9.3 Control Charts for Attributes Data

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.

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Attribute Data Classification: Defectives vs. Defects

Before selecting an attribute control chart, practitioners must distinguish between two fundamental categories of quality data:

1. 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: $p$-chart (fraction defective) and $np$-chart (number defective).
2. 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: $c$-chart (count of defects) and $u$-chart (defects per unit).

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$p$-Chart (Fraction Nonconforming)

The $p$-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 ($n_i$).

Statistical Foundation & Limit Derivations

Under the Binomial distribution with mean parameter $p$:

• Subgroup Fraction Nonconforming ($p_i$): $p_i = \frac{d_i}{n_i}$ (where $d_i$ is the count of defective units in subgroup $i$).
• Overall Average Fraction Nonconforming ($\bar{p}$):

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

• Standard Error of Proportion: $SE(p_i) = \sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}}$

Control Limits Formulas

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

Note: If subgroup sample size $n_i$ varies across days, control limits $UCL_p$ and $LCL_p$ will vary inversely with $\sqrt{n_i}$, resulting in "stepped" control limits. If $LCL_p$ calculates to a negative number, it is set to $0.0$.

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$np$-Chart (Number Nonconforming)

The $np$-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 ($0, 1, 2, 3 \dots$).

Operational Constraint & Formulas

The $np$-chart strictly requires a constant sample size ($n$) across all subgroups.

1. Center Line (CL): $CL_{np} = n\bar{p} = \frac{\sum_{i=1}^{k} d_i}{k}$
2. Control Limits: $$UCL_{np} = n\bar{p} + 3\sqrt{n\bar{p}(1-\bar{p})}$$ $$LCL_{np} = \max\left(0, \, n\bar{p} - 3\sqrt{n\bar{p}(1-\bar{p})}\right)$$

If subgroup sample sizes vary, an $np$-chart cannot be used, and the practitioner must default to a $p$-chart.

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$c$-Chart (Defects per Unit - Constant Sample Size)

The $c$-chart monitors the total count of individual defects ($c$) found across a sample unit with a strictly constant area of opportunity (e.g., defects per $100 \text{ m}^2$ of glass sheet, errors per $50$-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 ($\sigma^2 = \mu = \bar{c}$).

1. Average Defect Count ($\bar{c}$):

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

1. Standard Error: $SE(c) = \sqrt{\bar{c}}$
2. Control Limits: $$UCL_c = \bar{c} + 3\sqrt{\bar{c}}$$ $$LCL_c = \max\left(0, \, \bar{c} - 3\sqrt{\bar{c}}\right)$$

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$u$-Chart (Defects per Unit - Variable Sample Size)

When the area of opportunity or subgroup sample size ($n_i$) varies from sample to sample, the $u$-chart monitors the rate of defects per unit ($u_i = c_i / n_i$).

Statistical Foundation & Limit Derivations

1. Subgroup Defect Rate ($u_i$): $u_i = \frac{c_i}{n_i}$
2. Overall Average Defect Rate ($\bar{u}$):

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

1. Standard Error: $SE(u_i) = \sqrt{\frac{\bar{u}}{n_i}}$
2. Control Limits: $$UCL_u = \bar{u} + 3 \sqrt{\frac{\bar{u}}{n_i}}$$ $$LCL_u = \max\left(0, \, \bar{u} - 3 \sqrt{\frac{\bar{u}}{n_i}}\right)$$

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Attributes Control Chart Selection Matrix

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Worked Numerical Calculation Example: $p$-Chart

A call center audits $k = 20$ daily subgroups of $n = 100$ customer service calls each. Over the 20 days, a total of $\sum d_i = 120$ defective calls (compliance errors) are identified.

1. Overall Fraction Nonconforming: $\bar{p} = \frac{120}{20 \times 100} = \frac{120}{2000} = 0.060$ ($6.0\%$).
2. Standard Error: $SE(p) = \sqrt{\frac{0.060(1 - 0.060)}{100}} = \sqrt{\frac{0.060 \times 0.940}{100}} = \sqrt{0.000564} = 0.02375$.
3. Upper Control Limit: $UCL_p = 0.060 + 3(0.02375) = 0.060 + 0.07125 = 0.13125$ ($13.13\%$).
4. Lower Control Limit: $LCL_p = \max(0, 0.060 - 0.07125) = 0.0$.

• Interpretation: The process is in control as long as daily compliance error rates remain between $0.0\%$ and $13.13\%$. Any day exceeding $13.13\%$ represents a special cause out-of-control signal requiring root cause investigation.

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Detailed Mathematical Comparisons of Attribute Charts

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Worked Attribute Chart Calculation: $u$-Chart

A quality team inspects printed circuit boards for solder defects across variable daily production lots.

• Day 1: $n_1 = 50$ boards, $c_1 = 15$ defects $\implies u_1 = 15/50 = 0.30$
• Day 2: $n_2 = 100$ boards, $c_2 = 22$ defects $\implies u_2 = 22/100 = 0.22$
• Day 3: $n_3 = 80$ boards, $c_3 = 18$ defects $\implies u_3 = 18/80 = 0.225$

1. Overall Average Defect Rate ($\bar{u}$): $$\bar{u} = \frac{\sum c_i}{\sum n_i} = \frac{15 + 22 + 18}{50 + 100 + 80} = \frac{55}{230} = 0.2391\text{ defects/board}$$
2. Calculate Control Limits for Day 1 ($n_1 = 50$): $$SE(u_1) = \sqrt{\frac{0.2391}{50}} = \sqrt{0.004782} = 0.06915$$ $$UCL_{u1} = 0.2391 + 3(0.06915) = 0.2391 + 0.2074 = 0.4465$$ $$LCL_{u1} = \max(0, 0.2391 - 0.2074) = 0.0317$$