Control Charts

SPC Control Charts (I-MR, X̄-R, X̄-S)

Control chart core values (CL, UCL, LCL, σ̂) for I-MR, X̄-R and X̄-S plus Nelson Rule evaluation and capability from σ̂.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Control Charts > Variables Charts for Individuals / Subgroups

Description

The SPC engine covers the three classic Shewhart control charts: I-MR (individuals + moving range), X̄-R (subgroup mean + range, 2 ≤ n ≤ 10), and X̄-S (subgroup mean + std-dev, 2 ≤ n ≤ 25). Control limits are computed from the classic AIAG/Montgomery SPC constants (d2, A2, A3, D3, D4, B3, B4, c4). σ̂ is derived from the baseline subgroup; `baselineEnd` enables a Phase-1/Phase-2 analysis. The analyze(inputs) dispatcher additionally supports mode 'nelson' for the eight Nelson rules (1–6 enabled by default) and 'capability' for Cp/Cpk from a supplied CL and σ̂. The flat output shape (e.g. i.cl, xbar.ucl) matches the fixture reference values 1:1.

Formulas

I (Individuals)

CL_{I} = \overset{x}{ˉ}, UCL_{I} = \overset{x}{ˉ} + 3 \overset{σ}{^}_{I}, LCL_{I} = \overset{x}{ˉ} - 3 \overset{σ}{^}_{I}, \overset{σ}{^}_{I} = \frac{MR}{d _{2}}

Individuals subchart of I-MR

MR (Moving Range)

CL_{MR} = \overline{MR}, UCL_{MR} = D_{4} \overline{MR}, LCL_{MR} = D_{3} \overline{MR}

Moving range subchart of I-MR

X̄-R

UCL_{\overset{ˉ}{X}} = \overset{ˉ}{\overset{x}{ˉ}} + A_{2} \overset{ˉ}{R}, LCL_{\overset{ˉ}{X}} = \overset{ˉ}{\overset{x}{ˉ}} - A_{2} \overset{ˉ}{R}, \overset{σ}{^}_{\overset{x}{ˉ}} = \frac{R ˉ}{d _{2} n}

X̄ limits and σ̂ from mean range (2 ≤ n ≤ 10)

UCL_{R} = D_{4} \overset{ˉ}{R}, LCL_{R} = D_{3} \overset{ˉ}{R}

Range subchart

X̄-S

UCL_{\overset{ˉ}{X}} = \overset{ˉ}{\overset{x}{ˉ}} + A_{3} \overset{s}{ˉ}, LCL_{\overset{ˉ}{X}} = \overset{ˉ}{\overset{x}{ˉ}} - A_{3} \overset{s}{ˉ}, \overset{σ}{^}_{\overset{x}{ˉ}} = \frac{s ˉ}{c _{4} n}

X̄ limits and σ̂ from mean std-dev (2 ≤ n ≤ 25)

UCL_{S} = B_{4} \overset{s}{ˉ}, LCL_{S} = B_{3} \overset{s}{ˉ}

Std-dev subchart

Assumptions

Data is approximately normally distributed (especially relevant for I-MR)
Baseline measurements are independent and from a stable process
Subgroups are rationally formed: within-subgroup variation < between-subgroup

Limitations

Subgroup size n must match chart type: I-MR n=1, X̄-R 2 ≤ n ≤ 10, X̄-S 2 ≤ n ≤ 25
Autocorrelation or non-normality distorts the control limits; I-MR is especially sensitive
Nelson rules 7 and 8 are disabled by default — enabling them raises the false-alarm rate

References

Montgomery, Introduction to Statistical Quality Control, 8th Ed. — Chapters 6-7
AIAG — Statistical Process Control (SPC) Reference Manual, 2nd Edition
Nelson, L. S. (1984), The Shewhart Control Chart — Tests for Special Causes, Journal of Quality Technology, 16(4), 237–239