Unit 2: Control charts
Quality Toolkit for Managers notes · PTU syllabus (MBA 954-18)
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Unit summary
Every process varies; control charts separate normal variation from problems that need action. This unit covers the concept of variability, assignable and chance causes, specifications and tolerances, control charts for variables and attributes, Type I and Type II errors, and process capability.
After this unit you can
- Explain variability and its causes
- Construct control charts for variables and attributes
- Explain Type I and Type II errors in control charts
- Determine process capability
PTU syllabus topics
- Concept of variability
- assignable and chance causes
- specifications and tolerances
- control charts for variables and attributes
- Type I and Type II errors
- process capability determination
X̄ chart limits
X̄̄ ± A2 R̄
R chart limits
D3 R̄ to D4 R̄
p chart limits
p̄ ± 3 √[p̄(1 − p̄) / n]
Process capability
Cp = (USL − LSL) / 6σ
Cp ≥ 1.33 is usually considered capable
Topic 1
Concept of variability
- No two units are exactly alike; variation comes from materials, machines, methods, people, measurement and environment.
Nature
Many small random factors inherent in the process
Identifiable, specific factors
Pattern
Stable, predictable variation
Unusual shifts, trends, outliers
Action
Change the system (management)
Find and remove the cause (operators, engineers)
Example
Minor temperature changes
Worn tool, wrong material batch
Topic 2
Specifications and tolerances vs control limits
- Specification limits (USL, LSL): set by design and customers — what is acceptable.
- Control limits (UCL, LCL): calculated from process data (usually ± 3σ) — what the process actually does.
- A process can be in statistical control yet produce out-of-specification items if its natural variation is too wide.
Topic 3
Control charts for variables and attributes
x̄ chart
UCL = x̿ + A2 R̄; LCL = x̿ − A2 R̄
R chart
UCL = D4 R̄; LCL = D3 R̄
p chart
p̄ ± 3√(p̄(1 − p̄)/n)
np chart
n p̄ ± 3√(n p̄(1 − p̄))
c chart
c̄ ± 3√c̄
Example
Ten samples of size 5 give x̿ = 50 and R̄ = 4; with A2 = 0.577, x̄-chart limits are 50 ± 0.577 × 4 = 47.69 to 52.31.
- Variables charts (x̄ and R) track measurable characteristics; attribute charts (p, np for defectives; c for defects) track counts.
- A process is out of control if points fall outside limits or show runs, trends or cycles.
x̄ and s chart
UCL = x̿ + A3 s̄; LCL = x̿ − A3 s̄
u chart (defects per unit)
ū ± 3√(ū ÷ n)
Topic 4
Type I and Type II errors
Meaning
Concluding the process is out of control when it is in control
Concluding the process is in control when it has shifted
Effect
Unnecessary investigation and adjustment
Defects continue undetected
With 3σ limits
About 0.27% false alarms
Depends on size of shift and sample size
- Wider limits reduce Type I but raise Type II errors; larger samples reduce Type II errors.
Topic 5
Process capability
Cp
(USL − LSL) ÷ 6σ
Cpk
Minimum of (USL − μ) ÷ 3σ and (μ − LSL) ÷ 3σ
Interpretation
Cp or Cpk ≥ 1.33 capable; 1.0 marginal; below 1 not capable
Example
Specification 25.00 ± 0.06 mm, process mean 25.02, σ = 0.01: Cp = 0.12 ÷ 0.06 = 2.0; Cpk = min(0.04 ÷ 0.03, 0.08 ÷ 0.03) = 1.33 — capable, though the mean is off-centre.
- Six Sigma corresponds to Cp = 2.0 (with a 1.5σ shift allowance, 3.4 defects per million).
Key terms
- Chance cause
- Random variation inherent in a process
- Assignable cause
- Specific, identifiable source of variation
- Control limits
- Limits calculated from process variation
- Type I error
- False alarm that a process is out of control
- Cpk
- Capability index allowing for off-centre processes
Quick revision
- Sources of variation; chance vs assignable causes.
- Specification limits vs control limits.
- x̄, R, s charts; p, np, c, u charts; out-of-control signals.
- Type I and Type II errors.
- Cp and Cpk; 1.33 benchmark; Six Sigma.
Important exam questions
Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).
Short-answer questions
- Q1.Distinguish chance and assignable causes.
- Q2.Distinguish specification limits and control limits.
- Q3.When is a p chart used?
- Q4.What is a Type II error in control charts?
- Q5.State the formula for Cp.
- Q6.What does Cpk below 1 indicate?
Long-answer questions
- Q1.Explain the concept of variability and its causes.
- Q2.Explain control charts for variables with a numerical example.
- Q3.Explain control charts for attributes and Type I and Type II errors.
- Q4.Explain process capability and its indices.
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