Cp is greater than Cpk when the process is not perfectly centered within the specification limits. Cpk accounts for the process centering, while Cp only considers the process spread.
Here's a more detailed breakdown:
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Cp (Process Capability Index): Measures the potential capability of a process if it were perfectly centered between the upper specification limit (USL) and lower specification limit (LSL). It only reflects the spread (variation) of the process data. The formula for Cp is:
Cp = (USL - LSL) / (6 * σ)
where σ is the standard deviation of the process.
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Cpk (Process Capability Index, corrected): Measures the actual capability of a process, taking into account both the spread and the centering of the process. It indicates how close the process is running to its target value and relative to its specification limits. Cpk is calculated as the minimum of two values:
Cpk = min [ (USL - μ) / (3 σ) , (μ - LSL) / (3 σ) ]
where μ is the process mean.
Key Differences Summarized:
| Feature | Cp | Cpk |
|---|---|---|
| Focus | Potential capability (spread only) | Actual capability (spread and center) |
| Centering | Assumes perfect centering | Accounts for process centering |
| Calculation | Based on overall spread (6σ) | Based on distance to nearest limit |
| Interpretation | Shows potential best-case scenario | Shows real-world performance |
Why Cp > Cpk when the process is off-center:
When the process is perfectly centered, the mean (μ) is exactly halfway between the USL and LSL. In this ideal case, the distances from the mean to each specification limit are equal, and Cp = Cpk.
However, if the process mean shifts away from the center (becomes off-center), one of the values used to calculate Cpk (either (USL - μ) / (3 * σ) or (μ - LSL) / (3 * σ)) will decrease. Since Cpk is the minimum of these two values, Cpk will be smaller than it would be if the process were centered. Meanwhile, Cp remains unchanged because it doesn't consider centering.
Example:
Imagine a process with USL = 10, LSL = 2, and σ = 1.
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Scenario 1: Perfectly Centered Process (μ = 6)
Cp = (10 - 2) / (6 * 1) = 1.33
Cpk = min [ (10 - 6) / (3 1) , (6 - 2) / (3 1) ] = min [1.33, 1.33] = 1.33
In this case, Cp = Cpk.
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Scenario 2: Off-Center Process (μ = 5)
Cp = (10 - 2) / (6 * 1) = 1.33
Cpk = min [ (10 - 5) / (3 1) , (5 - 2) / (3 1) ] = min [1.67, 1.00] = 1.00
Here, Cp (1.33) > Cpk (1.00). The off-centering reduces the Cpk value, reflecting the reduced capability due to the process shift.
In conclusion, Cp provides an optimistic view of process capability, assuming perfect centering, while Cpk gives a more realistic assessment by accounting for both the spread and the actual location of the process. Therefore, Cp is greater than Cpk when the process is not perfectly centered.
