ENSH 304 Probability Notes

Chapter 5: Statistical Quality Control

Syllabus hours: 5 | Exam weight: 5 marks | Marks breakdown: basics 1, control charts 3, six sigma 1

Difficulty type: Medium | Version / Last Updated: 2026-04-18 | Not in syllabus: advanced design of experiments and multivariate SPC

Outcome: decide whether a process is in control, choose the right control chart, and interpret quality signals from engineering data.

1. Fundamental Concepts

Quality control checks whether a process is stable and producing acceptable output.
Control charts separate common-cause variation from assignable-cause variation.
Variables charts track measurements like length, weight, or thickness.
Attribute charts track counts or proportions of defects or defectives.
A process is in control when the plotted points stay within control limits and show no abnormal pattern.

2. Core Methods and Formulas

When to use: use control charts when the question asks whether a process is stable over time or whether production variation is acceptable.

When not to use: do not use control charts for a one-time comparison where no process history is given.

\bar x\text{-chart center line }=\bar{\bar x}

UCL_{\bar x}=\bar{\bar x}+A_2\bar R,\quad LCL_{\bar x}=\bar{\bar x}-A_2\bar R

UCL_R=D_4\bar R,\quad LCL_R=D_3\bar R

p=\frac{d}{n},\quad UCL_p=\bar p+3\sqrt{\frac{\bar p(1-\bar p)}{n}}

LCL_p=\bar p-3\sqrt{\frac{\bar p(1-\bar p)}{n}}

Six sigma idea: smaller process spread means fewer defects and better capability

3. Standard Models / Topics

Topic 1: Quality Control in Engineering

Basic notes: quality control is used to monitor whether a process behaves predictably. If the process is in control, variation is mostly due to common causes.

Conditions / use: use this topic when the question asks for process stability, control, or variation source.

Formula recap: in-control means the chart points stay inside limits and show no unusual runs or trends.

Seen-Before Check: “stable process,” “in control,” “assignable cause,” and “common cause” are the cues.

[Core] Problem 1: What is the purpose of quality control?

Answer: to monitor and maintain consistent product quality by detecting abnormal variation.

[Core] Problem 2: Distinguish common-cause and assignable-cause variation.

Answer: common cause is natural random variation; assignable cause comes from a specific problem or change in the process.

Interpretation checklist: state whether the process is stable and whether an observed signal requires investigation.

Topic 2: Control Charts for Variables and Proportions

Basic notes: variables charts are used for measurable quantities; p-charts are used for defective proportions. X-bar charts track the mean level, R-charts track spread, and p-charts track defect rate.

Conditions / use: use X-bar and R charts for subgroup measurements; use p-charts when the output is classified as defective/non-defective.

Formula recap: $UCL_{\bar x}=\bar{\bar x}+A_2\bar R$ , $LCL_{\bar x}=\bar{\bar x}-A_2\bar R$ , $UCL_p=\bar p+3\sqrt{\bar p(1-\bar p)/n}$ .

Seen-Before Check: if the question mentions sample subgroups, defectives, or chart limits, this topic is active.

[Core] Problem 1: A process has x̄̄ = 20 and R̄ = 4. Find the X-bar chart center line.

Answer: 20.

[Advanced] Problem 2: If p̄ = 0.08 and n = 100, find the p-chart center line and control limits.

CL=0.08,\quad UCL=0.08+3\sqrt{0.08(0.92)/100}

LCL=0.08-3\sqrt{0.08(0.92)/100}

[Advanced] Problem 3: For subgroup size 5, the subgroup means are 18, 21, 20, 19, 22 and the subgroup ranges are 5, 4, 3, 4, 4. Find the X-bar and R-chart limits using A2 = 0.577, D3 = 0, and D4 = 2.114.

\bar{\bar x}=(18+21+20+19+22)/5=20,\quad \bar R=(5+4+3+4+4)/5=4

UCL_{\bar x}=20+0.577(4)=22.31,\quad LCL_{\bar x}=20-0.577(4)=17.69

UCL_R=2.114(4)=8.46,\quad LCL_R=0(4)=0

Answer: plot subgroup means against 22.31 and 17.69, and ranges against 8.46 and 0.

[Advanced] Problem 4: A plotted point falls outside UCL. What does that indicate?

Answer: the process may be out of control and should be investigated for assignable causes.

[PYQ-Trap] Problem 5: Why are chart limits not the same as specification limits?

Answer: control limits monitor process behavior; specification limits define acceptable product requirements.

Interpretation checklist: identify the chart type, say whether a point is inside or outside limits, and connect it to process action.

Topic 3: Six Sigma Concepts and Interpretation

Basic notes: six sigma is a process-improvement idea focused on minimizing defects and variation. The more capable the process, the fewer defective outputs are expected.

Conditions / use: use six sigma language when the question asks about capability, defect reduction, or process improvement strategy.

Formula recap: the practical idea is narrower spread around the target and fewer outputs beyond the tolerance band.

Seen-Before Check: “defects per million,” “process capability,” or “sigma level” point to this topic.

[Core] Problem 1: What is the goal of six sigma?

Answer: to reduce process variation and defects to a very low level.

[Core] Problem 2: Why is process centering important in quality control?

Answer: even a low-variation process can create defects if its mean is shifted away from the target.

[Advanced] Problem 3: A process has USL = 32, LSL = 28, mean = 30.5, and σ = 0.5. Find Cp and Cpk.

C_p=\frac{32-28}{6(0.5)}=1.33,\quad C_{pk}=\min\left(\frac{32-30.5}{3(0.5)},\frac{30.5-28}{3(0.5)}\right)=1.00

Answer: Cp is 1.33 and Cpk is 1.00, so the process is capable but not perfectly centered.

Interpretation checklist: state whether the process is capable, centered, or both.

4. Applied Problem Solving

[Core] Choose the correct chart from process wording alone.
[Core] Decide whether the process is in control from plotted values.
[PYQ-Trap] Distinguish between “process control” and “product specification.”

5. System-Level Understanding

Quality control is the operational layer that keeps manufacturing stable.
Control charts are decision tools, not just calculation tools.
Six sigma is the continuous-improvement mindset that sits above individual charts.

6. Quick Reference

$\bar x\text{-chart}$ for process mean.

$R\text{-chart}$ for spread.

$p\text{-chart}$ for defective proportion.

Seen-Before Check: measurement, defectives, limits, or process stability.

7. Exam Tips

Always identify whether the chart is for variables or attributes before calculating limits.
State the decision in words: in control, out of control, needs investigation, or stable.
Seen-Before Check: if the problem mentions defectives, use p-chart logic; if it mentions measurements, use X-bar and R logic.

8. Common Pitfalls

Confusing control limits with specification limits.
Using p-chart formulas for variable data.
Assuming one outside point proves the process is permanently bad without checking for assignable cause.

9. Tools and Guides

Variables charts = X-bar and R.
Attributes charts = p-chart.
Quality control asks one question first: is the process behaving normally?