Quality, Reliability & Design of Experiments

Statistics that run factories and test labs. Learn statistical process control (x̄-R charts, control limits, rules), process capability Cp and Cpk, tolerance analysis, reliability statistics (MTBF, hazard, bathtub curve), and design of experiments (factorial designs, main effects and interactions).

Six Sigma, SPC charts on shop floors, reliability demonstration tests and designed experiments are how engineers keep processes stable, predict failures and find the best settings efficiently.

An analogy

A control chart is like a heart monitor for a process: normal beats stay within limits; spikes or trends signal something needs attention.

Control vs specification

Control limits come from the process's own variation (is it stable?). Specification limits come from the customer (is it good enough?). A stable process can still be incapable.

SPC

Plot subgroup means and ranges; points outside 3σ limits, runs of 7 on one side, or trends signal special causes to investigate. Common-cause variation needs system changes, not tweaking.

Capability

Cp measures potential (spread vs tolerance); Cpk also accounts for centring. Cpk ≥ 1.33 is a common requirement; Six Sigma targets Cpk = 2 (≈ 3.4 ppm with a 1.5σ shift).

Reliability and DOE

Bathtub curve: infant mortality, useful life (constant hazard), wear-out. Designed experiments vary factors systematically (2ᵏ factorials, fractional factorials, Taguchi) to find main effects and interactions with few runs.

Formulas

• x̄ chart limits: x̄̄ ± 3σ/√n
• Capability: Cp = (USL − LSL)/(6σ); Cpk = min(USL − μ, μ − LSL)/(3σ)
• Defect rate (normal): ppm ≈ 10⁶[Φ(−3Cpk_upper) + Φ(−3Cpk_lower)]
• Main effect (2-level): Effect = ȳ(high) − ȳ(low)

Worked example: Capability

Given: μ = 12.1, σ = 0.27 mm; spec 11.2–12.8 mm

Find: Cp, Cpk.

• Cp = 1.6/1.62 = 0.99
• Cpk = min(0.7, 0.9)/0.81 = 0.864

Answer: Cpk 0.86 — not capable; centre and reduce variation

Worked example: Control limits

Given: x̄̄ = 12.1, σ = 0.27, subgroups n = 5

Find: UCL, LCL.

• 3 × 0.27/√5 = 0.362

Answer: UCL 12.46, LCL 11.74

Common mistakes

• Wrong: Using spec limits as control limits. — Right: Control limits come from process data.
• Wrong: Changing one factor at a time. — Right: Factorial designs reveal interactions efficiently.

Applications

• Manufacturing: SPC and capability studies.
• Reliability testing: Life tests and warranty prediction.
• Process optimisation: Factorial experiments.

Key takeaways

• Control limits ≠ spec limits.
• Cpk = min(USL − μ, μ − LSL)/3σ.
• Bathtub curve for reliability.
• Factorial DOE finds effects and interactions.

Going further

• CUSUM & EWMA charts: Detecting small shifts.
• Response surface methods: Optimising continuous factors.
• Accelerated life testing: Arrhenius and inverse-power models.