Six Sigma, Process Capability & DMAIC

Reducing variation to near-perfect quality. Learn the Six Sigma philosophy, defects, DPU, DPMO and sigma levels (with the 1.5σ shift), process capability (Cp, Cpk, Pp, Ppk), the DMAIC roadmap and its tools (SIPOC, CTQ trees, MSA/Gauge R&R, Pareto, fishbone, 5 Whys, hypothesis tests, regression, DOE, FMEA, control plans), belts and project selection — with Python computing Cp, Cpk, ppm and sigma level.

Six Sigma, developed at Motorola in the 1980s and popularised by GE, is a data-driven method to reduce variation and defects. A 'six sigma' process produces about 3.4 defects per million opportunities — so rare that customers essentially never see one.

Its power comes from combining statistics with a disciplined project structure (DMAIC) and a focus on what customers consider critical to quality (CTQ).

An analogy

Process capability is like parking a car in a garage: the garage width is the specification, the car's side-to-side wobble is the process variation. Cp asks 'is the car narrow enough to fit?'; Cpk asks 'is it also parked in the centre, or scraping one wall?'.

Variation is the enemy

Customers feel variation, not averages: a delivery that is 'on time on average' may be late half the time. Common-cause variation is inherent to the process; special causes are assignable events (a worn tool, new supplier). SPC detects special causes; Six Sigma projects reduce common-cause variation and shift the mean to target.

DMAIC

The five-phase improvement roadmap:

Capability indices

Cp compares spec width with process spread; Cpk also accounts for centring. Common targets: Cpk ≥ 1.33 (existing processes), ≥ 1.67 (safety/new processes in automotive PPAP). Pp/Ppk use overall (long-term) σ; Cp/Cpk use within-subgroup (short-term) σ. Capability requires a stable process and roughly normal data (otherwise transform or use non-normal methods).

Organisation and extensions

Roles: champions select projects aligned with strategy; Black Belts lead full-time; Green Belts part-time; Master Black Belts coach. Lean Six Sigma combines flow/waste reduction with variation reduction. Design for Six Sigma (DMADV/IDOV) builds capability into new products using QFD, robust design (Taguchi) and tolerance design. The 1.5σ shift is a convention for long-term drift, debated but widely used.

Formulas

Worked example: Cp and Cpk

Given: USL 25.05, LSL 24.95, μ 25.012, σ 0.012

Find: Cp, Cpk.

Answer: Cp 1.39, Cpk 1.06

Worked example: DPMO

Given: 38 defects in 5000 units, 4 opportunities each

Find: DPMO.

Answer: 1900 → ≈ 4.4σ

Worked example: Rolled throughput yield

Given: Three steps with yields 98%, 95%, 99%

Find: RTY.

Answer: 92.2% first-pass through the whole process

Capability, ppm and sigma level

import math
def phi(z):
    return 0.5 * (1 + math.erf(z / math.sqrt(2)))
# Process capability and DPMO for a shaft diameter
USL, LSL = 25.05, 24.95
mean, s = 25.012, 0.012
Cp = (USL - LSL) / (6 * s)
Cpk = min(USL - mean, mean - LSL) / (3 * s)
p_out = (1 - phi((USL - mean) / s)) + phi((LSL - mean) / s)
print(f"Cp = {Cp:.2f}, Cpk = {Cpk:.2f}, expected defects = {p_out*1e6:.0f} ppm")
# Sigma level from observed DPMO (with 1.5-sigma shift convention)
defects, units, opp = 38, 5000, 4
dpmo = defects / (units * opp) * 1e6
lo, hi = 0.0, 8.0
for _ in range(100):
    z = (lo + hi) / 2
    if (1 - phi(z)) * 1e6 > dpmo: lo = z
    else: hi = z
print(f"DPMO = {dpmo:.0f} -> process sigma level = {z + 1.5:.2f}")
# Cp = 1.39, Cpk = 1.06, expected defects = 771 ppm
# DPMO = 1900 -> process sigma level = 4.39

The process is capable in principle (Cp 1.39) but off-centre (Cpk 1.06), so almost all 771 ppm of defects are oversize. Simply re-centring the mean at 25.000 would give Cpk = Cp = 1.39 and about 30 ppm — often the cheapest improvement. Reducing σ is the next step.

Common mistakes

Applications

Key takeaways

Going further