Statistical Process Control & Process Capability (Cpk)

Process Settings

Process mean μ (mm) 50.00

Process std-dev σ (mm) 0.15

Spec width (USL−LSL) ±1.00

Process Capability

Cpk ≥ 1.33 — process capable

Cp (potential)

—

Cpk (actual)

—

Parts produced

Est. defect rate

0 ppm

UCL / LCL (X̄)

—

Out-of-control pts

How it works

Every part comes off the line with a dimension drawn from a normal distribution N(μ, σ) around the process mean. ISO 9001-style quality control asks two separate questions: is the process stable (control), and is it capable of meeting the spec (capability)?

Control chart (X̄ chart, subgroup n=5):
  Center line CL = μ̂ (grand average)
  UCL = μ̂ + 3·(σ̂/√n)   LCL = μ̂ − 3·(σ̂/√n)
  → a subgroup average outside these lines flags a
    special-cause signal, not just natural variation.

Process capability (compares spread to spec):
  Cp  = (USL − LSL) / (6σ)
  Cpk = min( (USL−μ)/(3σ), (μ−LSL)/(3σ) )
  → Cp:  can the spread fit inside the spec at all?
  → Cpk: does it actually fit, accounting for centering?

Cpk ≥ 1.33 is the common ISO/automotive threshold for "capable" (≈63 ppm defects at worst case); Cpk ≥ 1.67 is "six-sigma-adjacent".
Widening σ or shifting μ off-center lowers Cpk even if Cp (raw spread vs spec) stays the same — capability penalizes poor centering.
The bars falling outside the red spec lines on the 3D line are the parts that would fail final inspection; the chart below plots each subgroup's average against the control limits exactly as a quality engineer would on the shop floor.

Real-world relevance: this X̄-chart-plus-Cpk workflow is the core statistical-process-control method behind ISO 9001 clause 9.1 (monitoring & measurement) and IATF 16949 supplier audits — the same math run on injection-molded parts, machined shafts, or PCB solder-paste thickness.

In-spec part Out-of-spec (rejected)

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