Cp

368. Cp

368.1. (Process Capability Index)

Measure how well a process can produce outputs within specified limits

Assumption: Process is centered within the specification limits

Value	Meaning	Quality Impact
	Process variation exceeds specs	Defects likely, even if centered
	Process barely fits within specs	OK if perfectly centered
	Process well within specs	Robust, fewer defects
	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		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

Step 3: Interpretation

Process		Within Spec	Defective	DPMO
Process 1	1.009
Process 2	0.682
Process 3	0.393
Six Sigma	2.0

Higher 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}]