Q: How is the PPM defect rate calculated?

The PPM (Parts Per Million) defect rate is calculated assuming a normal distribution. Upper defect probability = 1−Φ((USL−µ)/σ); lower defect probability = Φ((LSL−µ)/σ). The total defect rate is converted to PPM (×1,000,000). For example, at Cpk=1.33 (4σ), one-sided defects ≈ 31.7 PPM (two-sided ≈ 63.4 PPM). At Cpk=2.0 (6σ), less than 0.001 PPM one-sided.

Question 1

What is the difference between Cp and Cpk?

Accepted Answer

Cp (capability index) = (USL−LSL)/(6σ) measures the smallness of variation relative to the specification range. However, Cp does not account for the offset between the process center and specification center. Cpk does account for this offset: Cpk = min((USL−µ)/3σ, (µ−LSL)/3σ). If Cp is high but Cpk is low, the process center is offset from the specification center and centering adjustment is needed. Cpk ≥ 1.33 (4σ equivalent) is a common acceptance target.

Question 2

How do Pp and Ppk differ from Cp and Cpk?

Accepted Answer

Cp and Cpk use the 'within-subgroup σ' derived from Shewhart's R̄/d2 or s̄/c4. Pp and Ppk use the sample standard deviation s (overall σ) of all data. If Pp ≤ Cp, it indicates drift or shift in the process mean. When Cp ≈ Pp, the process is considered stable in the long term.

Question 3

What is Cpm (Taguchi capability index)?

Accepted Answer

Cpm incorporates the concept of Taguchi's loss function. Cpm = (USL−LSL)/(6τ), where τ² = σ² + (µ−T)² (T is the target value). When µ deviates from target T, τ > σ and Cpm < Cp. This means Cpm evaluates not only variation but also the deviation from target as a quality loss.

Question 4

How is the PPM defect rate calculated?

Accepted Answer

The PPM (Parts Per Million) defect rate is calculated assuming a normal distribution. Upper defect probability = 1−Φ((USL−µ)/σ); lower defect probability = Φ((LSL−µ)/σ). The total defect rate is converted to PPM (×1,000,000). For example, at Cpk=1.33 (4σ), one-sided defects ≈ 31.7 PPM (two-sided ≈ 63.4 PPM). At Cpk=2.0 (6σ), less than 0.001 PPM one-sided.

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