Process CapabilityJulian Kalac, P.Eng

Lean Six Sigma Master Black Belt

2

Normal Distribution

-6s -5s -4s -3s -2s -1s 0 1s 2s 3s 4s 5s 6s

One Standard

Deviation (s)

This is a Normal distribution where:

Mean = 0

Standard Deviation = 1

Process Capability & “Sigma”

Cp Sigma* Defect Rate

0.67 ± 2σ 31%

1.0 ± 3σ 6.7%

1.33 ± 4σ 0.6%

1.66 ± 5σ 0.02%

2.0 ± 6σ 0.003%

Sigma vs. Defect Rate• 1 sigma – 690,000 DPMO – 31%

• 2 sigma – 308,537 DPMO – 69.14%

• 3 sigma – 66807 DPMO – 93.32%

• 4 sigma – 6210 DPMO – 99.38%

• 5 sigma – 233 DPMO – 99.97%

• 6 sigma – 3.4 DPMO – 99.99%

When is good is good enough?

6.7% rejects

69% rejects

31% rejects

6

Cp

USLLSL

Voice of the

Customer

Voice of The Process

Voice of the Customer

Voice of the Process

Capability Ratio - compares the capability of a process (voice of the

process) to the specification limits (voice of the customer):

=USL - LSL

6s= Cp

Cp = 1: The process is

barely capable (Just fits into

the tolerance window).

Cp = 2: The process is a

six sigma process (The

tolerance window is twice

the process capability).

7

C Min(X -LSL

3

USL-X

3pk =

s s, )

CX -LSL

3pL =

s

USL-X

3sCpU =

Cpk accounts for process centering and spread.

Process Capability Ratios

8

Cp & Cpk for an Off-Center Process

Cp= 1.3

Cpk = 1.3

Cp= 1.3

Cpk = 0.8

Cp= 1.3

Cpk = 0.0

Process Capability & “Sigma” & Defect Per Mil

Cp Sigma* Defect/Mil

0.67 ± 2σ 308,537

1.0 ± 3σ 66807

1.33 ± 4σ 6210

1.66 ± 5σ 233

2.0 ± 6σ 3.4

10

Cpk

4.5

3.0

2.0

1.67

1.33

1.0

0.9

0.8

0.67

0.5

TARGET USL UCLLCL LSL

Z (Sigma) Value

6.05.0

4.0

3.0

2.0

PPM

<<1 PPBillion

.0018

< 1

230

2,700

6,210

16,400

66,800

308,000

3.4 PPM

Yield ST

99.99966 %99.977%

99. 379%

≈ 99%

≈ 95 %

Process Variation Shifts over time

What are some of the ways that we can easily indicate the dispersion

(spread) characteristic of the population?

Three measures that have historically been used:

range, variance and standard deviation

-6s -5s -4s -3s -2s -1s 0 1s 2s 3s 4s 5s 6s

Process Capability Cpk=2 Process Shift

Cpk= 1.33 after 1.5s Process shift

USLLSL

LCL UCLCpk = 2 before 1.5Ϭ shift

Cpk = 1.33 after 1.5Ϭ shift, within spec

-6s -5s -4s -3s -2s -1s 0 1s 2s 3s 4s 5s 6s

Process Capability Cpk =1 6.7% Out of Spec Process Shift 1.5 Out of spec= 52%

Control LimitsCpk= 1 -> 3s Process

LSL USL

LCL UCL

-6s -5s -4s -3s -2s -1s 0 1s 2s 3s 4s 5s 6s

Process Capability Cpk =2—38% Out of Spec

Control LimitsCpk= 0.67 -> 2s Process

LSL USL

LCL UCL

14

Accuracy and Precision

Accurate but not precise - On average,

the shots are in the center of the target

but there is a lot of variability

Precise but not accurate - The

average is not on the center, but

the variability is small

Source: iSixSigma

Variation and Mean Shift

LSL USL LSL USL

LSL USL

Off-Target, Low Variation

High Potential Defects

Good Cp but Bad Cpk

On Target

High Variation

High Potential Defects

No so good Cp and Cpk

On-Target, Low Variation

Low Potential Defects

Good Cp and Cpk

Variation reduction and process

centering create processes with

less potential for defects.

The concept of defect reduction

applies to ALL processes (not just

manufacturing)