P, NP, C, and U Charts

P, NP, C, and U Charts: Complete Guide for Six Sigma Black Belt

Introduction to Control Charts

Control charts are essential statistical tools used in the Control Phase of Six Sigma to monitor process performance and detect variations. Among the most commonly used are P charts, NP charts, C charts, and U charts. These charts help distinguish between common cause variation and special cause variation in your processes.

Why These Charts Are Important

Understanding and implementing P, NP, C, and U charts is critical because:

• Early Detection: These charts help identify process issues before they result in defective products or services
• Data-Driven Decisions: They provide objective evidence for process improvement decisions
• Regulatory Compliance: Many industries require statistical process control documentation
• Cost Reduction: By catching defects early, organizations reduce waste and rework costs
• Continuous Improvement: These tools support the continuous monitoring necessary for maintaining Six Sigma gains

Overview: What Are These Charts?

These four control charts are used for monitoring attribute data (pass/fail, defective/non-defective) rather than continuous measurement data:

P Charts (Proportion Chart)

What It Measures

A P chart monitors the proportion or percentage of defective items in a sample. It answers the question: "What percentage of items produced are defective?"

When to Use

• Sample size varies from sample to sample
• You're tracking the proportion of defectives (non-conforming items)
• Each item can be classified as either defective or non-defective
• Data comes from different batches or time periods with varying sizes

How It Works

Key Formula:

p = (Number of Defective Items) / (Total Sample Size)

Center Line (CL):

CL = p̄ (average proportion of defectives across all samples)

Control Limits:

UCL = p̄ + 3√(p̄(1-p̄)/n)

LCL = p̄ - 3√(p̄(1-p̄)/n)

Interpretation Example

Suppose you inspect 100 widgets each day for 20 days and find an average of 5% are defective (p̄ = 0.05). You would plot the daily defect proportions against the control limits. If Day 15 shows 12% defects, this point falls outside control limits, indicating a special cause that requires investigation.

NP Charts (Number of Defectives Chart)

What It Measures

An NP chart monitors the actual number of defective items rather than the proportion. It's simpler to use when sample sizes are constant.

When to Use

• Sample size is constant across all samples
• You prefer to track the actual count of defectives rather than proportions
• Easier interpretation for non-technical audiences ("5 defects" vs. "5%")
• Sample sizes must remain the same

How It Works

Key Formula:

np = Number of Defective Items in Sample

Center Line (CL):

CL = np̄ (average number of defectives)

Control Limits:

UCL = np̄ + 3√(np̄(1-p̄))

LCL = np̄ - 3√(np̄(1-p̄))

Interpretation Example

Inspecting batches of exactly 50 units per day, you find an average of 2.5 defective items per batch. The control limits would be calculated using the constant sample size of 50. If one batch contains 7 defects, it would likely exceed the upper control limit, signaling an out-of-control process.

C Charts (Count of Defects Chart)

What It Measures

A C chart monitors the total number of defects in a sample or per unit. Unlike P and NP charts that deal with defective items, C charts count the number of defects within items (multiple defects can exist in one item).

When to Use

• Sample size (or area of opportunity) is constant
• You're counting multiple types of defects on each unit
• Tracking defects per transaction, page, inspection area, etc.
• A single item can have multiple defects

How It Works

The C chart is based on the Poisson distribution.

Center Line (CL):

CL = c̄ (average number of defects)

Control Limits:

UCL = c̄ + 3√c̄

LCL = c̄ - 3√c̄

Interpretation Example

In a textile manufacturing process, you inspect each roll of fabric (constant area) and count total defects (tears, color variations, weaving errors). Over 25 rolls, you find an average of 4 defects per roll. Control limits would be: UCL = 4 + 3√4 = 4 + 6 = 10 defects. If roll #12 has 11 defects, it exceeds the UCL and requires investigation.

U Charts (Defects Per Unit Chart)

What It Measures

A U chart monitors the rate of defects per unit (defects per unit of measurement). It's the variable sample size version of the C chart.

When to Use

• Sample size (or area of opportunity) varies
• You need to account for different inspection areas or sample sizes
• Tracking defects per page (documents of varying lengths), per square meter, etc.
• More flexible than C charts when standardization isn't possible

How It Works

Key Formula:

u = (Total Defects) / (Sample Size or Area)

Center Line (CL):

CL = ū (average defects per unit)

Control Limits:

UCL = ū + 3√(ū/n)

LCL = ū - 3√(ū/n)

Important Note: The control limits change for each sample because n (sample size) varies.

Interpretation Example

You're reviewing documents of varying lengths. Document 1 is 5 pages with 2 errors (u = 0.4 defects/page). Document 2 is 10 pages with 3 errors (u = 0.3 defects/page). You track the u value for each document, with control limits adjusting based on document length. This allows fair comparison despite different document sizes.

Key Differences Summary

Step-by-Step Process for Creating These Charts

General Steps

1. Collect Data: Gather attribute data over a representative period (at least 25 samples)
2. Calculate Statistics: Compute the average proportion/count/rate (p̄, np̄, c̄, or ū)
3. Calculate Control Limits: Use appropriate formulas based on chart type
4. Plot Points: Graph each sample's value on the chart
5. Draw Control Lines: Add center line and control limits
6. Analyze Patterns: Look for points outside control limits or non-random patterns
7. Investigate and Act: Find root causes of special variation and implement corrections

Exam Tips: Answering Questions on P, NP, C, and U Charts

Tip 1: Identify the Correct Chart Type

Decision Logic:

• Is data about defective items or defects? If defective items → P or NP. If defects → C or U.
• Is sample size constant? If variable → P or U. If constant → NP or C.
• Create a mental decision tree: Start with "defective items?" then move to "constant sample size?"

Tip 2: Know the Formulas Cold

Memorize:

• P chart: UCL/LCL = p̄ ± 3√(p̄(1-p̄)/n)
• NP chart: UCL/LCL = np̄ ± 3√(np̄(1-p̄))
• C chart: UCL/LCL = c̄ ± 3√c̄
• U chart: UCL/LCL = ū ± 3√(ū/n)

Practice Tip: Write these formulas from memory at the start of your exam to avoid memory stress during problem-solving.

Tip 3: Distinguish Between "Defectives" and "Defects"

Exam Trick: Questions often use these terms interchangeably, but the distinction is crucial:

• Defective = Non-conforming item (the entire item fails) → Use P or NP charts
• Defect = Specific flaw or error (one item can have multiple defects) → Use C or U charts
• Read questions carefully for keywords like "number of items," "percentage of units," or "flaws per item"

Tip 4: Sample Size Matters

Remember:

• If the question mentions "constant sample size" → Immediately think NP or C
• If sample sizes vary or aren't mentioned as constant → Think P or U
• For U and P charts, you must recalculate control limits for each different sample size

Tip 5: Calculation Strategy for Exam

When solving problems:

• Step 1: Identify which chart type is needed
• Step 2: Calculate the average (p̄, np̄, c̄, or ū)
• Step 3: Plug into the correct control limit formula
• Step 4: Compare sample points to control limits
• Step 5: Identify out-of-control points and suggest root cause investigation

Tip 6: Interpret Control Limits Correctly

Common Exam Questions:

• "Is the process in control?" → Check if all points fall within control limits AND no patterns exist
• "What does this point indicate?" → If outside limits = special cause; if inside = common cause
• "Should we investigate?" → Yes, if any point exceeds control limits or patterns appear (runs, trends)

Tip 7: Handle Negative Lower Control Limits

Important Note: When calculating control limits for C and U charts, you may get negative LCL values. Never report negative counts or rates!

• If LCL calculates to negative → Set LCL = 0
• This is mathematically correct since you cannot have negative defects
• Always note this in your answer

Tip 8: Watch for Red Flags in Questions

Phrases to watch:

• "Percentage of..." → Likely P chart
• "Number of..." with constant batches → Likely NP chart
• "Defects per..." with varying units → Likely U chart
• "Count of defects" with constant area → Likely C chart
• "Multiple defects possible" → Definitely C or U, not P or NP

Tip 9: Practice Interpretation

Exam questions often ask:

• What process condition exists (in control vs. out of control)?
• What should the organization do next?
• What might have caused an out-of-control signal?

Your answers should include:

• The statistical evidence (which points are out of limits)
• The implication (special cause variation is present)
• The recommended action (investigate and correct the root cause)

Tip 10: Time Management

During the exam:

• Spend 30 seconds identifying the chart type before solving
• Use a calculator efficiently for formula calculations
• Double-check your formula choice matches the chart type
• Review your answer: Does it make practical sense?

Common Exam Scenarios

Scenario 1: Deciding Between P and NP Charts

Question Type: "A quality manager inspects 100 units daily. Which chart should be used to track daily defect percentages?"

Answer: P chart, because although sample size is constant in this case, the question asks for "percentages," which is the defining feature of P charts. Both could work since n is constant, but the language indicates P chart.

Scenario 2: Deciding Between C and U Charts

Question Type: "A call center tracks the number of errors per call. Call lengths vary. What chart is most appropriate?"

Answer: U chart, because defects per unit (errors per call) are tracked with varying sample size (call duration). A U chart normalizes to defects per standardized unit.

Scenario 3: Control Limit Calculation

Question Type: "Over 20 inspections of constant-size units, an average of 3.5 defects per unit was found. Calculate control limits using a C chart."

Solution:
c̄ = 3.5
UCL = 3.5 + 3√3.5 = 3.5 + 3(1.87) = 3.5 + 5.61 = 9.11
LCL = 3.5 - 5.61 = -2.11 → Set to 0 (cannot have negative defects)
Control Limits: [0, 9.11]

Scenario 4: Identifying Out-of-Control Process

Question Type: "Given a P chart with UCL = 0.08 and LCL = 0.02, which sample indicates the process is out of control: Sample A (0.07), Sample B (0.09), Sample C (0.04)?"

Answer: Sample B at 0.09, because it exceeds UCL of 0.08, indicating special cause variation.

Advanced Topics for Exam Mastery

Subgrouping and Rational Subgrouping

Proper subgrouping is crucial. Samples should be taken under similar conditions to detect true process changes. Mixing conditions (different shifts, machines, operators) within a sample masks real variation.

Rules for Out-of-Control Conditions

Beyond points exceeding control limits, look for:

• Run Rule: 8 or more consecutive points on one side of center line
• Trend Rule: 6 points steadily increasing or decreasing
• Cycling: Regular up-and-down patterns
• Clustering: Points consistently near center line (indicates data may be wrong)

Process Capability vs. Control Charts

Control charts assess stability (is variation predictable?), while capability indices assess conformance (does output meet specifications?). A process can be in control but not capable of meeting specifications.

Practical Application Example

Scenario: A software company reviews code for defects during testing. Each code module can have multiple defects (syntax errors, logic errors, security issues). Module sizes vary. Management wants to track quality trends.

Analysis:

• Data Type: Defects (multiple per module)
• Sample Size: Variable (module sizes differ)
• Chart Choice: U chart
• Metric: Defects per 100 lines of code
• Calculation: For each module, divide total defects by module size (in 100-line units)
• Control Limits: Adjust for each module's size
• Action: If a module's defect rate exceeds control limits, investigate coding practices used for that module

Conclusion and Key Takeaways

Mastering P, NP, C, and U charts requires understanding:

• The fundamental distinction: Defectives (P/NP) vs. Defects (C/U)
• Sample size implications: Constant (NP/C) vs. Variable (P/U)
• Formula accuracy: Each chart has specific mathematical requirements
• Practical interpretation: Control charts guide decision-making, not just statistics
• Continuous application: These tools support sustained improvement in the Control Phase

By thoroughly understanding these four chart types and practicing varied exam scenarios, you'll confidently tackle any Six Sigma Black Belt exam question on attribute control charts.