Control charts for attributes: p, np, c and u

Why it matters

Many quality characteristics are not measured but counted: a casting is good or bad, a go/no-go gauge accepts or rejects, a painted panel has a number of blemishes. Attribute control charts monitor such data cheaply, often straight from routine inspection records, and they are the natural first chart for assembly lines, electronics and service processes. Choosing the right one of the four charts is itself a frequent exam question.

Key ideas

Defective versus defect. This distinction decides the chart.

• A defective (non-conforming unit) is a whole unit that fails to meet a requirement. Each unit is either defective or not, so the count of defectives in a sample of n follows the binomial distribution. Charts: p (fraction defective) and np (number defective).
• A defect (non-conformity) is an individual flaw; one unit can carry several. Counts of defects in a given area of opportunity follow the Poisson distribution. Charts: c (defects per inspection unit or per sample of constant size) and u (defects per unit, sample size may vary).

Choosing the chart

Counting	Sample size constant	Sample size varies
Defectives (binomial)	np or p	p
Defects (Poisson)	c or u	u

p chart. Plots the fraction defective p = d/n of each sample. Its limits depend on n, so with varying sample sizes each point gets its own limits (or an average n is used when sizes vary by less than about 25 %).

np chart. Plots the number defective d directly; easier for shop-floor staff, but only valid for a constant sample size.

c chart. Plots the number of defects in each inspection unit, where the inspection unit is the same every time (one car, one 10 m² of cloth, one circuit board).

u chart. Plots defects per unit u = c/n when the number of units inspected changes from sample to sample.

Limits. All four use ±3 standard deviations of the plotted statistic about the centre line. Because counts cannot be negative, a computed LCL below zero is set to zero. A non-zero LCL is useful: a point below it means the process has become better (or the inspection has become careless) and should also be investigated.

Limitations compared with variables charts. Attribute charts need much larger samples (often 50–200 units) to detect a change, carry no information about how far a part is from the limit, and react only after defectives are already being made. A variables chart on the key dimension is preferred wherever measurement is practical; attribute charts are preferred for many characteristics at once, visual checks and go/no-go gauging.

Formulas

p̄ = Σd / Σn

• p̄ = average fraction defective; d = number of defectives in a sample; n = sample size (units).

UCL_p, LCL_p = p̄ ± 3·√[p̄(1 − p̄)/n]

• p chart limits (dimensionless fraction); use each sample's own n when sizes vary.

UCL_np, LCL_np = n·p̄ ± 3·√[n·p̄(1 − p̄)]

• np chart limits (number of defectives); constant n only.

c̄ = Σc / k and UCL_c, LCL_c = c̄ ± 3·√c̄

• c = number of defects in one inspection unit; k = number of inspection units (samples). Limits in number of defects.

ū = Σc / Σn and UCL_u, LCL_u = ū ± 3·√(ū / n)

• u = c/n = defects per unit; n = number of units in the sample (limits change with n).

In all cases a negative LCL is replaced by 0.

Worked examples

Example 1 (standard): p and np charts Given: 20 samples of 200 bearings each are inspected with go/no-go gauges; a total of 320 defectives is found. Find the p chart and np chart limits.

1. p̄ = Σd / Σn = 320 / (20 × 200) = 320 / 4000 = 0.080.
2. σ_p = √[p̄(1 − p̄)/n] = √(0.08 × 0.92 / 200) = √0.000368 = 0.01918.
3. UCL_p = 0.080 + 3 × 0.01918 = 0.1375; LCL_p = 0.080 − 0.0575 = 0.0225.
4. n·p̄ = 200 × 0.08 = 16; σ_np = √(16 × 0.92) = 3.837.
5. UCL_np = 16 + 3 × 3.837 = 27.51; LCL_np = 16 − 11.51 = 4.49.

p chart: 0.0225 to 0.1375 (CL 0.080); np chart: 4.49 to 27.51 defectives (CL 16). A sample with 28 or more defectives, or 4 or fewer, is a signal.

Example 2 (GATE level): c and u charts Given: 40 painted car bodies are inspected and 120 paint defects are found in total. (a) Find c chart limits for inspecting one body at a time. (b) On a later day 5 bodies are inspected together and 30 defects found. Using a u chart, is the process in control?

1. c̄ = 120 / 40 = 3.0 defects per body.
2. (a) UCL_c = 3 + 3√3 = 3 + 5.196 = 8.196; LCL_c = 3 − 5.196 < 0, so LCL = 0.
3. (b) ū = 120 / 40 = 3.0 defects per body; for n = 5, √(ū/n) = √(3/5) = 0.7746.
4. UCL_u = 3 + 3 × 0.7746 = 5.324; LCL_u = 3 − 2.324 = 0.676.
5. Observed u = 30 / 5 = 6.0 defects per body, which is above 5.324.

(a) c chart: 0 to 8.20 defects per body (CL 3). (b) u = 6.0 > UCL 5.32: out of control; look for an assignable cause in the paint shop that day.

Common mistakes

• Using a p or np chart for defects, or a c chart for defectives; first ask whether you are counting bad units or flaws.
• Using an np or c chart when the sample size varies; switch to p or u.
• Forgetting to divide by n inside the square root for p and u charts, or dividing twice.
• Reporting a negative LCL instead of setting it to zero.
• Ignoring points below a non-zero LCL; they may mean real improvement or an inspection failure.
• Mixing percentage and fraction: if p̄ is in %, the formula p̄(1 − p̄) needs a fraction.

For GATE PI

Typical questions give the number of samples, sample size and total defectives (or total defects) and ask for the centre line, UCL or LCL of a p, np, c or u chart; or ask which chart fits a described situation. Remember the binomial basis of p/np and the Poisson basis of c/u, and that LCL is set to zero when negative.

Quick check

1. Which chart would you use for the number of scratches per sheet of glass of fixed size?
2. With c̄ = 16, what are the c chart limits?
3. 25 samples of 100 units contain 100 defectives in all. What is p̄?
4. Which chart handles defects with a varying number of units inspected?

Answers: 1. c chart. 2. UCL = 28, LCL = 4. 3. 0.04. 4. u chart.