Which statement correctly defines Cp and Cpk and notes their key difference?

Which statement correctly defines Cp and Cpk and notes their key difference?

• A. Cp = (USL−LSL)/(3σ); Cpk = min[(USL−μ)/(3σ), (μ−LSL)/(3σ)].
• B. Cp = (USL−LSL)/(6σ); Cpk = [(USL−μ)/(3σ) + (μ−LSL)/(3σ)].
• C. Cp = (USL−LSL)/(6σ); Cpk = max[(USL−μ)/(3σ), (μ−LSL)/(3σ)].
• D. Cp = (USL−LSL)/(6σ); Cpk = min[(USL−μ)/(3σ), (μ−LSL)/(3σ)].

Answer: (D)

Cp and Cpk gauge how well a process fits within specification limits. Cp reflects potential capability if the process were perfectly centered, using the total tolerance spread over 6σ: (USL − LSL) / (6σ). Cpk reflects actual capability by accounting for where the process mean sits relative to the limits; it uses the smaller distance from the mean to either limit, expressed in units of σ: min[(USL − μ)/(3σ), (μ − LSL)/(3σ)]. The key difference is that Cp ignores the mean position and shows what the process could achieve if centered, while Cpk incorporates any mean shift and shows what the process is actually capable of, with the worse side determining the limit. The statement with Cp divided by 6σ and Cpk using the minimum of the two side distances correctly captures both spread and centering. Why the other options don’t fit: using 3σ in Cp misstates potential capability, since Cp should compare the total spread to the full 6σ. Using an average for Cpk isn’t how Cpk is defined, which relies on the smaller distance to a limit. Using the maximum for Cpk would imply the better side governs capability, which isn’t correct—the weaker side limits actual capability.

Cp and Cpk gauge how well a process fits within specification limits. Cp reflects potential capability if the process were perfectly centered, using the total tolerance spread over 6σ: (USL − LSL) / (6σ). Cpk reflects actual capability by accounting for where the process mean sits relative to the limits; it uses the smaller distance from the mean to either limit, expressed in units of σ: min[(USL − μ)/(3σ), (μ − LSL)/(3σ)].

The key difference is that Cp ignores the mean position and shows what the process could achieve if centered, while Cpk incorporates any mean shift and shows what the process is actually capable of, with the worse side determining the limit. The statement with Cp divided by 6σ and Cpk using the minimum of the two side distances correctly captures both spread and centering.

Why the other options don’t fit: using 3σ in Cp misstates potential capability, since Cp should compare the total spread to the full 6σ. Using an average for Cpk isn’t how Cpk is defined, which relies on the smaller distance to a limit. Using the maximum for Cpk would imply the better side governs capability, which isn’t correct—the weaker side limits actual capability.