6.1 Control Charts for Nonconforming Units: p and np Charts

6.1 Control Charts for Nonconforming Units: p and np Charts

Introduction to Attribute Control Charts

In statistical process control (SPC), quality characteristics are divided into two fundamental data categories: variables data and attributes data. While variables data involve continuous, quantitative measurements along a continuous scale (such as shaft diameters in millimeters, tensile strength in megapascals, or flange thickness in inches), attributes data involve qualitative classifications based on discrete criteria. Attributes inspection answers binary questions: Does the part meet the blueprint requirement or not? Is the unit conforming or nonconforming? Does the thread gage engage or bind?

As an ASQ Certified Quality Technician (CQT), you will encounter attribute inspection daily when operating go/no-go plug gages, optical profile sensors, visual surface inspection stations, and functional electrical continuity testers. Attribute control charts provide an indispensable statistical mechanism for monitoring process stability, detecting assignable causes of variation, and verifying whether quality improvements have successfully shifted baseline defect rates.

Nonconforming Units vs. Nonconformities: The Foundational Distinction

A critical conceptual distinction tested extensively on the ASQ CQT exam is the difference between a nonconforming unit and a nonconformity:

• Nonconforming Unit (Defective Part): An entire unit of product that fails to meet one or more specified requirements, causing the item to be classified as unacceptable. A part is either conforming (good) or nonconforming (bad)—a binary distinction. For example, a stamped bracket whose mounting hole is drilled undersize is categorized as one nonconforming unit.
• Nonconformity (Defect): A specific instance of a flaw, blemish, or departure from a quality requirement. A single product unit may contain multiple nonconformities (for example, a single painted enclosure might exhibit two scratches, one paint run, and three pinholes).

Control charts designed for nonconforming units (the focus of this section) are the $p$ chart (proportion nonconforming) and the $np$ chart (number nonconforming). Control charts designed for nonconformities are the $c$ chart and $u$ chart, detailed in Section 6.2.

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The Binomial Distribution: The Statistical Engine

Control charts for nonconforming units are mathematically founded upon the Binomial probability distribution. The binomial model applies when a process satisfies four specific statistical conditions:

1. The production or inspection process consists of a sequence of $n$ distinct trials (parts inspected).
2. Each trial results in only one of two mutually exclusive outcomes: conforming or nonconforming (success or failure).
3. The true probability of producing a nonconforming unit, denoted by $p$, remains constant from trial to trial.
4. All trials are statistically independent (the outcome of inspecting one part does not influence the outcome of inspecting any subsequent part).

Statistical Parameters of the Binomial Distribution

For a binomial distribution with sample size $n$ and fraction nonconforming $p$:

• Mean number of nonconforming units: $\mu = np$
• Variance of nonconforming counts: $\sigma^2 = np(1-p)$
• Standard deviation of nonconforming counts: $\sigma = \sqrt{np(1-p)}$

When evaluating the proportion (fraction) nonconforming, defined as the number of nonconforming items $x$ divided by sample size $n$ ($p = x/n$):

• Mean proportion nonconforming: $\mu_p = p$
• Variance of proportion nonconforming: $\sigma_p^2 = \frac{p(1-p)}{n}$
• Standard error of proportion nonconforming: $\sigma_p = \sqrt{\frac{p(1-p)}{n}}$

Normal Approximation to the Binomial

Because control charts utilize symmetric three-sigma ($\pm 3\sigma$) limits derived from normal distribution theory, the binomial distribution must be sufficiently symmetric for the control limits to be statistically valid. By rule of thumb, the normal approximation is acceptable when:

$$np \ge 5 \quad \text{and} \quad n(1-p) \ge 5$$

When the defect rate $p$ is very small (such as $p = 0.005$ in modern automated assembly), the subgroup sample size $n$ must be substantially larger (e.g., $n \ge 1,000$) to satisfy this condition and prevent severe limit distortion.

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The $p$ Chart: Proportion Nonconforming

The $p$ chart tracks the proportion (or percentage) of nonconforming units within a subgroup sample. The calculated subgroup proportion is:

$$p_i = \frac{np_i}{n_i} = \frac{\text{Number of nonconforming units in subgroup } i}{\text{Total number of units inspected in subgroup } i}$$

Primary Advantages of the $p$ Chart

• Flexibility with Sample Size: The $p$ chart can be utilized whether the subgroup sample size is constant across all inspection periods or varies from subgroup to subgroup (such as 100% sorting of daily production lots of fluctuating quantities).
• Normalized Metric: Because it reports a decimal fraction or percentage, performance can be directly compared across different production lines, work shifts, or facility divisions regardless of lot size differences.

Centerline Calculation

To establish baseline control limits, a technician typically collects preliminary data across $k$ rational subgroups (conventionally $k \ge 20$ to $25$ subgroups). The centerline of the $p$ chart, denoted as $\bar{p}$, represents the weighted average fraction nonconforming:

$$\bar{p} = \frac{\sum_{i=1}^k np_i}{\sum_{i=1}^k n_i} = \frac{\text{Total nonconforming units across all subgroups}}{\text{Total units inspected across all subgroups}}$$

Control Limit Formulas for Constant Sample Size ($n$)

When all subgroups contain an identical number of units $n$, the standard error $\sigma_p$ is constant across all subgroups:

$$\text{Standard Error: } \sigma_p = \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$$ $$\text{Upper Control Limit (UCL): } UCL_p = \bar{p} + 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$$ $$\text{Lower Control Limit (LCL): } LCL_p = \bar{p} - 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$$

The Zero Truncation Rule for LCL

Because a proportion cannot physically be less than zero, if the mathematical calculation yields a negative value for the lower control limit ($LCL_p < 0$), the technician must set the lower control limit to zero:

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

On an operational control chart, when $LCL = 0$, a subgroup plotting at zero nonconforming units indicates perfect performance, not an out-of-control condition. However, it also means the chart cannot detect statistically significant process improvements at the lower boundary unless subgroup sample sizes are enlarged.

Handling Variable Subgroup Sample Sizes ($n_i$)

In real-world inspection, subgroup sizes frequently fluctuate due to variable daily output, batch sizes, or 100% receiving inspection lots. When $n_i$ changes from subgroup to subgroup, the standard error changes inversely with $\sqrt{n_i}$. Quality technicians use three distinct methods to handle variable sample sizes:

1. Individual Exact Control Limits Method (Standard): Exact control limits are calculated individually for each subgroup based on its specific sample size $n_i$:

   $$UCL_{p_i} = \bar{p} + 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}}, \qquad LCL_{p_i} = \max\left(0, \bar{p} - 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}}\right)$$
   • When $n_i$ is larger than average, the control limits narrow (tighten).
   • When $n_i$ is smaller than average, the control limits widen (expand).
   • Visual Appearance: The control limits appear stepped or jagged across the chart.

2. Average Sample Size Method ($\pm 25\%$ Rule): To avoid drawing jagged limits, if individual subgroup sample sizes $n_i$ do not vary by more than $\pm 25\%$ from the average sample size $\bar{n}$ (where $\bar{n} = \frac{\sum n_i}{k}$), a single set of constant control limits may be established using $\bar{n}$:

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

   Important Operational Protocol: If any subgroup proportion plots near the average limit line (borderline points), the technician must calculate the exact individual control limit for that specific subgroup to verify whether it is truly in or out of control.

3. Standardized $p$ Chart ($Z$ Chart): Each subgroup proportion is converted into a standardized standard normal deviate ($Z$-score):

   $$Z_i = \frac{p_i - \bar{p}}{\sigma_{p_i}} = \frac{p_i - \bar{p}}{\sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}}}$$

   The standardized chart plots $Z_i$ with a constant Centerline $CL = 0$, an Upper Control Limit $UCL = +3$, and a Lower Control Limit $LCL = -3$.

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The $np$ Chart: Number of Nonconforming Units

The $np$ chart tracks the actual count (integer number) of nonconforming units within each subgroup, rather than a calculated proportion.

The Strict Requirement of the $np$ Chart

Why Shop Floors Prefer the $np$ Chart

• Operator Ease: Shop-floor technicians and machine operators do not need to perform division or handle decimal fractions. If an inspector checks 200 stamped brackets and finds 8 defective parts, they directly plot the number 8.
• Reduced Arithmetic Errors: Eliminating intermediate division reduces calculation mistakes during manual plotting.

Centerline and Control Limits for the $np$ Chart

Given $k$ subgroups of constant sample size $n$:

• Centerline: $CL = n\bar{p} = \frac{\sum_{i=1}^k np_i}{k}$
• Standard Deviation: $\sigma_{np} = \sqrt{n\bar{p}(1-\bar{p})}$
• Upper Control Limit: $UCL_{np} = n\bar{p} + 3\sqrt{n\bar{p}(1-\bar{p})}$
• Lower Control Limit: $LCL_{np} = n\bar{p} - 3\sqrt{n\bar{p}(1-\bar{p})}$ (If $LCL_{np} < 0 \implies LCL_{np} = 0$)

Notice the direct mathematical bridge between the two charts:

$$UCL_{np} = n \times UCL_p, \qquad LCL_{np} = n \times LCL_p, \qquad CL_{np} = n \times CL_p$$

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Comparison: $p$ Chart vs. $np$ Chart

| Feature | $p$ Chart | $np$ Chart | |---|---|---|| | Quality Characteristic Monitored | Proportion (fraction/percent) of nonconforming units | Absolute count (number) of nonconforming units | | Subgroup Sample Size ($n$) | Constant OR Variable | Strictly Constant | | Underlying Distribution | Binomial distribution | Binomial distribution | | Centerline ($CL$) | $\bar{p} = \frac{\sum np}{\sum n}$ | $n\bar{p} = \frac{\sum np}{k}$ | | Standard Deviation ($\sigma$) | $\sigma_p = \sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}}$ | $\sigma_{np} = \sqrt{n\bar{p}(1-\bar{p})}$ | | Upper Control Limit ($UCL$) | $\bar{p} + 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}}$ | $n\bar{p} + 3\sqrt{n\bar{p}(1-\bar{p})}$ | | Lower Control Limit ($LCL$) | $\bar{p} - 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n_i}}$ (truncate if $< 0$) | $n\bar{p} - 3\sqrt{n\bar{p}(1-\bar{p})}$ (truncate if $< 0$) | | Shop-Floor Practicality | Requires division; preferred by engineers and managers for cross-lot comparison | No division required; preferred by shop-floor operators for direct tally plotting |

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Step-by-Step Worked Numerical Examples

Worked Example 1: Establishing $p$ Chart and $np$ Chart Limits with Constant $n$

Scenario: A quality technician at a precision stamping plant monitors an automated progressive stamping press forming brass terminals. Every hour, a random sample of $n = 400$ terminals is inspected using an optical vision system. Over 25 consecutive production hours ($k = 25$), a total of 240 nonconforming terminals are identified ($\sum np = 240$). Total terminals inspected $= 25 \times 400 = 10,000$.

Step A: Calculate $p$ Chart Parameters

1. Centerline ($\bar{p}$): $$\bar{p} = \frac{\sum np}{\sum n} = \frac{240}{10,000} = 0.0240 \quad (2.40\%))$$
2. Standard Error ($\sigma_p$): $$\sigma_p = \sqrt{\frac{\bar{p}(1-\bar{p})}{n}} = \sqrt{\frac{0.0240 \times (1 - 0.0240)}{400}} = \sqrt{\frac{0.0240 \times 0.9760}{400}} = \sqrt{\frac{0.023424}{400}} = \sqrt{0.00005856} \approx 0.007652$$
3. Three-Sigma Spread ($3\sigma_p$): $$3\sigma_p = 3 \times 0.00765245 = 0.022957 \approx 0.0230$$
4. Control Limits: $$UCL_p = \bar{p} + 3\sigma_p = 0.0240 + 0.02296 = 0.04696 \approx 0.0470 \quad (4.70\%))$$ $$LCL_p = \bar{p} - 3\sigma_p = 0.0240 - 0.02296 = 0.00104 \approx 0.0010 \quad (0.10\%))$$

Step B: Calculate $np$ Chart Parameters for the Same Process

1. Centerline ($CL_{np}$): $$CL_{np} = n\bar{p} = 400 \times 0.0240 = 9.60 \text{ nonconforming terminals}$$
2. Standard Deviation ($\sigma_{np}$): $$\sigma_{np} = \sqrt{n\bar{p}(1-\bar{p})} = \sqrt{400 \times 0.0240 \times 0.9760} = \sqrt{9.3696} \approx 3.0610$$
3. Three-Sigma Spread ($3\sigma_{np}$): $$3\sigma_{np} = 3 \times 3.06098 = 9.1829 \approx 9.18$$
4. Control Limits: $$UCL_{np} = n\bar{p} + 3\sigma_{np} = 9.60 + 9.183 = 18.783 \approx 18.78 \text{ nonconforming terminals}$$ $$LCL_{np} = n\bar{p} - 3\sigma_{np} = 9.60 - 9.183 = 0.417 \approx 0.42 \text{ nonconforming terminals}$$

(Note: $400 \times UCL_p = 400 \times 0.04696 = 18.78$ terminals, verifying exact equivalence).

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Worked Example 2: Variable Subgroup Size $p$ Chart and Truncated Lower Limit

Scenario: A medical device cleanroom packages sterile catheter sets. Production lot sizes vary daily. The baseline average nonconforming fraction is established as $\bar{p} = 0.0120$ (1.20%). The technician must calculate individual control limits for two different shifts:

• Shift 1: Sample size $n_1 = 200$ units.
• Shift 2: Sample size $n_2 = 800$ units.

Calculations for Shift 1 ($n_1 = 200$):

$$\sigma_{p1} = \sqrt{\frac{0.0120 \times 0.9880}{200}} = \sqrt{\frac{0.011856}{200}} = \sqrt{0.00005928} \approx 0.007699$$ $$3\sigma_{p1} = 3 \times 0.007699 = 0.02310$$ $$UCL_{p1} = 0.0120 + 0.02310 = 0.03510 \quad (3.51\%))$$ $$LCL_{p1} = 0.0120 - 0.02310 = -0.01110$$

Rule Application: Since calculated $LCL_{p1} < 0$, set $LCL_{p1} = 0$.

Calculations for Shift 2 ($n_2 = 800$):

$$\sigma_{p2} = \sqrt{\frac{0.0120 \times 0.9880}{800}} = \sqrt{\frac{0.011856}{800}} = \sqrt{0.00001482} \approx 0.003850$$ $$3\sigma_{p2} = 3 \times 0.003850 = 0.01155$$ $$UCL_{p2} = 0.0120 + 0.01155 = 0.02355 \quad (2.36\%))$$ $$LCL_{p2} = 0.0120 - 0.01155 = +0.00045 \quad (0.045\%))$$

Technical Insight: Notice that for the larger sample ($n = 800$), the upper limit narrowed from 3.51% down to 2.36%, and the lower limit became a positive number ($0.045\%$), making it possible to detect a statistically significant process improvement if zero nonconforming parts are observed.

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Technician Inspection Scenarios & Common Exam Traps

Real-World Shop Scenario: Machining Bushings on Multi-Spindle Lathes

A technician oversees an automated cell turning hardened bronze bushings. The quality plan requires checking 150 bushings every two hours using a go/no-go plug gage for bore diameter. If an operator mistakenly records 140 parts on one shift and 160 on the next, an $np$ chart will generate false alarms or mask true shifts because the count is unadjusted for lot size. The technician must either strictly enforce the constant sample size of 150 or convert the charting system to a $p$ chart.

Common Exam Traps for CQT Candidates

• Exam Trap 1: Attempting an $np$ Chart with Variable Sample Sizes: Any exam question describing changing subgroup quantities (e.g., $n$ varies between 100 and 150) that suggests an $np$ chart is presenting an invalid distracter. Variable sizes require a $p$ chart.
• Exam Trap 2: Simple Averaging of Proportions: If given subgroups with different sample sizes, never sum the $p$ values and divide by $k$. Always compute $\bar{p} = \frac{\text{Total Rejects}}{\text{Total Inspected}}$.
• Exam Trap 3: Reporting a Negative LCL: Never leave a lower control limit as a negative number on the ASQ exam. A negative fraction nonconforming is physically impossible; it must always be rounded up to zero ($LCL = 0$).
• Exam Trap 4: Confusing $p$ with $u$: If the question describes counting scratches, pinholes, or surface cracks, it is asking about defects (nonconformities), which mandate a $c$ or $u$ chart—not a $p$ chart.