Control Charts

Attribute Control Charts (p, np, c, u)

Control chart core values (CL, UCL, LCL, σ̂) for p, np, c, u charts plus reduced Nelson rule evaluation.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Control Charts > Attributes Charts (P, NP, C, U)

Description

Engine for the four classic attribute Shewhart control charts: p (fraction defective, variable subgroup size), np (number defective, constant n), c (defects per unit, constant inspection opportunity), u (defects per unit, variable opportunity). For p and u with variable n the engine returns UCL/LCL/σ as arrays (variableLimits=true); with constant n as scalars. The analyze(inputs) dispatcher supports modes p, np, c, u and nelson.

Formulas

p-Chart

\overset{p}{ˉ} = \frac{\sum D _{i}}{\sum n _{i}}, \overset{σ}{^}_{p_{i}} = \frac{p ˉ ( 1 - p ˉ )}{n _{i}}, UCL_{i} = \overset{p}{ˉ} + 3 \overset{σ}{^}_{p_{i}}, LCL_{i} = max (0, \overset{p}{ˉ} - 3 \overset{σ}{^}_{p_{i}})

Fraction-defective chart with (possibly variable) per-sample limits

np-Chart

\overset{p}{ˉ} = \frac{\sum D _{i}}{nk}, CL = n \overset{p}{ˉ}, UCL = n \overset{p}{ˉ} + 3 n \overset{p}{ˉ} (1 - \overset{p}{ˉ})

Number-defective chart, constant n

c-Chart

\overset{c}{ˉ} = \frac{\sum c _{i}}{k}, UCL = \overset{c}{ˉ} + 3 \overset{c}{ˉ}, LCL = max (0, \overset{c}{ˉ} - 3 \overset{c}{ˉ})

Defects per unit, constant inspection opportunity (Poisson)

u-Chart

\overset{u}{ˉ} = \frac{\sum c _{i}}{\sum n _{i}}, \overset{σ}{^}_{u_{i}} = \overset{u}{ˉ} / n_{i}, UCL_{i} = \overset{u}{ˉ} + 3 \overset{σ}{^}_{u_{i}}

Defects per unit with (possibly variable) limits

Assumptions

p/np charts: defects are binomial; n·p̄ ≥ 5 for usable normal approximation
c/u charts: defect counts are Poisson; no overdispersion
Subgroups are independent and from a stable process

Limitations

With variable n on p/u charts, zone rules (5–8) are undefined; only rules 1–3 are evaluated
Overdispersion creates false-positive violations — Laney p'/u' not implemented

References

Montgomery, Introduction to Statistical Quality Control, 8th Ed. — Chapter 7
AIAG — Statistical Process Control (SPC) Reference Manual, 2nd Edition