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Attribute control charts: p, np, c, u

Attribute charts monitor data that is counted rather than measured. Choose the right chart based on what you are counting and whether subgroup size is constant.

Chart	What it counts	Subgroup size
p	Proportion defective (fraction of units that are defective)	Variable or constant
np	Number of defective units	Constant only
c	Number of defects per unit	Constant (unit size fixed)
u	Defects per unit (rate)	Variable unit size

Worked example 1 — p chart

20 subgroups of n = 100 each. Total defectives over all subgroups = 80.

p̄ = 80 / 2000 = 0.0400 σ̂_p = √(p̄ × (1 − p̄) / n) = √(0.04 × 0.96 / 100) = √0.000384 = 0.01960 UCL_p = p̄ + 3 × 0.01960 = 0.0988 LCL_p = max(0, p̄ − 3 × 0.01960) = 0

Worked example 2 — c chart

25 subgroups (one circuit board per subgroup). Total defects = 150.

c̄ = 150 / 25 = 6 UCL_c = c̄ + 3 × √c̄ = 6 + 3 × 2.449 = 13.35 LCL_c = max(0, c̄ − 3 × 2.449) = 0

u chart formula (variable unit size)

ū = total defects / total units inspected UCL_uᵢ = ū + 3 × √(ū / nᵢ) (recalculated per subgroup when n varies)

Practice problem

np chart: 10 subgroups of n = 50 each. Total defectives = 30. Compute np̄, UCL, LCL.

Show answer

np̄ = 30/10 = 3.0. σ = √(50 × 0.06 × 0.94) = √2.82 = 1.679. UCL = 3.0 + 3×1.679 = 8.04. LCL = max(0, 3.0 − 5.04) = 0.

Common exam traps

p vs np: p is the rate; np is the count. Use np only when n is constant (same sample size every subgroup).
c vs u: c counts defects per fixed unit. Use u when the unit size varies (e.g., inspecting rolls of different lengths).
LCL cannot be negative. Attribute counts cannot be negative. If the formula gives a negative LCL, set it to 0.
Variable n means variable limits on p and u charts. The control limits change with each subgroup when n varies. Plot each subgroup’s limits individually.

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