John A. Conte, P.E.When Cpk is greater than 1, the process is in better shape to perform at acceptable levels. As a rule of thumb, a minimum Cp and Cpk value of 1.33 (4 sigma) is desired

John A. Conte, P.E.

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Objectives Excited to be here!

Students, faculty, engineers

Share my engineering career

Some thoughts on Six Sigma

Some thoughts on Process Capability

Cp, Cpk, Pp and Ppk

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How Did I Get Here? Graduated with University of Missouri with BSIE in 1966 Lived in Kansas City, Denver, Dallas Worked for AT&T Western Electric for 25 years Worked for DSC Communications for 10 years ASQ Senior Instructor for past 10 years May 2008 responded to “my story” request August 2008 presented Engineering Seminar This year, invited to be a Professor for a Day by Dr. Occena Actually visited Dr. Chang’s class in August 2008

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Most valuable course The most valuable course during my four years here was “Quality Control and Industrial Statistics”

Lead to my job with AT&T Western Electric Served me the first 15 years The basis for 20 years of using Bayesian Metrics The focus for the courses I currently teach

ASQ CQE Exam Preparation ASQ BB Quality Engineering Statistics Villanova on‐line University Lean Six Sigma MBB

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September 1962 Defoe Hall McNair House

2/22/20125

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Job Offer upon graduation

I remember in the interview I was told “I am sorry but we do not have any current openings for an industrial engineer but we do have an opening for a quality engineer. I see from your college transcript that you have taken a course entitled Quality Control and Industrial Statistics. Would you like to work for us as a Quality Control Engineer?”

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Semi‐Retirement Website www.contesolutions.com

Published Papers on Bayesian Metrics Courses Taught for the American Society for Quality

ASQ CQE Exam Preparation Six Sigma Engineering Statistics Statistical Process Control Introduction to Quality Engineering CQPA, CQI, CSSGB Exam Preparation

Teach an on‐line course for Villanova University

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What is Six Sigma A methodology A defined Body of Knowledge Different from Quality Engineering?

DMAIC Project Champion, Project Management Team Approach Test for Normality Gage R&R Variation Minitab and Implementations without Statistics

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1972 ASQ CQE BoK

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ASQ CQE BoK

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Comparison of Common Process Capability Measures

Percent defective ‐ The percent of product that is nonconforming or defective.

Process yield ‐ The percent of product that meets its requirements.

PPM / DPMO ‐When the number of defective products produced is small, it is often shown as a ratio of the number of defectives in 1 million parts or the number of defects in 1 million opportunities. This is referred to as “PPM” for “parts per million defective” or “DPMO” for “defects per million opportunities.”

For example, a process that is producing 99.73% good product is producing .27% defective product, or 2700 parts per million (PPM) defective.

Process Capability Indices (Cp, Cpk) – Measure the relationship of the process “bell curve” to the specification limits; use “short‐term” / within‐group variation.

Process Performance Indices (Pp, Ppk) – Measure the relationship of the process “bell curve” to the specification limits; use “long‐term” / between‐group variation.

Sigma Levels ‐ A sigma level is nothing more than a z‐score – it measures the number of standard deviations of the process that can fit between the process average and the nearest performance limit or target.

For example, if a particular process’ average is 3 standard deviations from the nearest performance limit, the process is operating at a “3 sigma” level. A “6 sigma” level would mean that the process is centered six standard deviations away from the nearest limit.

PERFORMANCE TARGETS

+/- 1.5 SIGMA

+/- 3 SIGMA

+/- 6 SIGMA

A z‐score is nothing more than a distance measure ‐ in all cases it represents the number of standard deviations between the mean of the data set and some target value or point of interest.

If the point of interest is the nearest specification limit, the z score can be called a Sigma level!

x z

z Score / Sigma level Example:

The process is stable!! The process is normally distributed!! Process average = 12.24 Process standard deviation = 1.13759 Upper spec limit = 14 Lower spec limit = 10

What is the z score / sigma level for the upper spec limit?

x z



1.55 1 z

13759.1

24.124x

0.4

0.3

0.2

0.1

0.0

X

Den

sity

14

0.0609

12.2

Distribution PlotNormal, Mean=12.24, StDev=1.13759

% defective upper = 6.06%

z = 1.55



What is the z score / sigma level for the lower spec limit?

x z



1.97 - 1 z

13759.1

24.120x

What is the z score / sigma level for the lower spec limit?

0.4

0.3

0.2

0.1

0.0

X

Den

sity

10

0.0245

12.2


Remember that the Normal curve is

symmetric – the area under the curve is the same on each side.

Z = -1.97

% defective lower = 2.44%

0.4

0.3

0.2

0.1

0.0

X

Den

sity

10

0.915

1412.2


% defective lower = 2.44%

% defective upper = 6.06%

% good = 100 – percent defective

= 100 – 2.44 – 6.06

= 100 – 8.5 = 91.5%

0.4

0.3

0.2

0.1

0.0

X

Den

sity

10

0.915

1412.2


2.44% defective = 2.44 x 10,000 =

24,400 PPM defective

6.06% defective = 6.06 x 10,000 =

60,600 PPM defective

Parts per Million Defective (PPM) is simply the % defective multiplied by 10,000 (to equate the number to 1,000,000 instead of 100):

1413121110

USLLSL

LS L 10Target *U S L 14S ample M ean 12.24S ample N 20S tD ev (Within) 1.00141S tD ev (O v erall) 1.13759

P rocess D ata

Z.Bench 1.63Z.LS L 2.24Z.U S L 1.76C pk 0.59

Z.Bench 1.37Z.LS L 1.97Z.U S L 1.55P pk 0.52C pm *

O v erall C apability

P otential (Within) C apability

P P M < LS L 0.00P P M > U S L 50000.00P P M Total 50000.00

O bserv ed P erformanceP P M < LS L 12648.50P P M > U S L 39415.05P P M Total 52063.54

E xp. Within P erformanceP P M < LS L 24471.99P P M > U S L 60915.50P P M Total 85387.49

E xp. O v erall P erformance

W ithinO verall

Process Capability of Data

1413121110

USLLSL

LSL 10Target *USL 14Sample Mean 12.24Sample N 20StDev (Within) 1.00141StDev (O v erall) 1.13759

Process Data

Z.Bench 1.63Z.LSL 2.24Z.USL 1.76C pk 0.59

Z.Bench 1.37Z.LSL 1.97Z.USL 1.55Ppk 0.52C pm *


Potential (Within) C apability

PPM < LSL 0.00PPM > USL 50000.00PPM Total 50000.00

O bserv ed PerformancePPM < LSL 12648.50PPM > USL 39415.05PPM Total 52063.54

Exp. Within PerformancePPM < LSL 24471.99PPM > USL 60915.50PPM Total 85387.49

Exp. O v erall Performance

WithinOverall

Process Capability of Data

“Within” variation is based on the Range / MR / Sigma Chart average

“Overall” variation is based on the standard deviation across all results

The 1.5 Sigma Motorola Shift

Process Capability Indices: Cp, Cpk (Or Cpu And Cpl) and Cr

The Cp and Cpk family of process capability indices predict the potential capability of the process to meet its requirements ‐ they reflect how the process could perform if the shifts and drifts of the process were to be eliminated.

There is disagreement over the use of the term “short” term vs. “long” term:

Some people also refer to the Cp and Cpk as the "potential" or "long‐term" process capability indices because they reflect how the process could perform if it were completely stable.

Others call these the “short term” indices because they are based on the average amount of variation within each sample group, which is a smaller period of time than the entire data set.

Because these indices are being used to predict potential process performance, the standard deviation used ( ) is always estimated from the Range (or Sigma or Moving Range) control chart as follows:

R‐bar is the centerline value for the accompanying Range chart

d2 is a calculation factor based on sample size and is found on the following table:

2dR

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The Cp is limited in that it does not consider the location of the center of the process distribution.

A process centered outside the specification limits, and therefore highly defective, could still score a very good Cp if its variation is small enough.

The Cp is therefore considered to be a preliminary measure only. If the amount of variation is acceptable, then the Cpk must be calculated to assess the centering of the process relative to the performance limits.

11.4

75

11.250

11.0

25

10.8

00

10.5

75

10.3

50

10.1

259.9

00

LSL USL

LSL 10.5Target *USL 11.5Sample Mean 9.99089Sample N 100StDev (Within) 0.0458766StDev (O v erall) 0.0473643

Process Data

C p 3.63C PL -3.70C PU 10.96C pk -3.70

Pp 3.52PPL -3.58PPU 10.62Ppk -3.58C pm *







WithinOverall

Process Capability of C1

The Cpk compares the actual process center and spread to the nominal or target process center and spread. Cpk is based on the distance from the process mean to the nearest, and therefore riskiest, performance target, so the smallest value is always selected.

Cpk = the smaller of:

or

σ 3

x - Limit ePerformanc Upperˆ

σ 3

Limit ePerformanc Lower - xˆ

With both the Cp and Cpk indices, larger values are always better:

When Cpk is greater than 1, the process is in better shape to perform at acceptable levels.

As a rule of thumb, a minimum Cp and Cpk value of 1.33 (4 sigma) is desired by most American companies today, although many companies are seeking Cpk values of 2, 3, 4, and larger before they consider their processes to be capable.

When the Cpk is equal to 1 (3 sigma), the process may be acceptable at the moment, but any shifting of the process or increase in process variation will immediately drive the process to unacceptable performance levels.

When the Cpk is less than 1, the process is incapable of performing at an acceptable level and should be given immediate attention. PERFORMANCE TARGETS

+/- 1.5 SIGMA

+/- 3 SIGMA

+/- 6 SIGMA

A Cp of 0.5 means that the specification limits are half the width of the total predicted process variation.

A Cpk of 0.5 means that half of a Normal curve will fit between the average and either performance limit, so this process is at a 1.5 sigma level.

10.59.07.56.04.53.0

LSL USL

LSL 5Target *USL 9Sample Mean 7Sample N 125StDev (Within) 1.333

Process DataC p 0.50C PL 0.50C PU 0.50C pk 0.50




Exp. Within Performance


Special Process Capability Circumstances

What does it mean when the both Cp and Cpk are all the same?

Equal Cp and Cpk Values: The Cp and Cpk values will only be equal when a process is centered at the target or nominal value.

Note that the Cpk value for a given process can never be greater than the Cp value, so if the Cp is unacceptable, the Cpk be unacceptable as well.

10.59.07.56.04.53.0

LSL USL









What does a negative Cpk tell you?

Negative Cpk Values:

Companies will sometimes find their processes centered outside of the performance limits. When this occurs, the Cpk calculation still holds true. The only difference is that if the process mean is outside the performance limits (less than the lower performance limit or greater than the upper performance limit), one Cpk value will be a negative number.

The negative value is therefore chosen as the Cpk value since it will always be the lower of the two calculations.

11.475

11.250

11.025

10.800

10.575

10.350

10.125

9.900

LSL USL

LSL 10.5Target *USL 11.5Sample Mean 9.99089Sample N 100StDev(Within) 0.0458766StDev(Overall) 0.0473643

Process Data

Cp 3.63CPL -3.70CPU 10.96Cpk -3.70

Pp 3.52PPL -3.58PPU 10.62Ppk -3.58Cpm *

Overall Capability

Potential (Within) Capability


Observed PerformancePPM < LSL 1000000.00PPM > USL 0.00PPM Total 1000000.00


Exp. Overall Performance

WithinOverall



What does a Cpk = 0 tell you?

Cpk = 0: Processes centered on a performance limit will have a Cpk value of zero since there is no difference between the process average and the performance limit.

12.010.59.07.56.04.53.0

LSL USL








The Pp and Ppk indices use the overall variation as calculated by the standard deviation for all the data collected across all sample subgroups:

1n)x(x s

2

Pp and Ppk

When the process is stable, the Cp / Cpk indices and the Pp / Ppk indices will be very similar.

If the process is not stable, the use of the Cp / Cpk indices is definitely not recommended and the Pp / Ppk indices are suspect as well.

Mathematically how do the formulas for Cpk and Ppk differ?

Page 303, The Certified Quality Inspector Handbook, H. Fred Walker, Ahmad K. Elshennawy, Bhisham C. Gupta, and Mary McShane‐Vaughn, ASQ Press 2009

“Kotz and Lovelace (1998,253) strongly argue against the use of Pp and Ppk. They have written:

We highly recommend against using these indices when the process is not in statistical control. Under these conditions, the P‐numbers are meaningless with regard to process capability, have no tractable statistical properties, and infer nothing about long‐term capability of the process. Worse still, they provide no motivation to the user‐companies to get their process in control. The P‐numbers are a step backwards in the efforts to properly quantify process capability, and a step towards statistical terrorism in its undiluted form.

Montgomery (2005, 349) agrees with Kotz and Lovelace. He writes "The process performance indices Pp and Ppk are more than a step backwards. They are a waste of engineering and management effort‐‐they tell you nothing." The authors wholeheartedly agree with Kotz and Lovelace and Montgomery. No one can judge a process when its future behavior is so unpredictable.”

Revised Process Prep times in Minutes

1.536 1.589 1.482 1.561 1.547 1.502 1.506 1.500 1.550 1.542 1.432 1.573 1.592 1.557 1.502 1.534 1.482 1.492 1.600 1.518

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191715131197531

1.7

1.6

1.5

1.4

Observation

Ind

ivid

ua

l V

alu

e

_X=1.5298

UC L=1.6712

LC L=1.3885

191715131197531

0.16

0.12

0.08

0.04

0.00

Observation

Mo

vin

g R

an

ge

__MR=0.0532

UC L=0.1737

LC L=0

I-MR Chart of C4

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1.641.601.561.521.481.44

USL

LSL *Target *USL 1.5Sample Mean 1.52985Sample N 20StDev (Within) 0.0459255StDev (O v erall) 0.0432207

Process Data

C p *C PL *C PU -0.22C pk -0.22

Pp *PPL *PPU -0.23Ppk -0.23C pm *



PPM < LSL *PPM > USL 750000.00PPM Total 750000.00

O bserv ed PerformancePPM < LSL *PPM > USL 742142.93PPM Total 742142.93

Exp. Within PerformancePPM < LSL *PPM > USL 755104.50PPM Total 755104.50


WithinOverall


MBB Process Capability problem

Not knowing what the specification limits Using the SPC IX & MR chart control limits as your specification limits Failing to compute Cpk with (MR or R‐bar) and D2 value

Get same value for Cpk and Ppk Assuming Lower Specification Limit when none was given

They compute a value for Cp and Pp when it is NA When they use Minitab to solve problem

not knowing what the sub‐group size When asked to compute the percent within and beyond spec

they correctly get 24.35% within spec, but go on to tell me that some number other than 756,500 PPM are beyond specification

Questions from Students

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Documents

John A. Conte, P.E.When Cpk is greater than 1, the process is in better shape to perform at acceptable levels. As a rule of thumb, a minimum Cp and Cpk value of 1.33 (4 sigma) is desired