Triola Elementary Statistics · 11th Edition
Chapter 14: Statistical Process Control
Every key formula from Triola Elementary Statistics Chapter 14, in one searchable page. Click a card to study it — worked examples included.
14.2 · R CHART (VARIATION)
Monitors process variation by plotting sample ranges over time.
Centerline = R̄ (mean of sample ranges)
UCL = D4R̄, LCL = D3R̄
D3, D4 from Table 14-2 by subgroup size n.
WORKED EXAMPLE
Bottle energy contents, n = 10, R̄ = 0.3777:
D4 = 1.777, D3 = 0.223 ⇒
UCL = 0.6712, LCL = 0.0842.
D4 = 1.777, D3 = 0.223 ⇒
UCL = 0.6712, LCL = 0.0842.
14.2 · x̄ CHART (MEAN)
Monitors the process center by plotting sample means over time.
Centerline = x̄̄ (mean of all sample means)
UCL = x̄̄ + A2R̄, LCL = x̄̄ − A2R̄
A2 from Table 14-2 by subgroup size n.
WORKED EXAMPLE
x̄̄ = 14.019, R̄ = 0.3777, A2 = 0.308 ⇒
UCL = 14.135, LCL = 13.903 ⇒ mean is out of control.
UCL = 14.135, LCL = 13.903 ⇒ mean is out of control.
14.2 · OUT-OF-CONTROL SIGNALS
Process is NOT stable if:
- A pattern, trend, or cycle is obviously not random
- A point lies above the UCL or below the LCL
- Run of 8: 8 consecutive points all above or all below the centerline
KEY NOTES
- Extra business rules: 6-in-a-row trend, 14 alternating, 2-of-3 beyond 2σ, 4-of-5 beyond 1σ.
14.3 · p CHART (ATTRIBUTES)
For attribute data: proportion defective in each sample.
Centerline = p̄ = total defectstotal items sampled, q̄ = 1 − p̄
UCL = p̄ + 3√(p̄q̄/n) (cap at 1)
LCL = p̄ − 3√(p̄q̄/n) (floor at 0)
WORKED EXAMPLE
Defective defibrillators, 20 batches of 10,000:
p̄ = 200/200,000 = 0.001 ⇒
UCL = 0.001948, LCL = 0.000052.
p̄ = 200/200,000 = 0.001 ⇒
UCL = 0.001948, LCL = 0.000052.
14.3 · READING THE CHART
KEY NOTES
- Control limits describe the PROCESS, not the goal — they are unrelated to spec limits.
- Use x̄/R charts for measurements, the p chart for yes/no attributes.
- Plots are sequential over time — order matters.
A process “in control” is stable — it may still fail specs! Control charts answer “is it stable?”, not “is it good?”.
WATCH OUT!
Mistakes that cost points
- Control limits ≠ specification limits — don’t confuse them
- Use the constant matching your SUBGROUP size n in Table 14-2
- x̄̄ (center of the x̄ chart) ≠ x̄ (one sample mean)
- p chart limits: cap UCL at 1, floor LCL at 0
- An “in control” process can still make bad products
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