Short-Term vs Long-Term Capability & Sigma Shift

Short-Term vs Long-Term Capability & Sigma Shift (CSSGB BoK III.F.4 — Evaluate)

Quick Answer: Short-term capability (Cp/Cpk, σ_within) describes common-cause variation in a controlled snapshot. Long-term performance (Pp/Ppk, s_overall) includes additional variation over time. When you only have short-term data, Six Sigma practice often assumes a 1.5σ shift so long-term Z ≈ short-term Z − 1.5—use the concept carefully and label assumptions.

Why Two Time Horizons Exist

A process measured for two hours on one setup under one operator can look extremely capable. The same process measured over three months—with material lots, environmental cycles, maintenance events, and multiple crews—usually shows more total variation. Customers live in the long term. Engineers need the short-term entitlement to know what improvement is even possible.

Evaluate skill: match the metric to the decision. Vendor process qualification under ideal conditions ≠ plant annual PPM.

Short-Term Capability — Assumptions and Use

Short-term data usually means:

• Rational subgroups from a narrow window
• One dominant common-cause system
• Process held in statistical control during collection
• σ estimated from within-subgroup dispersion (R̄/d₂, S̄/c₄)

Assumptions when reporting short-term Cpk:

1. The stable conditions during the study represent the intended operating mode.
2. Special causes were absent (or removed and fixed).
3. The measurement system is adequate.
4. Distributional assumptions for interpretation hold.
5. You are not claiming that month-to-month delivery will match this Cpk without further control.

Short-term capability is ideal for:

• Diagnosing whether variation reduction or centering is the bottleneck
• Comparing machine setups under comparable conditions
• Setting entitlement goals for Improve (“if we eliminate shifts, Cpk could approach …”)

Long-Term Capability / Performance

Long-term data span enough time and conditions to include:

• Mean shifts and slow drifts
• Lot-to-lot and shift-to-shift differences
• Tool wear cycles, seasonal humidity, supplier changes (if not excluded)

s_overall is larger when those effects appear, so Ppk typically falls below Cpk. That gap is diagnostic gold: it quantifies the price of not holding the process at short-term entitlement.

Worked comparison:
Short-term: σ_within = 1.0, μ on target, USL/LSL such that Cpk = 2.0 ⇒ Z_st = 3 × 2.0 = 6.0
Long-term: s_overall = 1.5, still centered ⇒ Ppk = 2.0 × (1.0/1.5) ≈ 1.33 ⇒ Z_lt = 3 × 1.33 ≈ 4.0

Same specs; long-term margin collapsed because total σ grew 50%.

When Only Short-Term Data Are Available

Projects and machine runoff studies often cannot wait months. You still need a long-term risk story for the charter or customer PPAP-style discussion.

Honest options:

1. Report short-term indices only, clearly labeled, with a plan to collect long-term data under Control.
2. Estimate long-term Z from short-term Z using an agreed sigma shift model (below).
3. Stress the process deliberately (multiple lots, operators) in a compressed DOE-like window to approximate between-source variation—still not a full year, but better than one happy shift.

Dangerous option: publish short-term Cpk as if it were annual customer quality without disclaimer.

The Sigma Shift Concept

Six Sigma methodology popularized a 1.5σ shift: over time, process means are assumed to drift or shift by about 1.5 standard deviations relative to the short-term centered ideal. Therefore:

Z_long ≈ Z_short − 1.5

Classic branding:

• A process with short-term distance to nearest spec of 6σ (Z_st = 6) is treated as 4.5σ long-term after the shift.
• Defect tables that quote 3.4 DPMO for “Six Sigma” quality use that long-term 4.5σ normal tail (one-sided), not the raw short-term 6σ tail (~0.001 DPMO).

Connecting to indices:

• Z_st ≈ 3 × Cpk (with short-term σ, nearest-spec definition)
• Z_lt ≈ 3 × Ppk (with long-term s), or
• Z_lt ≈ Z_st − 1.5 when you impute long-term from short-term only

Worked example — shift arithmetic:
Cpk = 1.67 ⇒ Z_st = 3 × 1.67 ≈ 5.0
Assumed long-term after 1.5 shift: Z_lt ≈ 5.0 − 1.5 = 3.5
Equivalent rough Ppk if the only change were a 1.5σ mean shift: Ppk ≈ 3.5 / 3 ≈ 1.17

Worked example — “Six Sigma” label:
Z_st = 6.0 ⇒ after 1.5 shift, Z_lt = 4.5. From standard normal, P(Z > 4.5) ≈ 3.4 × 10⁻⁶ = 3.4 DPMO (one-sided). That is the textbook bridge candidates are expected to recognize.

Assumptions and Limitations of the 1.5σ Shift

The 1.5 value is a convention rooted in historical manufacturing observations and Motorola-era practice—not a law of physics for every service process.

Assume when using it:

• Short-term σ remains relevant as the unit of shift
• Mean drift dominates the short-to-long degradation (or is a fair average penalty)
• Normal model still frames tail probabilities
• Stakeholders understand the number is a planning assumption

Do not assume:

• Every process shifts exactly 1.5σ
• Reducing σ automatically multiplies by the same DPMO table without checking real long-term data
• Service or transactional processes with different drift mechanisms owe allegiance to 1.5
• You can skip control charts because “we already subtracted 1.5”

If long-term data exist, prefer measured Ppk/Z_lt over an imputed shift. Use 1.5 when the BoK/exam scenario or early-project estimate requires a standard bridge from short-term only.

Decision Framework for Green Belts

Practical Evaluation Checklist

1. Is the reported index short-term or long-term (within vs overall σ)?
2. Was a 1.5σ shift applied? Is it stated?
3. Does the claim match the data window (hour vs quarter)?
4. After Improve, did long-term Ppk actually rise—or only a one-day Cpk demo?
5. Is Control designed to prevent the shifts that create the long-term penalty?

Bottom Line for III.F.4

Short-term capability is the process entitlement under common cause in a controlled snapshot (Cp/Cpk). Long-term performance reflects total variation customers feel (Pp/Ppk). When only short-term data exist, Six Sigma’s 1.5σ shift estimates long-term Z ≈ short-term Z − 1.5 (so 6σ short-term ↔ ~4.5σ long-term and the 3.4 DPMO story). Evaluate which horizon the question needs, state assumptions, and close the Cpk–Ppk gap in real projects rather than only subtracting 1.5 on paper.