4.3 Attribute Control Charts

4.3 Attribute Control Charts

Attribute control charts evaluate discrete qualitative quality characteristics—such as go/no-go gauging, pass/fail functional tests, surface visual blemishes, or clerical errors. Attribute charts monitor process stability when quantitative variable measurements are either impossible or cost-prohibitive to obtain.

Attribute Classification: Defectives vs. Defects

Selecting the correct attribute chart requires distinguishing between two distinct statistical distributions:

1. Nonconforming Unit (Defective / Binomial Distribution):

   • Refers to an entire product or assembly that fails to meet specification requirements (e.g., a cracked casting, a leaking container, an non-functional PCB).
   • Outcome is binary: conforming (0) or nonconforming (1).
   • Governed by the Binomial Distribution.
   • Associated Control Charts: p-chart and np-chart.

2. Nonconformity (Defect / Poisson Distribution):

   • Refers to an individual flaw or failure instance on an item (e.g., scratches on an appliance door, solder bridges, paint runs, invoice typos).
   • A single item can contain multiple nonconformities.
   • Governed by the Poisson Distribution (counting independent occurrences over a continuous area of opportunity).
   • Associated Control Charts: c-chart and u-chart.

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Attribute Control Chart Selection Framework

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Mathematical Formulas and Worked Calculations

1. p-Chart (Fraction Nonconforming)

Monitors the proportion (p) of defective units in samples of size $n_i$. When subgroup size $n_i$ varies, control limits must be calculated individually for each sample size.

$$\text{Centerline: } \bar{p} = \frac{\text{Total Defective Units}}{\text{Total Inspected Units}} = \frac{\sum_{i=1}^k d_i}{\sum_{i=1}^k n_i}$$ $$UCL_p = \bar{p} + 3 \sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}}, \quad LCL_p = \max\left(0, \, \bar{p} - 3 \sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}}\right)$$

Worked Example: p-Chart with Constant Sample Size

An electronics plant inspects (k = 20) lots of (n = 500) circuit boards each. A total of (d = 150) defective boards are detected across all lots.

1. Calculate Average Proportion Defective ($\bar{p}$): $$\bar{p} = \frac{150}{20 \times 500} = \frac{150}{10,000} = 0.0150$$
2. Calculate Control Limits: $$\text{SE}_p = \sqrt{\frac{0.0150 \times (1 - 0.0150)}{500}} = \sqrt{\frac{0.014775}{500}} = \sqrt{0.00002955} = 0.005436$$ $$UCL_p = 0.0150 + 3(0.005436) = 0.0150 + 0.01631 = 0.03131 \text{ (or 3.13\%)}$$ $$LCL_p = 0.0150 - 0.01631 = -0.00131 \rightarrow \mathbf{0.0000} \text{ (truncated to zero)}$$

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2. np-Chart (Number Nonconforming)

Monitors actual count of defective items. Subgroup size (n) must be strictly constant across all samples.

$$\text{Centerline: } \bar{np} = \frac{\sum d_i}{k}$$ $$UCL_{np} = \bar{np} + 3 \sqrt{\bar{np}(1-\bar{p})}, \quad LCL_{np} = \max\left(0, \, \bar{np} - 3 \sqrt{\bar{np}(1-\bar{p})}\right)$$

Worked Example: np-Chart

Using data from above ((n = 500), $\bar{p} = 0.0150$, $\bar{np} = 500 \times 0.0150 = 7.50$ defectives per lot):

$$\text{SE}_{np} = \sqrt{7.50 \times (1 - 0.0150)} = \sqrt{7.3875} = 2.718$$ $$UCL_{np} = 7.50 + 3(2.718) = 7.50 + 8.154 = 15.654 \text{ defective units}$$ $$LCL_{np} = 7.50 - 8.154 = -0.654 \rightarrow \mathbf{0} \text{ defective units}$$

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3. c-Chart (Count of Nonconformities)

Monitors total defect counts (c) in a fixed inspection unit or area of opportunity (e.g., defects per single vehicle body panel).

$$\text{Centerline: } \bar{c} = \frac{\sum_{i=1}^k c_i}{k}$$ $$UCL_c = \bar{c} + 3 \sqrt{\bar{c}}, \quad LCL_c = \max\left(0, \, \bar{c} - 3 \sqrt{\bar{c}}\right)$$

Worked Example: c-Chart

A textile mill inspects (k = 25) standard fabric rolls (1 roll = 1 unit). Total defects observed across all rolls is $\sum c_i = 100$.

1. Centerline ($\bar{c}$): $$\bar{c} = \frac{100}{25} = 4.00 \text{ defects per roll}$$
2. Control Limits: $$UCL_c = 4.00 + 3\sqrt{4.00} = 4.00 + 3(2.00) = 10.00 \text{ defects}$$ $$LCL_c = 4.00 - 3(2.00) = 4.00 - 6.00 = -2.00 \rightarrow \mathbf{0} \text{ defects}$$

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4. u-Chart (Nonconformities Per Unit)

Monitors average defects per unit (u) when sample size $n_i$ (number of inspection units per subgroup) varies.

$$\text{Centerline: } \bar{u} = \frac{\text{Total Defects}}{\text{Total Inspected Units}} = \frac{\sum_{i=1}^k c_i}{\sum_{i=1}^k n_i}$$ $$UCL_u = \bar{u} + 3 \sqrt{\frac{\bar{u}}{n_i}}, \quad LCL_u = \max\left(0, \, \bar{u} - 3 \sqrt{\frac{\bar{u}}{n_i}}\right)$$

Worked Example: u-Chart

An automotive plant inspects painted vehicle panels over (k = 10) shifts. Total inspected panels (= 500), total paint defects (= 250).

1. Average Defects per Unit ($\bar{u}$): $$\bar{u} = \frac{250}{500} = 0.500 \text{ defects per panel}$$
2. Control Limits for a shift where (n = 50) panels were inspected: $$UCL_u = 0.500 + 3 \sqrt{\frac{0.500}{50}} = 0.500 + 3\sqrt{0.010} = 0.500 + 3(0.100) = 0.800 \text{ defects/panel}$$ $$LCL_u = 0.500 - 0.300 = 0.200 \text{ defects/panel}$$