Cp, Cpk, Pp, Ppk & Cpm

Cp, Cpk, Pp, Ppk & Cpm (CSSGB BoK III.F.3 — Evaluate)

Quick Answer: Cp and Cpk use within-subgroup (short-term) σ; Pp and Ppk use overall (long-term) s. Cp/Pp ignore centering; Cpk/Ppk include it. Cpm further penalizes distance from target T. Green Belts must compute indices from given data, compare them, and link the nearest-spec distance to sigma level.

Shared Building Blocks

For a two-sided continuous characteristic:

• Tolerance width = USL − LSL
• Process mean μ (estimated by X̄)
• Within-subgroup standard deviation σ_within (often from R̄/d₂ or S̄/c₄ on rational subgroups)
• Overall standard deviation s_overall (sample SD of all individuals in the study period)

One-sided specs: use only the applicable half of Cpk/Ppk (CPU or CPL); Cp is not defined in the usual two-sided form without a full width.

Define the one-sided ratios:

CPU = (USL − μ) / (3σ)
CPL = (μ − LSL) / (3σ)
Cpk = min(CPU, CPL)

(same structure for Ppk with s_overall).

Cp — Potential Capability (Spread Only)

Cp = (USL − LSL) / (6 σ_within)

Cp asks: If the process were perfectly centered, would the natural 6σ spread fit inside the specs? It does not use μ. A process with huge Cp can still ship defects if the mean sits on one fence.

Interpretation guide (common industrial language, not a universal law):

Cpk — Actual Short-Term Capability (Spread + Location)

Cpk = min[ (USL − μ) / (3 σ_within) , (μ − LSL) / (3 σ_within) ]

Cpk uses the nearest specification. Always Cpk ≤ Cp. Equality holds when the process is centered between LSL and USL (for symmetric use of the same σ).

Worked example — Cp and Cpk:
LSL = 10.00, USL = 16.00, μ = 12.50, σ_within = 0.80

• Cp = (16 − 10) / (6 × 0.80) = 6 / 4.8 = 1.25
• CPU = (16 − 12.50) / (3 × 0.80) = 3.50 / 2.40 = 1.458
• CPL = (12.50 − 10) / (3 × 0.80) = 2.50 / 2.40 = 1.042
• Cpk = min(1.458, 1.042) = 1.04 (approx.)

Reading: Potential Cp = 1.25, but the mean is closer to LSL, so Cpk ≈ 1.04. Improvement priority: center upward toward mid-spec (13.0) and/or reduce σ.

Second worked example — centered:
Same specs, μ = 13.00, σ_within = 0.80

• Cp = 1.25 still
• CPU = CPL = (3.00) / 2.40 = 1.25
• Cpk = 1.25 = Cp

Centering recovered the full potential.

Pp and Ppk — Performance with Overall Variation

Pp = (USL − LSL) / (6 s_overall)
Ppk = min[ (USL − μ) / (3 s_overall) , (μ − LSL) / (3 s_overall) ]

Use the same formulas as Cp/Cpk but replace σ_within with the overall standard deviation of all data points in the period. Overall s includes between-subgroup shifts, drifts, and longer-term noise, so typically:

s_overall ≥ σ_within ⇒ Pp ≤ Cp and Ppk ≤ Cpk (when mean estimates align).

Worked example — Pp/Ppk:
LSL = 10, USL = 16, μ = 13.0, s_overall = 1.20 (σ_within was 0.80)

• Pp = 6 / (6 × 1.20) = 6 / 7.2 = 0.83
• Ppk = (3.0) / (3 × 1.20) = 3 / 3.6 = 0.83 (centered)

Short-term Cpk was 1.25; long-term Ppk is 0.83. The gap flags extra variation over time—investigate mean shifts, multiple streams, or unstable inputs—not just “operators need to try harder on each piece.”

Cpm — Taguchi / Target-Centered Capability

When the customer cares about closeness to target T, not only staying inside LSL–USL, use:

Cpm = (USL − LSL) / (6 √(σ² + (μ − T)²))

Here σ is the process standard deviation used for the study context (often overall or the σ specified by the method). The term (μ − T)² inflates the denominator when the mean misses the target—even if Cpk looks acceptable because the mean is still far from both fences.

When Cpm is used:

• Target is not optional (fit, color, matching, calibrated outputs)
• Loss increases as you move away from T (Taguchi loss thinking)
• You need a single index that punishes both excess variation and target miss

Worked sketch: USL − LSL = 6, σ = 0.5, μ = T ⇒ √(σ² + 0) = 0.5 ⇒ Cpm = 6/(6×0.5) = 2.0. If μ − T = 0.5 with same σ, √(0.25+0.25)=√0.5≈0.707 ⇒ Cpm ≈ 6/(6×0.707) ≈ 1.41. Cpk might still be high if specs are wide; Cpm fell because of the target offset.

Relationships to Remember

1. Cpk ≤ Cp and Ppk ≤ Pp
2. Pp ≤ Cp and Ppk ≤ Cpk when overall variation exceeds within variation
3. Cpk ≈ Cp (and Ppk ≈ Pp) when well centered
4. Large Cp − Cpk gap ⇒ centering problem
5. Large Cpk − Ppk gap ⇒ long-term / between variation problem
6. Cpm falls when μ moves from T even if still inside specs

From Capability to Sigma Level

Distance from the mean to the nearest specification, in sigma units:

Z_min = min[ (USL − μ)/σ , (μ − LSL)/σ ]

Because Cpk = Z_min / 3 when the same σ is used:

Z_min = 3 × Cpk (short-term when σ = σ_within)
Z_min = 3 × Ppk (long-term when σ = s_overall)

Worked example: Cpk = 1.33 ⇒ Z_min = 3 × 1.33 ≈ 4.0. Under a normal model centered such that the nearest fence is 4σ away, short-term nonconformance in that tail is about Φ(−4) ≈ 0.0032% on that side (before any long-term shift discussion in III.F.4).

Many Six Sigma deployments convert long-term defect rates to a sigma level table (with or without a 1.5σ shift). For the exam, know the Z = 3 × Cpk (or Ppk) bridge and that higher indices mean more standard deviations of margin to the nearest spec.

Evaluation Pitfalls

• Reporting Cp alone when the process is off-center
• Mixing within and overall σ in the same formula
• Treating Cpk as a percent yield without a distributional assumption
• Comparing vendor Cpk to plant Ppk without matching definitions
• Using Cpm without a meaningful customer target T

Bottom Line for III.F.3

Memorize the four core formulas, the Cpm target penalty, the inequalities among indices, and Z_min = 3 × Cpk/Ppk. Evaluate which index answers the question being asked: potential vs. actual, short-term vs. long-term, specs-only vs. target-focused.