Six Sigma & SPC Constants — OpenSpecEngineering

— Six Sigma & SPC Constants

Control-chart factors, capability-index relationships, and the process sigma / DPMO / yield conversions used in statistical process control and Six Sigma work.

Choosing a Control Chart

Data	Subgroup	Chart
Variable (measured)	n = 1	I–MR (individuals & moving range)
Variable	2 ≤ n ≤ ~9	X̄–R (mean & range)
Variable	n ≥ ~10	X̄–S (mean & standard deviation)
Attribute — fraction defective	variable n	p chart
Attribute — count defective	constant n	np chart
Attribute — defects, constant area	constant n	c chart
Attribute — defects, variable area	variable n	u chart

Shewhart Control Chart Constants

n	A₂	A₃	d₂	D₃	D₄	B₃	B₄	c₄
2	1.880	2.659	1.128	0.000	3.267	0.000	3.267	0.7979
3	1.023	1.954	1.693	0.000	2.574	0.000	2.568	0.8862
4	0.729	1.628	2.059	0.000	2.282	0.000	2.266	0.9213
5	0.577	1.427	2.326	0.000	2.114	0.000	2.089	0.9400
6	0.483	1.287	2.534	0.000	2.004	0.030	1.970	0.9515
7	0.419	1.182	2.704	0.076	1.924	0.118	1.882	0.9594
8	0.373	1.099	2.847	0.136	1.864	0.185	1.815	0.9650
9	0.337	1.032	2.970	0.184	1.816	0.239	1.761	0.9693
10	0.308	0.975	3.078	0.223	1.777	0.284	1.716	0.9727
11	0.285	0.927	3.173	0.256	1.744	0.321	1.679	0.9754
12	0.266	0.886	3.258	0.283	1.717	0.354	1.646	0.9776
13	0.249	0.850	3.336	0.307	1.693	0.382	1.618	0.9794
14	0.235	0.817	3.407	0.328	1.672	0.406	1.594	0.9810
15	0.223	0.789	3.472	0.347	1.653	0.428	1.572	0.9823

X̄ limits: X̄̄ ± A₂R̄ (from ranges) or X̄̄ ± A₃s̄ (from std devs). R limits: D₃R̄ and D₄R̄. s limits: B₃s̄ and B₄s̄. Process spread estimate: σ̂ = R̄/d₂ = s̄/c₄. For I–MR, individuals limits use X̄ ± 2.66 M̄R and the moving-range UCL is 3.267 M̄R.

Process Sigma ↔ DPMO ↔ Yield

With the conventional Six Sigma ±1.5σ long-term shift applied (i.e. process sigma = Z_LT + 1.5). A "6σ process" delivers 3.4 defects per million opportunities.

Process sigma	DPMO	Yield
1.0 σ	691,462	30.85%
2.0 σ	308,538	69.15%
3.0 σ	66,807	93.32%
3.5 σ	22,750	97.73%
4.0 σ	6,210	99.379%
4.5 σ	1,350	99.865%
5.0 σ	233	99.9767%
5.5 σ	32	99.99968%
6.0 σ	3.4	99.999660%

Standard Normal z-values

Two-sided confidence	One-sided area	z
80%	0.90	1.282
90%	0.95	1.645
95%	0.975	1.960
95.45%	0.9772	2.000
98%	0.99	2.326
99%	0.995	2.576
99.73%	0.99865	3.000

Capability Indices

Index	Definition
C_p	(USL − LSL) / 6σ_within — potential capability, centred
C_pk	min[ (USL − μ)/3σ_within , (μ − LSL)/3σ_within ] — actual capability
P_p, P_pk	Same forms using the overall (long-term) standard deviation
C_pm	(USL − LSL) / 6√(σ² + (μ − T)²) — Taguchi index, penalises off-target

C_pk	Interpretation
< 1.00	Not capable — producing out-of-spec product
1.00 – 1.33	Marginally capable — tight control needed
1.33 – 1.67	Capable (common minimum for ongoing production)
≥ 1.67	Highly capable (≥ 2.0 often required for new / critical processes)

DMAIC

Phase	Purpose
Define	Project charter, scope, customer CTQs, and the problem statement.
Measure	Validate the measurement system, baseline current performance, map the process.
Analyze	Identify and verify the root causes / key process inputs (X's).
Improve	Develop, pilot, and confirm solutions that address the verified causes.
Control	Control plan, SPC, mistake-proofing, and handoff to the process owner.