If all sample averages on an
𝑥
−
- chart fall within the control limits, all output will be conforming.

Question

If all sample averages on an 
𝑥
−
- chart fall within the control limits, all output will be conforming.

BlackTom AI · Accepted Answer

Control limits indicate whether a process is statistically stable (in control), not whether every item meets product specifications.

1) Relevant knowledge points and concepts
- x̄ (x-bar) chart: monitors the sample means over time to detect shifts caused by special causes. Control limits (usually ±3σ/√n around the process mean) reflect natural process variation.
- Specification limits: customer or design requirements; separate from control limits.
- Process capability (Cp, Cpk): relates process spread and mean to specification limits; a process can be in control but not capable of meeting specs.
- Individual item variation vs. sample-mean variation: x̄ charts track averages; individual items vary more than subgroup means.

2) Step-by-step reasoning
- If every sample average falls within the x̄ chart control limits, the process is likely free of special causes and is stable.
- Stability does not guarantee that the distribution of individual items lies within specification limits. A stable process with large variability or mean offset can produce nonconforming items even though subgroup means stay within control limits.
- Also, x̄ charts use subgroup size n, so subgroup means have smaller spread (σ/√n) than individual measurements (σ). Thus, samples can appear “in control” while individual measurements exceed specs.

3) Example calculation (intermediate steps)
- Suppose individual measurements ~ N(μ=0, σ=1). For subgroup size n=4, x̄ standard deviation = σ/√n = 1/2 = 0.5. Control limits for x̄ ≈ μ ± 3·0.5 = ±1.5.
- Suppose specification limits for individual items are ±2. An individual value exceeding ±2 is nonconforming.
- Probability an individual item exceeds spec: P(|X|>2) = 2(1 − Φ(2)) ≈ 2(1 − 0.97725) = 0.0455 (≈4.55%). Even though all x̄ values may lie within ±1.5 (in control), about 4.55% of individual items fail specs.

4) Why the correct answer is correct
- The statement conflates being statistically in control with meeting specifications. Control limits do not ensure every output conforms; they only indicate the absence of assignable causes of variation.

5) Why the other option (True) is incorrect
- Marking “True” would ignore differences between control limits and specification limits and ignore individual-item variability; it would falsely claim guaranteed conformance.

6) Conclusion
- Therefore the correct answer is False, matching the provided answer.

If all sample averages on an 𝑥 − - chart fall within the control limits, all output will be conforming.判断题

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