Cpk

372. Cpk

372.1. $𝐶_{𝑝 𝑘}$ (Process Capability Index with Centering)

Minimum distance from mean to nearest spec limit divided by 3 standard deviations. Process capability considering centering and variation.

If process is centered:

𝐶_{𝑝 𝑘} = 𝐶_{𝑝} = \frac{USL - LSL}{6 𝜎}

If process mean is off center, the smaller side of the process-to-limit ratio determines $𝐶_{𝑝 𝑘}$ :

𝐶_{𝑝 𝑘} = Min (\frac{USL - \bar{𝑥}}{3 𝜎}, \frac{\bar{𝑥} - LSL}{3 𝜎})

If $𝐶_{𝑝 𝑘} < 𝐶_{𝑝}$ , process is not centered

$𝐶_{{𝑝 𝑘}}$ Value	Meaning	Quality Impact
$𝐶_{{𝑝 𝑘}} < 1$	Process not capable, output falls outside specs	Many defects, poor performance
$𝐶_{{𝑝 𝑘}} = 1$	Process just meets specification limits	Acceptable only if well centered
$𝐶_{{𝑝 𝑘}} > 1$	Process is capable, centered or nearly so	Fewer defects, reliable quality
$𝐶_{{𝑝 𝑘}} = 2$	Excellent capability, Six Sigma level	Near-perfect quality

Example

Suppose a company manufactures metal rods, and the specification limits for the diameter of the rods are:

Target: 10 mm
Lower Specification Limit (LSL): 9.8 mm
Upper Specification Limit (USL): 10.2 mm

The process has:

A standard deviation $𝜎$ of 0.05 mm.
A process mean $𝜇$ of 10.1 mm.

Step 1: Calculate the distance from the mean to the USL:

\frac{USL - 𝜇}{3 𝜎} = \frac{10.2 mm - 10.1 mm}{3 \times 0.05 mm} = \frac{0.1 mm}{0.15 mm} = 0.67

Step 2: Calculate the distance from the mean to the LSL:

\frac{𝜇 - LSL}{3 𝜎} = \frac{10.1 mm - 9.8 mm}{3 \times 0.05 mm} = \frac{0.3 mm}{0.15 mm} = 2.00

Step 3: Determine $𝐶_{𝑝 𝑘}$ :

𝐶_{𝑝 𝑘} = min (0.67, 2.00) = 0.67

Interpretation:

$𝐶_{𝑝}$ = 1.33 means the process spread (6 $𝜎$ ) fits 1.33 times within the tolerance range (the distance between the Upper Specification Limit and Lower Specification Limit.

$𝐶_{𝑝}$ = 1.00: Process mean is exactly at the midpoint of the specification limits, and the process variation fits exactly within these limits. 99.73% of the output will be within specifications, indicating a capable process (3 sigma process).
$𝐶_{𝑝} > 1.00$ : The higher the $𝐶_{𝑝 𝑘}$ , the more capable and stable the process is, meaning it can consistently produce parts within tolerance with minimal risk of defects.
$𝐶_{𝑝} < 1$ : The process mean is off-center or the variation is wider than the specification limits, or both. A significant portion of the output may fall outside the specification limits.

372. Cpk

372.1. 𝐶𝑝𝑘 (Process Capability Index with Centering)

372.1. $𝐶_{𝑝 𝑘}$ (Process Capability Index with Centering)