X-bar and R Chart, Capability, and a Supervisor Who Trusts the Specification
A bottling line fills 500 mL nominal to a specification of 500 ± 8 mL, at about 28,000 bottles per day. Fill volume is monitored with an X-bar and R chart, n = 5, k = 25 subgroups. The charts give X-double-bar = 502.75 mL and R-bar = 3.90 mL. Constants for n = 5: A2 = 0.577, D3 = 0, D4 = 2.114, d2 = 2.326. Subgroup 18 plots at 505.4 mL. Subgroup 7 has a range of 8.9 mL; a new operator started that day. Subgroups 19 through 25 all fall above the center line. The R chart is otherwise stable. The customer purchase order requires Cpk of at least 1.33 on all special characteristics, and fill volume is listed as one. The line supervisor says the process is fine because no bottle has ever been rejected for fill and "the specification is wider than your control limits."
1. Calculate the control limits. Is the process in statistical control? Justify with specific signals.
Read the worked answer
Limits:
- X-bar UCL = 502.75 + 0.577(3.90) = 502.75 + 2.25 = 505.00 mL
- X-bar LCL = 502.75 - 2.25 = 500.50 mL
- R UCL = 2.114(3.90) = 8.24 mL; R LCL = D3(R-bar) = 0
Not in control. Three separate signals:
- Subgroup 18 at 505.4 mL exceeds the X-bar UCL of 505.00 mL - a point beyond a control limit.
- Subgroup 7 range of 8.9 mL exceeds the R UCL of 8.24 mL - a point beyond a control limit, coincident with the new operator. Excess within-subgroup variation from a single day usually means technique or setup, not chance.
- Subgroups 19–25: seven consecutive points above the center line. A sustained one-sided run is a shift in the process mean, not noise.
Order matters. Deal with the range signal first: an inflated R inflates the X-bar limits (via R-bar) and can mask mean shifts. Investigate and, if the new operator is an assignable cause, exclude that subgroup with documented justification and recompute the limits before drawing conclusions about the mean.
The seven-point run is the more serious finding: something changed around subgroup 19 - a new filler head, a valve, product temperature, or a sensor. The chart is telling you the process moved, and no capability number computed from these 25 subgroups is valid until the cause is found and the process is stable again.
2. Assume, for this question only, that the process is stable at X-double-bar = 502.75 and R-bar = 3.90. Compute Cp, Cpk, and the estimated defect level. Does the process meet the contract?
Read the worked answer
Estimate sigma from the range chart:
sigma-hat = R-bar / d2 = 3.90 / 2.326 = 1.677 mL
Cp (potential, ignores centering):
Cp = (USL - LSL) / 6 sigma = (508 - 492) / (6 x 1.677) = 16 / 10.06 = 1.59
Cpk (actual, includes centering):
- USL - mean = 508 - 502.75 = 5.25 mL
- mean - LSL = 502.75 - 492 = 10.75 mL
- Cpk = 5.25 / (3 x 1.677) = 5.25 / 5.03 = 1.04
Defect estimate (upper side governs):
z = 5.25 / 1.677 = 3.13, one-tailed area = 0.00087, so about 870 ppm above the upper limit; the lower side is more than 6 sigma away and contributes essentially nothing. At 28,000 bottles/day that is roughly 24 out-of-tolerance bottles per day.
Verdict: Cp 1.59 shows the spread is adequate; Cpk 1.04 fails the 1.33 requirement. The gap between the two numbers is entirely centering. To reach Cpk 1.33 you need the mean at least 4 sigma (6.71 mL) below the USL, i.e. mean at or below 501.3 mL - about 1.5 mL lower. That is a set-point change, not a capital project, and it takes Cpk to roughly 1.6 at the current sigma.
Do not report these numbers to the customer until the special causes from the previous question are resolved and capability is recalculated on a stable process.
3. Respond to the supervisor's two claims: that no bottle has been rejected, and that the specification is wider than the control limits.
Read the worked answer
Claim 1 - "no bottle has ever been rejected."
This is an absence of detection, not evidence of quality. Three possibilities, all of which need checking before the claim is used:
- The plant may not be measuring every bottle. At 28,000/day, an estimated 870 ppm is about 24 bottles per day; if there is no check-weigher and no reject gate, nobody is generating the data that would show it.
- "Rejected" may only mean a downstream customer complaint, which is a lagging and insensitive indicator.
- Even at Cpk 1.04 the expected escape rate is small in percentage terms, so a human noticing rejects by eye is not a control.
The right question is not "has anyone complained" but "what is the demonstrated defect rate and where does the measurement come from?"
Claim 2 - "the specification is wider than the control limits."
This is the classic confusion between the voice of the process and the voice of the customer, and here it is arithmetically backwards. The X-bar limits (500.50 to 505.00) describe where subgroup averages should fall - they are roughly 3 sigma/root-n around the mean, so they are always much narrower than the specification, which applies to individual bottles. Comparing the two directly is invalid on its face.
Also note the geometry: the specification is 500 ± 8 and the process mean sits at 502.75, so the process is closer to the upper limit than to the lower. The control limits being inside the specification tells you nothing about Cpk. A process can be perfectly in control and still be incapable, or, as here, in control with respect to the wrong center.
What to do: show the supervisor the two numbers side by side - Cp 1.59 versus Cpk 1.04 - and the 870 ppm estimate. Ask for the set-point change to bring the mean to about 501 mL, then re-chart for 25 subgroups and re-run the capability.