Control Chart Selection & Construction

Control Chart Selection & Construction (CSSGB BoK VI.A.3 — Apply)

Quick Answer: Choose the chart from data type → subgroup size / constant n → defectives vs defects. Variables: I-MR (n=1), X-bar/R (small n), X-bar/s (larger n), median (small-n alternative). Attributes: p/np for defectives; c/u for defects. Construct with time order, process-based centerline, and process-based control limits; use location and dispersion charts together for variables data.

Decision Table — Which Chart?

Defectives vs defects:

• Defective unit = whole item fail/pass (one decision per unit).
• Defect = countable nonconformity; one unit can have many defects.

Variables Charts — Construction Essentials

X-bar and R (X̄–R)

Use when: continuous CTQ, rational subgroups of small n (often 3–5 in practice).

Plot:

• X-bar chart: subgroup averages vs time
• R chart: subgroup ranges vs time

Typical limits (constants from tables):

• X-bar: CL = X̿ , UCL/LCL = X̿ ± A₂R̄
• R: CL = R̄ , UCL = D₄R̄ , LCL = D₃R̄ (LCL may be 0 for small n)

Order of use: Evaluate the R chart first. If ranges are out of control, the within-σ estimate is unstable—do not trust X-bar limits until dispersion is stable.

X-bar and s (X̄–s)

Use when: continuous data with larger n (commonly n ≥ 10) so the sample standard deviation is a better dispersion statistic than R.

Plot: X-bar and s (subgroup sample SD).
Limits: X̿ ± A₃s̄ for averages; B₄s̄ / B₃s̄ style limits for s (table constants). Same philosophy as X̄–R with s replacing R.

I-MR / XmR (Individuals and Moving Range)

Use when: n = 1—batch processes, expensive tests, one reading per day, accounting metrics, or no natural rational subgroup.

Plot:

• I chart: each individual xᵢ
• MR chart: moving range MRᵢ = |xᵢ − xᵢ₋₁| (usually span 2)

Typical limits:

• I: CL = X̄ , UCL/LCL = X̄ ± E₂MR̄ (equivalently X̄ ± 3(MR̄/d₂) with d₂ ≈ 1.128 for MR of 2)
• MR: CL = MR̄ , UCL = D₄MR̄

Caution: Individuals charts are more sensitive to non-normality than X-bar charts (averages normalize by CLT). Still the correct default when n = 1.

Median Chart

Use when: small subgroup size and you want a location statistic less pulled by outliers than the mean, or when calculation simplicity on the floor matters.

Plot: subgroup median (and usually range). Slightly less statistically efficient than X-bar under normality, but Apply-level recognition: it is a valid variables option for small n—not the first default when X-bar is fine.

Attribute Charts — Construction Essentials

p Chart (fraction nonconforming)

Statistic: p̂ = (defectives in sample) / n
Use when: each unit is good/bad and n may change by period.
Limits: CL = p̄ , UCL/LCL = p̄ ± 3√[p̄(1−p̄)/nᵢ] — limits step when nᵢ changes.

np Chart

Statistic: number of defectives in the sample (np).
Use when: defectives data and n is constant.
Limits: based on n and p̄; simpler communication (“how many bad in each sample of 50”).

c Chart

Statistic: count of defects in a constant opportunity (e.g., defects per 100 m of cable, flaws per identical panel).
Limits: CL = c̄ , UCL/LCL = c̄ ± 3√c̄ (Poisson model). LCL floored at 0 when negative.

u Chart

Statistic: u = defects / opportunity size (defects per unit, per m², etc.) when opportunity varies.
Limits: ū ± 3√(ū/nᵢ) with nᵢ = opportunity size—limits vary with nᵢ.

Construction Checklist (All Charts)

1. Operational definition of the CTQ and of “defective” or “defect.”
2. Time-ordered sampling with rational subgroups (or n = 1 honestly).
3. Collect a baseline (often 20–25 subgroups for variables charts—follow your org’s standard).
4. Compute CL and trial control limits from process data.
5. Remove special-cause points only with cause identified (or stratify), then revise limits if policy allows.
6. Extend limits for ongoing control; react to signals with defined actions.
7. Never set control limits equal to specs “to make the chart look customer-focused.”

Worked Selection Scenarios

Scenario 1: Hourly, measure the diameter of four consecutive shafts from one lathe.
→ X-bar and R (continuous, small n = 4).

Scenario 2: One hardness reading per heat-treated batch; batches are sequential.
→ I-MR (n = 1).

Scenario 3: Each shift inspects exactly 200 switches; record how many fail continuity.
→ np chart (defectives, constant n). A p chart also works; np is natural when n fixed.

Scenario 4: Daily inspection sample size varies (80–150 claims); track fraction with coding errors.
→ p chart (defectives, varying n).

Scenario 5: Count paint defects on each identical door panel (same area every time).
→ c chart (defects, constant opportunity).

Scenario 6: Count scratches on glass sheets of different areas; report defects per m².
→ u chart (defects, varying opportunity).

Scenario 7: Subgroups of n = 12 thickness readings per roll.
→ X-bar and s (continuous, larger n).

Scenario 8: Small n = 3; floor staff already plot medians and want outlier resistance.
→ Median and range chart is acceptable; X-bar/R remains the more common efficient choice if means are fine.

Using Charts (Apply Behaviors)

• R or s (or MR) out of control → investigate variation sources (tool wear scatter, measurement, mixture) before over-interpreting averages.
• X-bar or I out of control with stable dispersion → mean shift: setup, material lot, temperature, method change.
• p or np rise → more defectives: process or incoming quality change.
• c or u rise → more defects per opportunity: handling, contamination, process chaos.
• After a sustained process improvement, recalculate limits so charts monitor the new common-cause system—not the old, worse baseline forever.

Common Selection Traps

Bottom Line for VI.A.3

Apply means you can walk from a process description to the right chart and know what to plot and why. Memorize the decision table: variables vs attribute, n = 1 vs small vs large n, constant vs varying opportunity, defectives vs defects. Construct with process-based limits, keep dual variables charts in the correct analysis order, and use signals to drive the right investigation.