Two Capability Indices, One Filling Line
A filling line targets 500.0 g with a label tolerance of 490–510 g. Over 30 days, 25 subgroups of n = 5 were plotted on an X̄–R chart and the chart shows no out-of-control points. From R̄ the within-subgroup sigma is estimated at 1.9 g. Pooling all 125 individual weights gives a mean of 501.2 g and a sample standard deviation of 2.6 g. Sales wants to advertise Cpk ≥ 1.33; the customer's supplier-quality engineer has asked for 'the capability index' with no further definition; the plant procedure does not distinguish short-term from long-term variation. A capital request for a new filler is being decided this week and these numbers will be quoted in the justification.
1. Calculate Cp, Cpk, Pp and Ppk, and estimate the nonconforming fraction implied by each sigma estimate.
Read the worked answer
Within-subgroup (sigma = 1.9 g)
- Cp = (USL − LSL) / (6·sigma) = 20 / 11.4 = 1.75
- Cpk = min(510 − 501.2, 501.2 − 490) / (3·1.9) = min(8.8, 11.2) / 5.7 = 8.8 / 5.7 = 1.54
Overall (s = 2.6 g)
- Pp = 20 / (6·2.6) = 20 / 15.6 = 1.28
- Ppk = 8.8 / (3·2.6) = 8.8 / 7.8 = 1.13
Nonconforming estimates
- With s = 2.6: z_U = 8.8 / 2.6 = 3.38 → upper tail ≈ 3.6 × 10^-4, about 360 ppm. Lower: z_L = −11.2 / 2.6 = −4.31, negligible.
- With sigma = 1.9: z_U = 4.63 → ≈ 1.8 ppm. Lower: z_L = −5.89, negligible.
The two answers differ by two orders of magnitude, and the gap is entirely the between-subgroup variation that R̄ cannot see. The point to make explicit: the chart is 'in control' only because the X̄ limits are 3·sigma_xbar = 3 × (1.9/√5) ≈ 2.5 g wide, so real lot-to-lot shifts of 1–2 g are invisible to it while still inflating the pooled standard deviation.
2. The plant manager proposes to simply move the mean to 500.0 g and claim Ppk ≥ 1.33. Evaluate that proposal quantitatively.
Read the worked answer
Required margin for Ppk = 1.33 with s = 2.6:
3 × 1.33 × 2.6 = 10.4 g
Available margin from a centered mean (500.0 g) is only 10.0 g. Therefore:
- Best achievable Ppk after centering = 10.0 / (3 × 2.6) = 1.28, which is short of 1.33.
- The shortfall is small (about 4% of sigma), which is exactly why the proposal is dangerous: it looks almost achievable, so someone will re-run the study until the number passes.
- The honest arithmetic says centering alone cannot deliver the target; sigma_overall must fall to at most 10.0 / (3 × 1.33) = 2.51 g at a centered mean, or to 8.8 / (3 × 1.33) = 2.21 g if the mean stays at 501.2 g.
- Eliminating the between-subgroup component (overall sigma → 1.9 g) takes Ppk at a centered mean to 10.0 / 5.7 = 1.75, which is the real prize and a much stronger capital justification.
So the capital request should be argued on the 1.9 g vs 2.6 g gap, not on a mean shift.
3. Which index would you publish externally, and what definitions would you write into the plant procedure?
Read the worked answer
Publish the Ppk (1.13), not the Cpk (1.54). The customer receives product from the long-term distribution, so Ppk is the index that describes what ships; quoting 1.54 would be accurate only for a process that does not exist yet.
What the procedure should fix:
- Define sigma_within explicitly as R̄/d2 from a chart whose subgroups are consecutive parts within a short time window, so only common-cause repeatability is captured.
- Define sigma_overall from a representative pooled sample of at least 25 subgroups / 100 individuals spanning shifts, lots and operators.
- State which index governs which decision: Cpk for internal improvement targets and machine acceptance, Ppk for customer submissions, regulatory filings and outgoing quality claims.
- Require a normality check and a histogram before any index is quoted; require the study period and the number of subgroups in the report.
Additional diagnostic the data demand: s = 2.6 against R̄/d2 = 1.9 is a stratification signature. Before publishing anything, plot an individuals chart of the 25 subgroup means and a run chart of subgroup ranges against time and against material lot. The between-subgroup source (fill density, product temperature, nozzle wear, lot change) is the actionable finding, and it is what the capital request should be aimed at.