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Six Sigma Process: the true definition

le 29 avril- Paris: Understanding the statistical foundation behind the terminology and the true definition.

Table of Contents
The Common Misconception

❌ Common Belief (INCORRECT)

"If a process has specification limits at +3σ and -3σ from the mean, it's a Six Sigma process."

There is a widespread misunderstanding among Lean Six Sigma practitioners: many assume that positioning specification limits at ±3 sigma from the process mean automatically qualifies the process as "Six Sigma."

This is incorrect.

The Math Behind ±3σ

If you consult a Gaussian (normal) distribution table also called a Z table, the probability of a data point falling outside ±3σ is approximately:

0.27%

= 2 × 0.135% (each tail)

= 2,700 DPMO

(Defects Per Million Opportunities)

That is far from the famous 3.4 DPMO associated with Six Sigma performance.

⚠️ Important Distinction

±3σ corresponds to a 6σ distance (3σ on each side = 6σ total width). That's all. It does NOT correspond to a 6σ process capability.

So What IS Six Sigma?

✅ Correct Definition

Under the Six Sigma convention (including the 1.5σ shift):

One-Sided Specification

A true Six Sigma process has a specification limit at:

4.5σ

from the mean

Two-Sided Specification

For symmetric specifications, limits must be at:

±4.65σ

from the mean

At ±4.65σ, the process achieves the 3.4 DPMO benchmark that defines Six Sigma performance (2 × 1.7 DPMO (each tail)).

Visual Comparison: ±3σ vs ±4.65σ

❌ ±3σ: 4.28σ Process (NOT Six Sigma)

LSL: -3.000Mean: 0.00USL: 3.000

2,700 DPMO

0.27% defect rate

Equivalent to a one-sided specification of 2.68σ

Image: Normal distribution with specification limits at ±3σ showing defect probability of 0.135% in each tail.

✅ ±4.65σ :6σ Process (True Six Sigma)

LSL: -4.650Mean: 0.00USL: 4.650

3.4 DPMO

0.00034% defect rate

Equivalent to a one-sided specification of 4.5σ ✓

Image: Normal distribution with specification limits at ±4.65σ showing defect probability of 0.00017% in each tail.

📊 See the Visual Proof

The carousel images from DMAIC Suite's Process Capability tool demonstrate this exact difference. The first slide shows ±3σ with 2,700 DPMO. The third slide shows ±4.65σ with 3 DPMO (accounting for the 1.5σ shift).

So if 3.4 DPMO corresponds to 4.5σ, why is it called a 6σ process?
Understanding the 1.5σ Shift Convention

The Six Sigma methodology includes a 1.5σ shift assumption to account for process drift over time which is added to any long term σ value. This is why:

📍

4.5σ +1.5σ shift = 6σ . We call it Z Short-term (or Z-Benchmark) = Z Long-term Z + 1.5 which gives Z Short-term = 6.00σ

Perfect process centering, no drift - Z being the number of σ

📊

Long-term (Z Long Term): 4.50σ

After accounting for 1.5σ process shift

Short-term (Z Short Term): 6.00σ

🎯

Result: 3.4 DPMO LT

= True Six Sigma performance level

When you apply the conventional 1.5σ long-term shift, specification limits at ±4.65σ from the mean - with Long-term data)- equate to 6σ short-term capability, which is Six Sigma performance.
Critical Assumption: Normal Distribution Required

⚠️ Important Prerequisite

Everything discussed above is only true if your data follows a normal (Gaussian) distribution. Before applying Six Sigma calculations, always verify normality using tests like Anderson-Darling, Shapiro-Wilk, or visual methods (Q-Q plots, histograms).

If your data is not normally distributed, the standard deviation and Z-value calculations will not accurately represent process capability. In such cases, you may need to:
- Transform the data (e.g., Box-Cox transformation) to achieve normality
- Use non-parametric methods or percentile-based capability indices
- Apply distribution-specific calculations (Weibull, Lognormal, etc.)
- Identify and remove special cause variation before assessing capability
💡 DMAIC Suite Advantage

DMAIC Suite's Process Capability tool automatically performs Anderson-Darling normality tests and displays the results with your capability analysis. You'll see a green "Data follows normal distribution" badge when your data is valid for standard Six Sigma calculations.
Long-Term vs Short-Term Z-Values

An important clarification that is often overlooked:

📊

If you calculate a Z-value using long-term data...

The resulting Z-value is a long-term Z-value (Z Long Term).

⚡

If you calculate a Z-value using short-term data...

The resulting Z-value is a short-term Z-value (Z Short Term or Z-Benchmark).

The distinction matters because the data collection term determines which Z-value you're calculating, not the formula itself. A Gaussian distribution table lookup will always give you a Z value result. Depending on whether your standard deviation was calculated from:
- Long-term data: include common cause variation and special cause variation - capturing process drift and shifts in addition to inherent process variation
- Short-term data: include common cause variation only, measuring inherent process variation
Going further... Two Methods to Calculate Z-Shift

To convert between long-term and short-term Z-values, you can use either a quick approximation or a precise calculation based on your actual data.

Method 1: Quick Six Sigma Convention (1.5σ Shift)

This is the standard industry approach used when you don't have detailed subgroup data:

Z Short-term = Z Long-term + 1.5

or equivalently:

Z Long-term = Z Short-term - 1.5

Example: If your long-term Z = 3.0σ, then your short-term Z-Benchmark = 3.0 + 1.5 = 4.5σ

This 1.5σ shift is an estimate based on empirical studies of process drift over time. It's widely accepted but is indeed a "makeshift solution" that may not reflect your actual process behavior.
Method 2: Precise Calculation from Rational Subgrouping

For a more accurate Z-shift specific to your process, calculate it from your actual data using rational subgroups:
Step 1: Calculate Two Standard Deviations

σ_overall (long-term): Standard deviation of all data points across all subgroups

σ_within (short-term): Pooled standard deviation within subgroups only

Step 2: Calculate Z-Values

One-sided spec:

Z Long-term = (spec - μ) / σ_overall

Z Short-term = (spec - μ) / σ_within

Two-sided specs (LSL & USL):

Z LSL Long-term = (LSL - μ) / σ_overall

Z USL Long-term = (USL - μ) / σ_overall

Obtain p(d)LT LSL and p(d)LT USL with Z table

p(d)LT total = p(d)LT LSL + p(d)LT USL

Get Z Long-term from p(d)LT total with Z table

Z LSL Short-term = (LSL - μ) / σ_within

Z USL Short-term = (USL - μ) / σ_within

Obtain p(d)ST LSL and p(d)ST USL with Z table

p(d)ST total = p(d)ST LSL + p(d)ST USL

Get Z Short-term from p(d)ST total with Z table

Step 3: Calculate Your Actual Z-Shift

Z-shift = Z Short-term - Z Long-term
Example: If your Z Long-term = 2.8σ and Z Short-term = 4.1σ, then your actual Z-shift = 1.3σ (not the assumed 1.5σ).

This method gives you the real shift in your process provided that your data follow a normal distribution. If your calculated shift is significantly different from 1.5σ, it indicates your process drift behavior differs from the Six Sigma convention assumption.
💡 Which Method Should You Use?

Use Method 1 (1.5σ shift) for quick estimates, benchmarking, or when you don't have rational subgroup data.

Use Method 2 (precise calculation) when you need accurate process capability assessment, have control chart data with rational subgroups, or are making critical decisions based on capability indices.
Key Takeaway

❌±3σ limits do NOT make a Six Sigma process. They result in 2,700 DPMO.

✅±4.65σ limits achieve Six Sigma performance. They result in 3.4 DPMO.

📐Understanding the statistical foundation behind the terminology matters. Precision in language reflects precision in thinking.

💡 Pro Tip

Use DMAIC Suite's Process Capability Analysis tool to instantly calculate Z-scores, DPMO, and visualize your process performance with automatic 1.5σ shift calculations.

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