SYE6047 Chap.8 Variable Control Charts and Capability
Variable Control Charts and Capability
Two charts answer two linked questions
When rational subgroups contain measurements, the x-bar chart monitors subgroup average and the R chart monitors within-subgroup range. The range chart is interpreted first because an unstable estimate of within-subgroup variation weakens the limits used for the average chart.
Together they distinguish shifts in location from changes in short-term spread while retaining process time order.
Build limits from the process
For equal subgroup size, calculate each subgroup mean and range, then their grand mean and average range. Standard constants convert those summaries into three-sigma limits.
These are control limits estimated from process data, not specification limits supplied by customers or design. Points beyond limits and prescribed nonrandom patterns prompt investigation; they do not identify the cause by themselves.
Revise only with evidence
An unusual point should remain visible while an assignable cause is investigated.
If a distinct event is verified and removed from future operation, the team may transparently exclude the affected subgroup and recalculate a baseline for the corrected process. Removing points solely because they widen limits manufactures apparent control and erases evidence about the system.
Capability follows stability
Potential capability compares specification width with six estimated standard deviations.
With a range-based estimate, sigma is related to the average range through the subgroup constant. The ratio describes potential spread fit, not centring. A stable process can have good consistency but poor conformance, and an unstable process can temporarily appear inside specifications. Control and capability therefore require separate conclusions.
What this chapter covers
- 01
Rational subgrouping
- 02
Subgroup means and ranges
- 03
Grand mean and average range
- 04
R-chart limits
- 05
X-bar chart limits
- 06
Signal investigation
- 07
Transparent baseline revision
- 08
Potential capability and its assumptions
Worked example · free
Calculate limits and keep capability separate
- 3Calculate the R-chart limits.
- 4Calculate the x-bar chart limits.
- 5Separate control from capability.
Key terms
- Rational Subgroup
- A small set formed so within-subgroup variation represents short-term common causes.
- Grand Mean
- The average of the subgroup means used as the x-bar chart centre line.
- Average Range
- The average of subgroup ranges used to estimate within-subgroup variation.
- Control Limit
- A process-derived boundary for expected variation in a plotted statistic.
- Specification Limit
- A requirement boundary established by customer or design needs.
Variable Control Charts and Capability FAQ
Why inspect the R chart first?
X-bar limits depend on an estimate of within-subgroup variation. If ranges are unstable, that estimate is not dependable, so interpretation of the average chart can be misleading. Investigate range signals and establish a defensible spread baseline before treating mean-chart limits as evidence of process location.
Can I delete any point outside a limit and recalculate?
No. A signal starts investigation. Exclusion is defensible only when a specific assignable event is verified, documented and removed or separated from the process represented by the new baseline.
Does a high Cp prove that output meets specifications?
No. Cp compares potential spread with specification width and ignores centring. It also presumes a stable process and a suitable distribution and variation estimate for the conditions studied.
Exam move
Write one fixed calculation order at the top of every exercise: subgroup summaries, R limits, x-bar limits, signal review, justified revision, capability. Label every boundary CL, UCL, LCL, USL or LSL so control limits cannot be mistaken for specifications. Build one blank calculation frame for subgroup means, ranges, grand mean, average range, constants and both pairs of limits.
Read the R chart first and explain why its stability supports the x-bar limits. When a point signals, write investigate before any deletion and list the operational evidence that would justify baseline revision. Recalculate both linked charts after a justified exclusion and preserve the original result. For capability, identify the source of sigma, check statistical control and write whether the ratio includes centring.
Practise four conclusion states: stable and capable, stable and incapable, unstable with observations inside specifications, and unstable with nonconformance. Each requires a different action. On the final pass, cover the chart labels and recover whether a boundary came from process behaviour or customer requirements solely from its symbol and calculation.
