Cp

371. Cp

371.1. $𝐶_{𝑝}$ (Process Capability Index)

Measure how well a process can produce outputs within specified limits

𝐶_{𝑝} = \frac{USL - LSL}{6 𝜎}

Assumption: Process is centered within the specification limits

$𝐶_{𝑝}$ Value	Meaning	Quality Impact
$𝐶_{𝑝} < 1$	Process variation exceeds specs	Defects likely, even if centered
$𝐶_{𝑝} = 1$	Process barely fits within specs	OK if perfectly centered
$𝐶_{𝑝} > 1$	Process well within specs	Robust, fewer defects
$𝐶_{𝑝} = 2$	Six Sigma level	World-class performance

Example

A manufacturing process produces metal rods with a target diameter of 10.0 mm. The specification limits are:

Target: 10 mm
Lower Specification Limit (LSL): 9.7 mm
Upper Specification Limit (USL): 10.3 mm
Specification Width (Tolerance): 0.6 mm

Let’s analyze three different processes with varying levels of capability.

Process	Mean	Std Dev	$𝐶_{𝑝}$	Assessment
Process 1	9.998	0.099	1.009	Good
Process 2	10.001	0.147	0.682	Marginal
Process 3	9.994	0.255	0.393	Poor

Process 1	Process 2	Process 3	Six Sigma
$\begin{aligned} 𝐶_{𝑝} & = \frac{USL - LSL}{6 𝜎} \\ = \frac{10.3 - 9.7}{6 \times 0.099} \\ = \frac{0.6}{0.6} \\ = 1.009 \end{aligned}$	$\begin{aligned} 𝐶_{𝑝} & = \frac{USL - LSL}{6 𝜎} \\ = \frac{10.3 - 9.7}{6 \times 0.147} \\ = \frac{0.6}{0.9} \\ = 0.682 \end{aligned}$	$\begin{aligned} 𝐶_{𝑝} & = \frac{USL - LSL}{6 𝜎} \\ = \frac{10.3 - 9.7}{6 \times 0.255} \\ = \frac{0.6}{1.5} \\ = 0.393 \end{aligned}$	$\begin{aligned} 𝐶_{𝑝} & = \frac{USL - LSL}{6 𝜎} \\ = \frac{10.3 - 9.7}{6 \times 0.05} \\ = \frac{0.6}{0.30} \\ = 2 \end{aligned}$

Step 3: Interpretation

Process	$𝐶_{𝑝}$	Within Spec	Defective	DPMO
Process 1	1.009	$96.64 %$	$3.36 %$	$33600$
Process 2	0.682	$84.13 %$	$15.87 %$	$159700$
Process 3	0.393	$50.00 %$	$50.00 %$	$500000$
Six Sigma	2.0	$99.9999998 %$	$0.0000002 %$	$0.002$

Higher $𝐶_{𝑝}$ means fewer defects and better process quality. and lower defects per million opportunities (DPMO)

c_p.py


cp_values = [2, 1, 0.1]

for cp in cp_values:
    z = 3 * cp  # spec limit in standard deviations
    prob_defective = 2 * (1 - norm.cdf(z))
    within_specs = 1 - prob_defective
    dpmo = prob_defective * 1_000_000

    print({
      "Cp": cp,
      "Sigma limit (±z)": z,
      "Percent within specs": within_specs * 100,
      "Percent defective": prob_defective * 100,
      "DPMO": dpmo
    })

Output:

[{'Cp': 2,
  'Sigma limit (±z)': 6,
  'Percent within specs': 99.99999980268245,
  'Percent defective': 1.9731754008489588e-07,
  'DPMO': 0.001973175400848959},
{'Cp': 1,
  'Sigma limit (±z)': 3,
  'Percent within specs': 99.73002039367398,
  'Percent defective': 0.2699796063260207,
  'DPMO': 2699.796063260207},
{'Cp': 0.1,
  'Sigma limit (±z)': 0.30000000000000004,
  'Percent within specs': 23.582284437790534,
  'Percent defective': 76.41771556220947,
  'DPMO': 764177.1556220946}]

371. Cp

371.1. 𝐶𝑝 (Process Capability Index)

Output:

371.1. $𝐶_{𝑝}$ (Process Capability Index)