6.2 Control Charts for Nonconformities: c and u Charts

6.2 Control Charts for Nonconformities: c and u Charts

Nonconformities (Defects) vs. Nonconforming Units (Defectives)

In quality assurance and testing, confusion between a nonconformity and a nonconforming unit is one of the most frequent sources of error on the ASQ CQT examination. To select and calculate control charts correctly, a technician must maintain an unshakeable understanding of this distinction:

• Nonconformity (Defect): A single nonfulfillment of an intended usage requirement or engineering specification. Nonconformities are countable physical occurrences of flaws on a product. Examples include a scratch on an automotive windshield, a cold solder joint on a printed circuit board (PCB), a pinhole in an electroplated coating, a tear in a composite fabric prepreg, or an air void in a structural adhesive bond.
• Nonconforming Unit (Defective Item): An entire unit of product that possesses one or more nonconformities, rendering it noncompliant with acceptance requirements.

Illustrative Industrial Analogy

Consider an aircraft composite wing skin undergoing non-destructive ultrasonic testing (NDT):

• The technician identifies 3 micro-voids, 1 dry fiber spot, and 2 surface scratches.
• The wing skin contains a total of 6 nonconformities (defects).
• However, because the wing skin has failed acceptance criteria, it represents exactly 1 nonconforming unit (defective).

If the quality objective is to track the proportion of rejected wing skins, the technician uses a $p$ chart. If the objective is to monitor the count and density of specific physical flaws across the composite skin to diagnose autoclave pressure and resin infusion parameters, the technician must use a $c$ chart or $u$ chart.

+-------------------------------------------------------------------------+
|                    NONCONFORMING UNIT (1 Defective Part)                |
|                                                                         |
|   [Scratch 1]          [Solder Bridge]          [Scratch 2]             |
|   (Defect #1)            (Defect #2)            (Defect #3)             |
|                                                                         |
|              [Surface Void]              [Pinhole]                      |
|                (Defect #4)              (Defect #5)                     |
|                                                                         |
|   TOTAL DEFECTS (Nonconformities) = 5                                   |
|   TOTAL DEFECTIVE UNITS (Nonconforming Units) = 1                       |
+-------------------------------------------------------------------------+

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The Poisson Distribution: Foundation for Defect Counts

While nonconforming units are governed by the binomial distribution, counts of nonconformities are modeled by the Poisson probability distribution. The Poisson distribution models the occurrence of discrete, rare events distributed randomly across a continuous continuum of space, area, volume, length, or time.

Mathematical Properties of the Poisson Distribution

A process follows a Poisson distribution when:

1. Nonconformities occur at random and independently across the inspection unit.
2. The probability of a nonconformity occurring in a very small segment is proportional to the size of that segment.
3. The probability of two or more nonconformities occurring at the exact same point is virtually zero.
4. The total opportunity for nonconformities is infinitely large, while the actual likelihood of a nonconformity occurring at any single point is very small.

The Defining Mathematical Hallmark

The defining mathematical characteristic of the Poisson distribution is that its variance equals its mean:

$$\text{Mean: } \mu = \lambda$$ $$\text{Variance: } \sigma^2 = \lambda = \mu$$ $$\text{Standard Deviation: } \sigma = \sqrt{\mu} = \sqrt{\lambda}$$

Because the variance is directly determined by the mean, calculating control limits for Poisson charts ($c$ and $u$) requires no independent estimate of process dispersion (unlike variables charts which require $\bar{R}$ or $s$). The standard deviation is derived entirely from the baseline defect average!

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The $c$ Chart: Nonconformities per Constant Inspection Unit

The $c$ chart monitors the total count of nonconformities ($c$) observed in an inspection unit of constant size.

Defining the Inspection Unit

An inspection unit is an established quantity of product against which nonconformities are tallied. It can represent:

• A single discrete item: One circuit board, one machined cylinder block, one appliance door.
• A defined continuous physical measure: Exactly 100 square meters of textile fabric, 50 feet of extruded rubber seal, 1,000 feet of insulated wire, or 10 linear feet of structural weld.
• A fixed collection of units: A standard tote of exactly 50 stamped medical brackets.

Centerline and Control Limits for the $c$ Chart

Given $k$ preliminary subgroups, each consisting of one constant inspection unit, where $c_i$ is the count of nonconformities in subgroup $i$:

• Centerline: $\bar{c} = \frac{\sum_{i=1}^k c_i}{k} = \frac{\text{Total nonconformities across all subgroups}}{\text{Total number of inspection subgroups}}$
• Standard Deviation: $\sigma_c = \sqrt{\bar{c}}$
• Upper Control Limit: $UCL_c = \bar{c} + 3\sqrt{\bar{c}}$
• Lower Control Limit: $LCL_c = \bar{c} - 3\sqrt{\bar{c}}$

The Zero Truncation Rule for $LCL_c$

Because you cannot count negative defects, if the calculated lower limit is less than zero ($LCL_c < 0$), set $LCL_c = 0$:

$$\text{If } LCL_c < 0 \implies LCL_c = 0$$

For an $LCL_c$ to be greater than zero, the process average must satisfy $\bar{c} - 3\sqrt{\bar{c}} > 0$, which mathematically requires $\bar{c} > 9.0$ nonconformities per unit. If $\bar{c} \le 9.0$, the calculated $LCL$ will always be zero or negative, truncating to $LCL = 0$.

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The $u$ Chart: Nonconformities per Unit (Variable Inspection Size)

The $u$ chart monitors the average number of nonconformities per unit ($u$). It is designed for situations where the inspection unit size varies from subgroup to subgroup, although it can also be applied when the unit size is constant.

Subgroup Metric: Defect Rate per Unit ($u_i$)

In each subgroup $i$, the technician inspects $n_i$ inspection units and counts a total of $c_i$ nonconformities. The plotted statistic is the defect density:

$$\nu_i = \frac{c_i}{n_i} = \frac{\text{Number of nonconformities in subgroup } i}{\text{Number of inspection units in subgroup } i}$$

Note: Here, $n_i$ does not have to be an integer. For example, if the standard inspection unit is defined as $100\text{ m}^2$ of rolled steel, inspecting a roll of $250\text{ m}^2$ represents $n_i = 2.5$ inspection units.

Centerline Calculation for the $u$ Chart

The centerline $\bar{u}$ represents the total defects observed divided by the total inspection units examined across all $k$ subgroups:

$$\bar{u} = \frac{\sum_{i=1}^k c_i}{\sum_{i=1}^k n_i} = \frac{\text{Total nonconformities across all subgroups}}{\text{Total inspection units inspected across all subgroups}}$$

Control Limits for Variable Subgroup Sizes ($n_i$)

Because the standard error of $u$ is $\sigma_{u_i} = \sqrt{\frac{\bar{u}}{n_i}}$, control limits must be calculated for each specific subgroup based on its inspection size $n_i$:

$$UCL_{u_i} = \bar{u} + 3\sqrt{\frac{\bar{u}}{n_i}}$$ $$LCL_{u_i} = \bar{u} - 3\sqrt{\frac{\bar{u}}{n_i}} \quad \text{(if } LCL < 0 \implies LCL = 0\text{)}$$

Dynamic Behavior of $u$ Chart Limits

• When inspecting a large sample (e.g., $n_i = 10$ units), the denominator in $\sqrt{\bar{u}/n_i}$ is large, making $\sigma_{u_i}$ small. The control limits narrow significantly around the centerline.
• When inspecting a small sample (e.g., $n_i = 1$ unit), the standard error is large, and the control limits expand widely.

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The Comprehensive 4-Quadrant Attribute Chart Selection Guide

Choosing the correct attribute control chart is one of the most vital competencies for an ASQ Certified Quality Technician. The selection process follows a simple, logical two-question decision framework:

1. What is being evaluated?
   • A nonconforming unit (binary: good vs. bad, conforming vs. nonconforming)? $\implies$ Use $p$ or $np$ chart.
   • A count of nonconformities (defects, flaws, blemishes per unit)? $\implies$ Use $c$ or $u$ chart.
2. Is the subgroup sample size constant or variable?
   • If nonconforming units: Constant $n \implies \mathbf{np}$ (or $p$); Variable $n \implies \mathbf{p}$ chart only.
   • If nonconformities: Constant inspection unit $\implies \mathbf{c}$ (or $u$); Variable inspection unit $\implies \mathbf{u}$ chart only.

The 4-Quadrant Attribute Chart Matrix

                              QUALITY CHARACTERISTIC EVALUATED
                 
                     NONCONFORMING UNITS                 NONCONFORMITIES
                    (Defective Parts: Binary)            (Defects: Counts)
                 +-----------------------------+-----------------------------+
                 |                             |                             |
    CONSTANT     |          np CHART           |           c CHART           |
    SUBGROUP     |  Tracks number of rejects   |   Tracks count of defects   |
    SIZE (n)     |  Binomial distribution     |   Poisson distribution      |
                 |  CL = n * p_bar             |   CL = c_bar                |
  S              |  Limits = n*p_bar ± 3*sigma |   Limits = c_bar ± 3*sqrt(c)||
  U              +-----------------------------+-----------------------------+
  B              |                             |                             |
  G VARIABLE     |           p CHART           |           u CHART           |
  R SUBGROUP     |  Tracks fraction nonconform |   Tracks defect rate/unit   |
  O SIZE (n_i)   |  Binomial distribution     |   Poisson distribution      |
  U              |  CL = p_bar                 |   CL = u_bar                |
  P              |  Limits = p_bar ± 3*sigma_i |   Limits = u_bar ± 3*sigma_i||
                 +-----------------------------+-----------------------------+

Practical Decision Tree for Quality Technicians

                              [Start Inspection Plan]
                                         |
                   Are you counting defects or classifying units?
                                         |
               +-------------------------+-------------------------+
               |                                                   |
       [Classifying Units]                                  [Counting Defects]
      (Conforming / Nonconforming)                        (Flaws / Nonconformities)
               |                                                   |
    Is sample size (n) constant?                        Is inspection unit constant?
         /           \                                       /           \
       YES            NO                                   YES            NO
        |              |                                    |              |
    [np Chart]     [p Chart]                            [c Chart]      [u Chart]
    (or p Chart)                                        (or u Chart)

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Step-by-Step Worked Numerical Examples

Worked Example 1: Establishing a $c$ Chart for PCB Wave Solder Defects

Scenario: An electronics assembly technician inspects high-reliability printed circuit boards after wave soldering. The inspection unit is defined as one completed circuit board ($n = 1$). Across 25 consecutively inspected boards ($k = 25$), the technician records a total of 175 soldering defects (bridges, insufficient solder, voids), yielding $\sum c = 175$.

1. Calculate the Centerline ($\bar{c}$): $$\bar{c} = \frac{\sum c}{k} = \frac{175}{25} = 7.00 \text{ defects per board}$$
2. Calculate the Standard Deviation ($\sigma_c$): $$\sigma_c = \sqrt{\bar{c}} = \sqrt{7.00} \approx 2.6458$$
3. Calculate the Three-Sigma Spread ($3\sigma_c$): $$3\sigma_c = 3 \times 2.64575 = 7.9373 \approx 7.94$$
4. Calculate Upper and Lower Control Limits: $$UCL_c = \bar{c} + 3\sqrt{\bar{c}} = 7.00 + 7.937 = 14.937 \approx 14.94 \text{ defects}$$ $$LCL_c = \bar{c} - 3\sqrt{\bar{c}} = 7.00 - 7.937 = -0.937$$
5. Apply Truncation Rule: Since $LCL_c < 0$, set $LCL_c = 0$.

Interpretation: If an inspected board exhibits 15 or more solder defects, it breaches the UCL, signaling an assignable cause (e.g., flux contamination, preheat temperature drop, conveyor speed surge). If a board has 0 defects, it plots at the LCL, which is common cause performance for this baseline.

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Worked Example 2: Establishing a $u$ Chart for Variable Area Coating Inspection

Scenario: A technician inspects coils of galvanized sheet metal for surface coating blemishes (dross pimples, blisters, uncoated spots). Because customer orders specify different roll lengths, the inspected area varies by shift. The standard inspection unit is defined as $100\text{ m}^2$ of surface area. Over 20 production shifts ($k = 20$), a total area of $15,000\text{ m}^2$ is inspected, representing $\sum n = 15,000 / 100 = 150.0$ inspection units. A total of 360 blemishes are counted across all shifts ($\sum c = 360$).

Step A: Calculate Centerline ($\bar{u}$)

$$\bar{u} = \frac{\sum c}{\sum n} = \frac{360}{150.0} = 2.40 \text{ blemishes per } 100\text{ m}^2 \text{ unit}$$

Step B: Calculate Individual Limits for Shift 1 (Large Lot)

On Shift 1, the mill produces $600\text{ m}^2$ of steel ($n_1 = 6.0$ units) and the inspector finds 18 blemishes ($c_1 = 18$).

1. Subgroup Metric ($u_1$): $$u_1 = \frac{c_1}{n_1} = \frac{18}{6.0} = 3.00 \text{ blemishes/unit}$$
2. Standard Error ($\sigma_{u1}$): $$\sigma_{u1} = \sqrt{\frac{\bar{u}}{n_1}} = \sqrt{\frac{2.40}{6.0}} = \sqrt{0.40} \approx 0.6325$$
3. Control Limits for Shift 1: $$UCL_{u1} = 2.40 + 3(0.6325) = 2.40 + 1.897 = 4.297 \approx 4.30 \text{ blemishes/unit}$$ $$LCL_{u1} = 2.40 - 3(0.6325) = 2.40 - 1.897 = 0.503 \approx 0.50 \text{ blemishes/unit}$$

Evaluation: Since $u_1 = 3.00$ falls between $0.50$ and $4.30$, Shift 1 is in statistical control.

Step C: Calculate Individual Limits for Shift 2 (Small Lot)

On Shift 2, a short run of only $150\text{ m}^2$ is produced ($n_2 = 1.5$ units) and the inspector finds 8 blemishes ($c_2 = 8$).

1. Subgroup Metric ($u_2$): $$u_2 = \frac{c_2}{n_2} = \frac{8}{1.5} = 5.333 \text{ blemishes/unit}$$
2. Standard Error ($\sigma_{u2}$): $$\sigma_{u2} = \sqrt{\frac{\bar{u}}{n_2}} = \sqrt{\frac{2.40}{1.5}} = \sqrt{1.60} = 1.2649$$
3. Control Limits for Shift 2: $$UCL_{u2} = 2.40 + 3(1.2649) = 2.40 + 3.795 = 6.195 \approx 6.20 \text{ blemishes/unit}$$ $$LCL_{u2} = 2.40 - 3.795 = -1.395 \implies LCL_{u2} = 0$$

Evaluation: Even though the defect rate on Shift 2 jumped to $5.33$ blemishes/unit (nearly double Shift 1), the point is in control because the smaller sample size produces wider limits ($UCL = 6.20$). The apparent spike is fully explainable by random common cause sampling variation!

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Technician Inspection Scenarios & Common Exam Traps

Real-World Shop Scenario: Automotive Primer Line Inspection

A technician inspecting automotive door panels for paint blemishes notes that some doors have 0 flaws, while others have 4 or 5 paint sags or dirt specs. An inexperienced technician mistakenly logs the data on an $np$ chart, entering the door as "defective" if it has $\ge 1$ flaw. This practice throws away vital quality information: a door with 1 minor spec requires 10 seconds of wet-sanding, whereas a door with 5 severe paint sags requires stripping and complete re-priming. Transitioning to a $c$ chart allows the engineering team to monitor total defect count, detect spray nozzle clogging, and correlate dirt counts directly with paint booth air filtration cycles.

Common Exam Traps for CQT Candidates

• Exam Trap 1: Confusing Nonconformities with Nonconforming Units: If an exam question asks which chart to use when "recording the number of scratches on a polished shaft," select a $c$ or $u$ chart. If it asks for "the number of shafts rejected for surface roughness," select an $np$ or $p$ chart.
• Exam Trap 2: Using a $c$ Chart with Fluctuating Inspection Area: If the inspection area, length, or sample quantity changes from batch to batch, a $c$ chart is invalid; the $u$ chart must be used.
• Exam Trap 3: Calculating $\bar{u}$ as the Simple Mean of $u_i$ Rates: Always calculate $\bar{u} = \frac{\sum c}{\sum n}$. Never average the subgroup $u_i$ rates directly unless all $n_i$ are identical.
• Exam Trap 4: Assuming Poisson Charts Require Range Data: Never look for $A_2$, $D_4$, or standard deviation tables when calculating $c$ or $u$ charts. The standard deviation is simply $\sqrt{\bar{c}}$ or $\sqrt{\bar{u}/n_i}$.