ($\sigma_c^2 = \bar{c} \implies \sigma_c = \sqrt{\bar{c}}$), the Shewhart $3\sigma$ control limits are:

$CL_c = \bar{c}$ $UCL_c = \bar{c} + 3 \sqrt{\bar{c}}$ $LCL_c = \max\left(0, \bar{c} - 3 \sqrt{\bar{c}}\right)$

The $u$-Chart (Nonconformities per Unit)

When the size of the inspection unit varies between samples (e.g., different rolls of fabric with varying yardage, or inspecting variable batch sizes of electronic subassemblies), the raw defect count $c$ cannot be compared directly. The $u$-chart tracks the average defect density per inspection unit ($u = c / n$):

$u_i = \frac{c_i}{n_i}$

Where $n_i$ represents the number of inspection units in subgroup $i$, and $c_i$ is the total count of defects observed across those $n_i$ units. The grand average defect density across $m$ subgroups is:

$\bar{u} = \frac{\sum_{i=1}^m c_i}{\sum_{i=1}^m n_i}$

Because $u_i = c_i / n_i$ and $\operatorname{Var}(c_i) = n_i \bar{u}$, the variance of $u_i$ is $\operatorname{Var}(u_i) = \frac{n_i \bar{u}}{n_i^2} = \frac{\bar{u}}{n_i}$. The standard error is $\sigma_{u,i} = \sqrt{\frac{\bar{u}}{n_i}}$.

The control limits for the $u$-chart are:

$CL_u = \bar{u}$ $UCL_{u,i} = \bar{u} + 3 \sqrt{\frac{\bar{u}}{n_i}}$ $LCL_{u,i} = \max\left(0, \bar{u} - 3 \sqrt{\frac{\bar{u}}{n_i}}\right)$

If the inspection unit size is constant across all subgroups ($n_i = n$), the limits simplify to fixed horizontal lines with denominator $\sqrt{\bar{u}/n}$.

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4. Attribute Chart Selection Decision Framework

Selecting the correct control chart is a frequent focus on the PE exam. The decision tree below structures the selection process:

Attribute Chart Selection Flowchart
                     What is being inspected?
                                |
        +-----------------------+-----------------------+
        |                                               |
   Nonconforming Units                             Nonconformities
   (Defective Items)                               (Defect Counts)
        |                                               |
   Is sample size n                                Is the area of
   strictly constant?                             opportunity constant?
        |                                               |
    +---+---+                                       +---+---+
    |       |                                       |       |
   YES      NO                                     YES      NO
    |       |                                       |       |
 np-chart  p-chart                               c-chart  u-chart
 (or p)   (variable limits)                      (n = 1)  (variable n_i)

Comprehensive Attribute Comparison Matrix

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5. Lower Control Limit Truncation and Zero-Bound Mechanics

In physical manufacturing systems, the count of defects, count of defective items, and fraction nonconforming have a natural physical boundary at zero ($p \ge 0, d \ge 0, c \ge 0$). However, Shewhart formulas calculate symmetrical limits using $3\sigma$ margins:

$LCL_{\text{calculated}} = CL - 3\sigma$

When process quality is high (low $\bar{p}$ or low $\bar{c}$) or sample size $n$ is small, $3\sigma$ frequently exceeds the center line value, yielding a negative mathematical result ($LCL < 0$).

Engineering Implications of $LCL = 0$

When $LCL = 0$, the control chart becomes one-sided for single-point violations. An operational breakthrough (e.g., an improved solder flux yielding zero defects) cannot register as a lower limit breach. To detect significant quality improvements on low-defect lines:

1. Increase the subgroup size $n$ until the calculated $LCL$ is strictly positive ($CL - 3\sigma > 0$):
   • For a $p$-chart: $n > \frac{9(1 - \bar{p})}{\bar{p}}$
   • For a $c$-chart: $\bar{c} > 9$
2. Apply run rules (e.g., 8 consecutive points below the center line) to confirm statistically significant quality improvements.

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6. Comprehensive Worked Numerical Problem: Multi-Chart Comparison

A quality engineer at an automated electronics manufacturing facility monitors surface-mount technology (SMT) assembly on a high-density telecommunications PCB. The engineer evaluates 25 historical production shifts during stable operations. Across these 25 shifts, exactly $n = 100$ PCBs were inspected per shift ($N_{\text{total}} = 25 \times 100 = 2,500$ boards).

Inspection records reveal:

• A cumulative total of $175$ boards were classified as nonconforming (defective) due to failure on automated optical inspection.
• Across all inspected boards, a cumulative total of $625$ solder defects (bridging, insufficient wetting, tombstoning) were recorded. Note that some defective boards contained multiple defects.

Task A: Construct the $p$-Chart and $np$-Chart Control Limits

1. Calculate baseline parameters: $\bar{p} = \frac{\sum d_i}{\sum n_i} = \frac{175}{2500} = 0.070$ $\text{Subgroup Size: } n = 100$

2. $p$-Chart Limits: $CL_p = \bar{p} = 0.070$ $\sigma_p = \sqrt{\frac{\bar{p}(1 - \bar{p})}{n}} = \sqrt{\frac{0.070 \times (1 - 0.070)}{100}} = \sqrt{\frac{0.070 \times 0.930}{100}} = \sqrt{0.000651} = 0.025515$ $3\sigma_p = 3 \times 0.025515 = 0.07654$ $UCL_p = \bar{p} + 3\sigma_p = 0.070 + 0.07654 = 0.14654 \approx 0.1465$ $LCL_p = \max(0, 0.070 - 0.07654) = \max(0, -0.00654) = 0$

3. $np$-Chart Limits (since $n = 100$ is constant): $CL_{np} = n\bar{p} = 100 \times 0.070 = 7.00\text{ defective boards}$ $\sigma_{np} = \sqrt{n\bar{p}(1 - \bar{p})} = \sqrt{100 \times 0.070 \times 0.930} = \sqrt{6.51} = 2.5515$ $3\sigma_{np} = 3 \times 2.5515 = 7.6545$ $UCL_{np} = 7.00 + 7.6545 = 14.6545 \approx 14.65\text{ boards}$ $LCL_{np} = \max(0, 7.00 - 7.6545) = 0\text{ boards}$

Notice that $UCL_{np} = n \times UCL_p = 100 \times 0.14654 = 14.654$ boards.

Task B: Construct the $u$-Chart Limits

Because the engineer is tracking the count of nonconformities (defects) across inspection units of $n = 100$ boards per shift, a $u$-chart monitors the average defects per board:

1. Calculate baseline defect rate per board ($\bar{u}$): $\bar{u} = \frac{\sum c_i}{\sum n_i} = \frac{625}{2500} = 0.250\text{ defects per board}$

2. $u$-Chart Limits for standard shift ($n = 100$ boards): $CL_u = \bar{u} = 0.250\text{ defects/board}$ $\sigma_u = \sqrt{\frac{\bar{u}}{n}} = \sqrt{\frac{0.250}{100}} = \sqrt{0.0025} = 0.050\text{ defects/board}$ $3\sigma_u = 3 \times 0.050 = 0.150\text{ defects/board}$ $UCL_u = \bar{u} + 3\sigma_u = 0.250 + 0.150 = 0.400\text{ defects/board}$ $LCL_u = \bar{u} - 3\sigma_u = 0.250 - 0.150 = 0.100\text{ defects/board}$

Notice that because $\sigma_u$ is small, the lower control limit is strictly positive ($LCL_u = 0.100 > 0$). If a shift exhibits an average defect rate below $0.100$ defects/board, it triggers an assignable cause alarm indicating a statistically significant quality improvement!

3. Evaluation of an Unusual Batch: On shift 26, line maintenance reduces production such that only $n_{26} = 36$ boards are assembled. Inspectors record $12$ total defects on these 36 boards. Is Shift 26 in statistical control?

• Observed rate: $u_{26} = 12 / 36 = 0.3333\text{ defects/board}$.
• Revised standard error for $n_{26} = 36$: $\sigma_{u,26} = \sqrt{\frac{\bar{u}}{n_{26}}} = \sqrt{\frac{0.250}{36}} = \frac{0.500}{6} = 0.08333\text{ defects/board}$ $UCL_{u,26} = 0.250 + 3(0.08333) = 0.250 + 0.250 = 0.500\text{ defects/board}$ $LCL_{u,26} = \max(0, 0.250 - 0.250) = 0\text{ defects/board}$

Since $u_{26} = 0.3333$ falls well within $[0, 0.500]$, Shift 26 remains in statistical control despite the smaller sample size.

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7. Common PE Exam Pitfalls for Attribute Charts

1. Using an $np$-Chart with Variable Subgroup Sizes: An $np$-chart tracks the count of defectives, which directly scales with sample size. If an exam problem states that lot sizes fluctuate from 80 to 120 units, selecting an $np$-chart is an immediate disqualifier; you must use a $p$-chart.
2. Confusing Defect Counts with Defective Unit Counts: A part with 4 solder bridges is 1 defective unit, but contributes 4 defects. Watch the problem statement terminology carefully: "fraction nonconforming" or "defective items" dictates $p$ or $np$; "defects per unit" or "imperfections per roll" dictates $c$ or $u$.
3. Failing to Truncate Negative Lower Control Limits: When calculations yield $LCL = -0.024$, never select an answer option displaying a negative limit. The lower limit for attribute charts is bounded by zero ($LCL = 0$).
4. Forgetting $\sqrt{n}$ in $p$-Chart Standard Error: The standard deviation of the proportion is $\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$. Examinees occasionally forget to divide by $n$ inside the radical or erroneously multiply by $n$, confusing $\sigma_p$ with $\sigma_{np}$.