11.3 Control Charts for Attributes & Run Rules

11.3 Control Charts for Attributes & Run Rules

Classification of Attribute Control Charts

Not all quality inspections yield continuous dimensional measurements. In countless shop scenarios, parts are evaluated using Go/No-Go plug gages, visual standards, leak testing, or surface finish scratch patterns. These inspections produce attribute data—qualitative data that is counted rather than measured on a continuous numerical scale.

A foundational competency for the ASQ Certified Quality Inspector is correctly categorizing attribute data into one of two fundamental statistical families:

1. Nonconforming Units (Defective Units / Binary Pass-Fail):
   • An entire physical component is judged as either conforming or nonconforming (acceptable or rejected, pass or fail, go or no-go). Regardless of whether a part has one scratch or ten scratches, it is counted as one nonconforming unit.
   • Governing Distribution: Binomial Distribution, where each part represents an independent Bernoulli trial with a probability $p$ of being nonconforming.
   • Applicable Charts: $p$-chart (fraction nonconforming) and $np$-chart (number of nonconforming units).
2. Nonconformities (Defects / Flaw Counts):
   • An inspection evaluates the count of specific individual flaws or defects occurring on a unit of product or across an "area of opportunity." A single printed circuit board might contain 3 solder bridges, 1 missing resistor, and 2 polarity reversals; this represents 6 total nonconformities on a single unit.
   • Governing Distribution: Poisson Distribution, which models the independent occurrence of rare discrete events across a continuous continuum (time, area, length, or volume).
   • Applicable Charts: $c$-chart (count of nonconformities) and $u$-chart (nonconformities per unit).

The Attribute Chart Selection Matrix

+-----------------------------------------------------------------------------+
|                   ATTRIBUTE CONTROL CHART SELECTION FLOWCHART               |
+-----------------------------------------------------------------------------+
|                        What type of attribute data?                         |
|                                     |                                       |
|            +------------------------+------------------------+              |
|            v                                                 v              |
|   NONCONFORMING UNITS                                NONCONFORMITIES        |
|   (Defective Parts - Binomial)                       (Defects Count - Poisson) |
|            |                                                 |              |
|      Is sample size n                                  Is inspection area   |
|     constant or variable?                             constant or variable? |
|      /             \                                   /             \      |
|   Constant       Variable                           Constant       Variable |
|      |               |                                 |               |    |
|      v               v                                 v               v    |
|   np-Chart        p-Chart                           c-Chart         u-Chart |
|   (Counts)      (Proportions)                       (Counts)       (Per Unit)|
+-----------------------------------------------------------------------------+

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$p$-Charts and $np$-Charts (Binomial Distribution)

The $p$-Chart (Fraction / Proportion Nonconforming)

The $p$-chart monitors the ratio of nonconforming parts to the total number of parts inspected in each subgroup. It is the most versatile attribute chart because it accommodates both constant and varying subgroup sample sizes.

Mathematical Formulas for $p$-Chart

1. Subgroup Fraction Nonconforming ($p_i$): $p_i = \frac{np_i}{n_i} = \frac{\text{Number of Nonconforming Units in Subgroup } i}{\text{Total Number of Units Inspected in Subgroup } i}$
2. Center Line ($\bar{p}$): $\bar{p} = \frac{\sum_{i=1}^{k} np_i}{\sum_{i=1}^{k} n_i} = \frac{\text{Total Nonconforming Units Across All Subgroups}}{\text{Total Parts Inspected Across All Subgroups}}$
3. 3-Sigma Control Limits: $UCL_p = \bar{p} + 3\sqrt{\frac{\bar{p}(1 - \bar{p})}{n_i}}$ $LCL_p = \max\left(0, \; \bar{p} - 3\sqrt{\frac{\bar{p}(1 - \bar{p})}{n_i}}\right)$

Handling Variable Sample Sizes on $p$-Charts

When daily production quantities vary, the sample size $n_i$ changes from day to day. Notice that $n_i$ appears in the denominator of the standard error term: $\sigma_p = \sqrt{\bar{p}(1-\bar{p})/n_i}$.

• Impact on Limits: When the sample size $n_i$ is large, the standard error decreases, causing the control limits to narrow. When the sample size is small, the standard error increases, causing the control limits to widen (creating "stepped" control limits).
• The $\pm 25%$ Rule: If individual subgroup sample sizes do not vary by more than $\pm 25%$ from the average sample size $\bar{n}$, industrial standards permit calculating a single set of standardized control limits using $\bar{n} = (\sum n_i) / k$, simplifying shop-floor chart interpretation.

The $np$-Chart (Number of Nonconforming Units)

The $np$-chart tracks the absolute count of nonconforming parts rather than a percentage. Machine operators frequently prefer the $np$-chart because it eliminates daily fractional division—the operator simply counts the rejected parts and plots the integer.

Mandatory Condition and Formulas for $np$-Chart

CRITICAL RULE: The $np$-chart strictly requires a constant sample size ($n$) for all subgroups. If sample size varies, the $np$-chart cannot be used!

1. Center Line: $CL_{np} = n\bar{p}$
2. 3-Sigma Control Limits: $UCL_{np} = n\bar{p} + 3\sqrt{n\bar{p}(1 - \bar{p})}$ $LCL_{np} = \max\left(0, \; n\bar{p} - 3\sqrt{n\bar{p}(1 - \bar{p})}\right)$

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$c$-Charts and $u$-Charts (Poisson Distribution)

The $c$-Chart (Count of Nonconformities)

The $c$-chart monitors the total number of defects found on a single inspection unit or across a strictly defined, constant area of opportunity.

• Applications: Number of surface blemishes on a standardized $4 \times 8\text{ ft}$ sheet of aluminum, number of weld porosity voids per linear foot of pipe weld, or number of paint imperfections on an automotive door.
• Center Line: $CL_c = \bar{c} = \frac{\sum_{i=1}^{k} c_i}{k}$
• 3-Sigma Control Limits (since for a Poisson distribution, variance equals the mean, $\sigma^2 = \bar{c}$, so $\sigma = \sqrt{\bar{c}}$): $UCL_c = \bar{c} + 3\sqrt{\bar{c}}$ $LCL_c = \max\left(0, \; \bar{c} - 3\sqrt{\bar{c}}\right)$

The $u$-Chart (Nonconformities Per Unit)

When the area of opportunity, quantity of material, or sample size varies from inspection to inspection, the count of defects must be normalized. The $u$-chart monitors the average number of nonconformities per inspection unit ($u = c / n$).

• Applications: Inspecting varying square yardage of fabric rolls, auditing varying lot quantities of machined gears for burrs, or inspecting structural beams of differing lengths.
• Subgroup Rate ($u_i$) and Center Line ($\bar{u}$): $u_i = \frac{c_i}{n_i}, \qquad CL_u = \bar{u} = \frac{\sum_{i=1}^{k} c_i}{\sum_{i=1}^{k} n_i}$
• 3-Sigma Control Limits: $UCL_u = \bar{u} + 3\sqrt{\frac{\bar{u}}{n_i}}$ $LCL_u = \max\left(0, \; \bar{u} - 3\sqrt{\frac{\bar{u}}{n_i}}\right)$

The Zero Lower Limit Rule for Attributes Charts

A mathematical universal across all four attribute charts ($p, np, c, u$) is the lower limit truncation rule. When the historical defect rate is low, subtracting $3\sigma$ frequently results in a negative number (e.g., $LCL = -0.014$). Because a machine cannot produce a negative fraction of defective parts or negative flaw counts, any calculated negative Lower Control Limit must be set to zero ($LCL = 0$).

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Out-of-Control Decision Rules: WECO & Nelson Rules

Statistical Process Control is not merely about waiting for a point to fall outside the 3-sigma control limits. A process can remain entirely within control limits while undergoing severe non-random shifts, trends, or operator tampering. To detect these disturbances early, quality professionals divide the area between the center line and control limits into standardized statistical zones.

+-----------------------------------------------------------------------------+
|                        CONTROL CHART ZONES FOR RUN RULES                    |
+-----------------------------------------------------------------------------+
| UCL (+3σ) ----------------------------------------------------------------- |
|              ZONE A  (Between +2σ and +3σ) -> 2.14% Probability             |
| +2σ       ----------------------------------------------------------------- |
|              ZONE B  (Between +1σ and +2σ) -> 13.59% Probability            |
| +1σ       ----------------------------------------------------------------- |
|              ZONE C  (Between CL and +1σ)  -> 34.13% Probability            |
| CENTER LINE =============================================================== |
|              ZONE C  (Between CL and -1σ)  -> 34.13% Probability            |
| -1σ       ----------------------------------------------------------------- |
|              ZONE B  (Between -1σ and -2σ) -> 13.59% Probability            |
| -2σ       ----------------------------------------------------------------- |
|              ZONE A  (Between -2σ and -3σ) -> 2.14% Probability             |
| LCL (-3σ) ----------------------------------------------------------------- |
+-----------------------------------------------------------------------------+

The Western Electric Company (WECO) Rules

Developed at AT&T's Western Electric Hawthorne Works, these four classic rules identify non-random patterns that have a vanishingly small probability of occurring by chance:

1. Rule 1 (Extreme Point): 1 point falls beyond Zone A (outside either the $UCL$ or $LCL$).
   • Statistical Probability: $p \approx 0.0027$ ($0.27%$). Indicates an immediate gross disturbance (e.g., broken tool, sudden material mix-up).
2. Rule 2 (Two of Three Rule): 2 out of 3 consecutive points fall in Zone A or beyond on the same side of the center line.
   • Statistical Probability: $p \approx 0.0015$ ($0.15%$). Signals a rapid shift in the process mean or a large expansion in variability.
3. Rule 3 (Four of Five Rule): 4 out of 5 consecutive points fall in Zone B or beyond on the same side of the center line.
   • Statistical Probability: $p \approx 0.0028$ ($0.28%$). Detects a moderate, persistent process shift of approximately $1.5\sigma$.
4. Rule 4 (Run Above/Below Center Line): 8 or 9 consecutive points fall on one side of the center line.
   • Statistical Probability: $(0.5)^8 \approx 0.0039$ ($0.39%$). Proves that the process mean has shifted permanently away from the historical center line.

Nelson Supplementary Rules

In 1984, Lloyd S. Nelson expanded the detection framework to identify specific mechanical failure modes:

• Rule 5 (Trend): 6 consecutive points steadily increasing or steadily decreasing.
  • Shop Physical Meaning: Classic diagnostic signature of progressive cutting tool wear, grinding wheel breakdown, machine thermal expansion, or chemical reagent depletion.
• Rule 6 (Alternating / Systematic Over-Adjustment): 14 consecutive points alternating up and down (saw-tooth pattern).
  • Shop Physical Meaning: Hallmark signature of process tampering (an operator adjusting a machine offset after every single part) or alternating between two un-synchronized tool spindles, pallets, or cavity streams.
• Rule 7 (Stratification / Hugging Center Line): 15 consecutive points in Zone C (within $\pm 1\sigma$) on either side of the center line.
  • Shop Physical Meaning: Artificially low variation. This occurs when an inspector computes control limits using mixed, highly variable data, or when subgroups are formed by blending parts from different machine heads, causing averages to artificially collapse toward center.
• Rule 8 (Mixture / Avoiding Center Line): 8 consecutive points on both sides of center line with none falling in Zone C.
  • Shop Physical Meaning: Bimodal distribution caused by mixing parts from two distinct, widely separated production streams.

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Real Shop Inspection Scenarios & Common Exam Traps

• Real Shop Scenario — The Sheet Metal Stamping Tear Mystery: An inspector on an automotive stamping line monitors tear defects on stamped floor pans. Because the stamping press run rate varies from 400 to 900 pans per shift, the inspector sets up an $np$-chart and plots the count of torn stampings per shift. On Tuesday, the count jumps from 4 to 9 tears, triggering panic and a die teardown. Metrological Diagnosis: The quality engineer investigates and points out that on Monday, only 400 pans were stamped (tear rate $= 4 / 400 = 1.0%$). On Tuesday, the line ran continuously, stamping 900 pans (tear rate $= 9 / 900 = 1.0%$). The defect proportion was completely identical! The apparent spike was purely an artifact of using an $np$-chart with variable sample sizes. The inspector must use a $p$-chart when sample sizes fluctuate.

• Exam Trap: Confusing Nonconforming Units with Defects: Exam questions often describe an inspection task: "An inspector examines painted housings. If a housing has any paint run, it is rejected. The sample size is 50 housings per hour. Which chart should be used?" Candidates see "paint runs" and instinctively think $c$-chart. However, notice the wording: "If a housing has any paint run, it is rejected"—the housing is classified as a binary pass/fail unit! Because sample size is constant ($n = 50$) and the metric is defective units, an $np$-chart (or $p$-chart) is correct, NOT a $c$-chart.

• Exam Trap: Negative Lower Control Limit Truncation: If a calculated $LCL$ for a $p$-chart equals $-0.015$, never select an answer choice stating $LCL = -0.015$. The correct answer is universally $LCL = 0$.

• Exam Trap: WECO Zone A Rule Requirements: Be careful on test questions regarding WECO Rule 2. The rule is two out of three consecutive points in Zone A or beyond on the same side of the center line. Distractor options will state "three consecutive points in Zone A" or "two points in Zone A on opposite sides." Both conditions are required: 2 of 3, and on the same side.