Bb Wk1 320 Process Capability

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Copyright © 2001-2005Six Sigma Academy International, LLC

All rights reserved; for use only in compliance with SSA license.

Process Capability



Copyright © 2001-2005

Six Sigma Academy International, LLC


Process Capability Pg 1

The Breakthrough Strategy® And Process Capability

Calculate baseline

Process Capability

Assess Process

Capability after

improvements are made

1. Select Output Characteristic and identify

key process input and output variables

2. Define Performance Standards

3. Validate Measurement System

4. Establish Process Capability

5. Define Performance Objectives6. Identify Variation Sources

7. Screen Potential Causes

8. Discover Variable Relationships

9. Establish Operating Tolerances

10. Validate Measurement System

11. Determine Final Process Capability

12. Implement Process Controls







Module Objectives

By the end of this module the participants should be able to:

• Explain where capability analysis fit into the 12 Step

Breakthrough Strategy

• Compute capability statistics with Variable Data

• Compute capability statistics with Attribute Data

• Discuss the concepts of stability as a prerequisite for

Capability Analysis

• Utilize MINITAB® for capability studies







What Is Process Capability?

Process Capability is a measure of how well the process output

(Voice Of Process) meets the customer requirements(Voice Of Customer).

Process Capability is analogous to measuring how well

the car gets parked in the garage, time after time.







Why Use Process Capability?

Process Capability answers the following questions:

• How is the process performing?

• How well could the process perform?

• What can be expected tomorrow, next week?

• Are customer‟s expectations being met?







Data Types

Attribute Data System:

• Capability is defined in terms of PASS/FAIL or categories

Continuous Data System:

• Capability is defined in terms of

defects under the curve and

outside of the specification limits

% Good % Bad

S P E C

L S L

T A R G E T

U S L

DefectsDefects







Calculating Capability

All processes can be described by the Z

score, making process comparisons easy.

Attribute Data Variables Data

Six Sigma Product ReportCapability Analysis and Sixpack

and/or Six Sigma Process Report

Sigma (Z)

DPMO DPMO







Voice Of Process,Voice Of Customer

• Specification limits represent the Voice Of Customer, VOC

• The process output represents Voice Of Process, VOP

• Process Capability measures how well VOP “fits” within VOC

LSL(Lower

specification

limit)

USL(Upper

specification

limit)

Voice Of Customer, VOC

Voice Of Process, VOP

CTQ, CTD, CTC

Now let‟s review some basic capability metrics.






Z score Transform







Z score Transformations

• If a data set is normally distributed, we use the mean and Standard Deviation to

determine the percentage (or probability) of observations within a selected range

• We can transform any normally distributed scale to its equivalent Z scale or score

using the formula below

• X will often represent a Lower or Upper Specification Limit (LSL and USL,

respectively) – It is the “Point of Interest”

• Z is the measure from the mean to the Point of Interest in Standard Deviations

X Z

Point of Interest

Mean

Z = 3

1

2

3







Z scale = units

are Standard

Deviations

10

86413

1412

-1-2-3 3210

USL

1610

Let's assume a process

µ = 10 and

σ = 2

Question 1: If my

Upper SpecificationLimit (USL) is 13,

how many mm

is the USL from

my mean?

Question 2: If theStdDev is 2,

how many StDevs

is my USL from

my mean?

X scale = units

are mm

Z = X - µσ

Z = 13 - 10

2

= 1.5

Z score Transformations







Z scale = units

are Standard

Deviations

864 1412

-1-2-3 3210

Z = X- µ

σ

1610

X scale = units

are Inches

X Z

? 1

10 ?

6 ?

? -3

? 1.5

? -2.25

13 ?15.5 ?

? 4

? -4

Z score Transformations – Exercise




Six Sigma Academy International, LLC All rights reserved; for use only in compliance with SSA license.


Z scale = units

are Standard

Deviations

864 1412

-1-2-3 3210

1610

X scale = units

are mm

Given ±1σ = 68%, ±2σ = 95%, ±3σ = 99.73%: Answer the following:

X z % of

area

to

rightof X

% of

area

to left

of X

10

12

14

168

6

4

Z score Transformations – Probability Exercise






P(z > -1.5) =

1-.0668 = .9332 Yield 93.3%

Probabilities Using Z Table

Standard Normal Probabilities:

The table is based on the area P

under the Standard Normal

Probability curve to the left of the

“Point of Interest” (Z = -1.5)

z 0 0 .0 1 0 .0 2 0 .0 3 0 .0

-4 .0 0 .0 0 0 0 3 0 .0 0 0 0 3 0 .0 0 0 0 3 0 .0 0 0 0 3 0 .0 0 0 0

-3 .9 0 .0 0 0 0 5 0 .0 0 0 0 5 0 .0 0 0 0 4 0 .0 0 0 0 4 0 .0 0 0 0

-3 .8 0 .0 0 0 0 7 0 .0 0 0 0 7 0 .0 0 0 0 7 0 .0 0 0 0 6 0 .0 0 0 0

-3 .7 0 .0 0 0 1 1 0 .0 0 0 1 0 0 .0 0 0 1 0 0 .0 0 0 1 0 0 .0 0 0 0

-3 .6 0 .0 0 0 1 6 0 .0 0 0 1 5 0 .0 0 0 1 5 0 .0 0 0 1 4 0 .0 0 0 1

-3 .5 0 .0 0 0 2 3 0 .0 0 0 2 2 0 .0 0 0 2 2 0 .0 0 0 2 1 0 .0 0 0 2

-3 .4 0 .0 0 0 3 4 0 .0 0 0 3 2 0 .0 0 0 3 1 0 .0 0 0 3 0 0 .0 0 0 2

-3 .3 0 .0 0 0 4 8 0 .0 0 0 4 7 0 .0 0 0 4 5 0 .0 0 0 4 3 0 .0 0 0 4

-3 .2 0 .0 0 0 6 9 0 .0 0 0 6 6 0 .0 0 0 6 4 0 .0 0 0 6 2 0 .0 0 0 6

-3 .1 0 .0 0 0 9 7 0 .0 0 0 9 4 0 .0 0 0 9 0 0 .0 0 0 8 7 0 .0 0 0 8

-3 .0 0 .0 0 1 3 5 0 .0 0 1 3 1 0 .0 0 1 2 6 0 .0 0 1 2 2 0 .0 0 1 1

-2 .9 0 .0 0 1 8 7 0 .0 0 1 8 1 0 .0 0 1 7 5 0 .0 0 1 6 9 0 .0 0 1 6

-2 .8 0 .0 0 2 5 6 0 .0 0 2 4 8 0 .0 0 2 4 0 0 .0 0 2 3 3 0 .0 0 2 2

-2 .7 0 .0 0 3 4 7 0 .0 0 3 3 6 0 .0 0 3 2 6 0 .0 0 3 1 7 0 .0 0 3 0

-2 .6 0 .0 0 4 6 6 0 .0 0 4 5 3 0 .0 0 4 4 0 0 .0 0 4 2 7 0 .0 0 4 1

-2 .5 0 .0 0 6 2 1 0 .0 0 6 0 4 0 .0 0 5 8 7 0 .0 0 5 7 0 0 .0 0 5 5

-2 .4 0 .0 0 8 2 0 0 .0 0 7 9 8 0 .0 0 7 7 6 0 .0 0 7 5 5 0 .0 0 7 3

-2 .3 0 .0 1 0 7 2 0 .0 1 0 4 4 0 .0 1 0 1 7 0 .0 0 9 9 0 0 .0 0 9 6

-2 .2 0 .0 1 3 9 0 0 .0 1 3 5 5 0 .0 1 3 2 1 0 .0 1 2 8 7 0 .0 1 2 5

-2 .1 0 .0 1 7 8 6 0 .0 1 7 4 3 0 .0 1 7 0 0 0 .0 1 6 5 9 0 .0 1 6 1

-2 .0 0 .0 2 2 7 5 0 .0 2 2 2 2 0 .0 2 1 6 9 0 .0 2 1 1 8 0 .0 2 0 6

-1 .9 0 .0 2 8 7 2 0 .0 2 8 0 7 0 .0 2 7 4 3 0 .0 2 6 8 0 0 .0 2 6 1

-1 .8 0 .0 3 5 9 3 0 .0 3 5 1 5 0 .0 3 4 3 8 0 .0 3 3 6 2 0 .0 3 2 8

-1 .7 0 .0 4 4 5 6 0 .0 4 3 6 3 0 .0 4 2 7 2 0 .0 4 1 8 1 0 .0 4 0 9

-1 .6 0 .0 5 4 8 0 0 .0 5 3 7 0 0 .0 5 2 6 2 0 .0 5 1 5 5 0 .0 5 0 5

-1 .5 0 .0 6 6 8 1 0 .0 6 5 5 2 0 .0 6 4 2 5 0 .0 6 3 0 1 0 .0 6 1 7

-1 .4 0 .0 8 0 7 6 0 .0 7 9 2 7 0 .0 7 7 8 0 0 .0 7 6 3 6 0 .0 7 4 9

-1 .3 0 .0 9 6 8 0 0 .0 9 5 1 0 0 .0 9 3 4 2 0 .0 9 1 7 6 0 .0 9 0 1-1 .2 0 .1 1 5 0 7 0 .1 1 3 1 4 0 .1 1 1 2 3 0 .1 0 9 3 5 0 .1 0 7 4

-1 .1 0 .1 3 5 6 6 0 .1 3 3 5 0 0 .1 3 1 3 6 0 .1 2 9 2 4 0 .1 2 7 1

LSL

Z Tables may be found in the “Black

Belt Memory Jogger” and most stat

textbooks. We will use MINITAB to

calculate probabilities

- . .

-1 . 5 0 . 0 6 6 8 1

-1 . 4 0 . 0 8 0 7 6






Probabilities Using MINITAB

• Run: Calc Probability Distributions Normal

• Select Cumulative probability and enter –1.5 in the Input constant box

• Use MINITAB

‟s Cumulative

Probability Function to

convert the Z score to

determine the area under

the curve

Session Window Output:

Cumulative Distribution Function

Normal with mean = 0 andStandard Deviation = 1.00000

x P( X <= x )

-1.5000 0.0668How many “Zs” from the

mean and in what direction






Calculating The Area ToThe Right Of Any Point

• From the same distribution curve, the area to the right of the Z score

(point of interest or right tail) can be calculated by subtracting the areato the left of the Z score from 1.0 (the total area under a standard curve,

i.e., 100%)

• Given Z = 0.94:

MINITAB provides ananswer of .8264 or

82.64% is to the left of

the Point of Interest and

17.36% to the right

P(x > z) = 1-.8264 =

.1736 or 17.36%

.1736

Z=0.94

Normal

C C F

3210-1-2-3

0.4

0.3

0.2

0.1

0.0

.8264






Relating Z score From Area Under Curve

• Up till now, we knew the Z score (or value) and found the area

• We can also calculate the Z score if given an area

• The question is P(Z < ?) = 0.8264 (or how many Zs and in what

direction to cover 82.64% of the area under the curve)

• Determine the Z score or Point of Interest that is represented by the

total area of interest under the curve to the left of Z

Normal

C

C F

Z=?

3210-1-2-3

0.4

0.3

0.2

0.1

0.0

82.64%






Calculating Z score Or InverseCumulative Probability

• Use MINITAB‟s Inverse cumulative probability function to convert the

area of interest under the curve to the left of some Point of Interest tothe Z score value of that Point of Interest

• Calc Probability Distribution Normal

- Click “Inverse cumulative probability”

- Input constant = .8264

• Answer is in the

session window:

Inverse Cumulative Distribution Function

Normal with mean = 0 and

Standard Deviation = 1.00000

P (X < = x) x

0.8264 0.9400






Z Table Exercises

• Find the area under the curve to the right and to the left of each of the

following Z values:

1.1, 2.4, 3.2, 0.45, -2.2, -1.75

• Given a process with a mean of 20 and a Standard Deviation of 4, find

the area under the curve to the right and to the left of each of the

following X values:22, 26, 20, 18, 14






The Z score Transformation:

Z X

Z

Z

47.5 45

12 5.

Z X

DEFECTS

to the rightof the USL

USL=47.5

Suppose the diameters

of shafts are normally

distributed with a mean

of (45) and a Standard

Deviation of (1). Thecustomer derived Upper

Specification Limit is

(47.5). What is the

DPMO for this process?

From MINITAB, the probability that a shaft is

less than (47.5) is 99.38% and the probability

of a defect is (1 - .9938) or .0062%.

DPMO = .0062 x 1,000,000 = 6,200

Knowing the distribution and the Specification Limits allows the

prediction of capability!

Z score TransformationPractical Application – DPMO Calculation






Application Of Z score Transform

Assume that the average number of days an account receivable is on the

books until funds are received is 25 days, with a Standard Deviation of four days…

• What are the chances that an account receivable is not closed before 30

days? Give answer as both a % and, assuming that 30 days is our

Upper Specification Limit, a DPMO.

• How many days does it take until 20% of the accounts receivable have

been paid?

• How many days does it take until 99.99% of the accounts receivable

have been paid?






ZBench Calculation (1)

• One of the key metrics used in Six Sigma is the Z Benchmark or

simply ZBench

• It is a single metric that takes into account defects both beyond the

Upper Specification Limit and below the Lower Specification Limit

• For our example, let us establish a process

• µ = 10

• σ = 2

• USL = 13

• LSL = 8

• We will first look at the Upper and Lower cases separately






10

86413

1412

-1-2-3 3210

USL

Z = 13 - 10

2

= 1.5

1610

Let's assume a process

µ = 10 and σ = 2

Question: If myUpper Specification

Limit is 13,what

% of my production

is defective? (Brown

area under the curve) Answer: Use Z Table

or MINITAB for

Z =1.5

Probability

of a defect

is 6.68%

ZBench Calculation (2) P (Defects) Upper

USL = 13






10

864 1412

-1-2-3 3210

Z = 8 - 10

2= -1

1610

Same processµ = 10

and σ = 2

Question: If myLower Specification

Limit is 8, what % of

my production is

defective?

(Yellow area under the curve)

Answer: Use

MINITAB for Z = -1

Probabilityof a defect

is 15.87%

LSL = 8

ZBench Calculation (3) P (Defects) Lower






10

864

13

1412

-1-2-3 3210

USL

1610

Question: If myUSL is 13 and my LSL

is 8, what % of my

production

is defective?

(Brown and yellowareas under

the curve)

Answer: Use

MINITAB

for Z =1.5 and Z = -1

and then add the

probabilities of

defect from both sides

Probability

of a defect

past USL

is 6.68%

LSL

Probabilityof a defect

below LSL

is 15.87%

ZBench Calculation (4) P (Defects) Total

Probability

of a defecttotal

is 22.55%






Question:P(Z> USL) = 6.68 %

P(Z< LSL) = 15.87%

P(Total) = 22.55%

If I threw all my

defects on one side,

how many StDevs

would fit between the

mean and the line

where the defects

start? Answer: Use

MINITAB for p=.2255

ZBench Calculation (5) ZBench

If we know the area that is equal to the TOTAL defects, we can find

its associated Z score ZBench (One number that tells it all).

10

864 1412

-1-2-3 3210

1610

C ( )






10

864 1412

-1-2-3 3210

1610

Total

probabilityof a defect

is 22.55%.

From Z table

or MINITAB

find Z = .75

11.5

0.75

Total probability of a defect is

22.55% (area under curve to right)

ZBench is 0.75, you can fit

+0.75 Standard Deviations

between the mean and thePoint of Interest.

ZBench Calculation (6) ZBench





Short Term vs. Long Term

P V i ti O Ti






Process Variation Over Time

Lot 1

Lot 2

C

T Q

Lot 3

Lot 4

Lot 5

Short Term Studies

Long Term Study

Sh t T L T






Short Term vs. Long TermSamples Of Data

Short Term Sample

• Free from assignable (special) causes

- Represents random (common) causes only

• Group of “like things”

• Collected across a narrow inference space- Frequently it is data from one lot of material, on one shift, on one

machine, with one operator

Long Term Sample

• Consists of random and assignable causes

• Collected across a broad inference space

- Data from several lots, many shifts, many machines and operators

Sh t T S l O S b






Short Term Samples Or Sub-groups

• Machine produces 60 jobs/minute

- Method 1: Collect sub-groups consisting of five consecutive pieces

taken on the hour

- Method 2: Collect sub-groups consisting of five pieces, each one

taken an hour apart

• Which method would you expect to have the least amount of variation?

• Which is a better estimate of short term variation?

Method 1 would provide a good estimate of short term variation.

Sh t T L T Z






• ZST (ZBench short term) is defined as the industry benchmark

• Represents 1.5 shift in process relative to specification target

- The rationale for a 1.5 sigma shift is based on a number of studies

conducted by Motorola. It is intended to be a rule of thumb or a best

estimate of the difference between short term and long term

capability.

Short Term vs. Long Term Z

Without a calculation, assume a 1.5 shift relative to target.

ShiftSTLT ZZZ

Sh t A d L T V i ti






Short And Long Term Variation

Long term variation is the combination of short term

variation, process shifts, and long term process drifts.

L S L

T A R G

E T

U S L

T i m e

Short Term

Variation

Long Term

Variation

Shift A d D ift Wh D It H ?






Shift And Drift, Why Does It Happen?

• Recall how our process output, Y, can be represented as follows:

• In other words, the process output (Y) is influenced by several input

variables (Xs)

• Some inputs may vary over long periods of time, others over short

periods of time

• Graphically, this might look like…

),...,,( 21 n x x x f Y

E f f e c t

Time

An X with “short

term” influence

An X with “long

term” influence

OK So Wh Worr Abo t Shift And Drift?






OK, So Why Worry About Shift And Drift?

• The long term capability represents actual performance of the process

over “long” periods of time • What time period is considered long?

- Long term in this case doesn‟t directly mean time

- It represents a time period where all the input variables have had a

chance to influence the process• For example…

- Multiple shifts, seasons, departments, lines, designers, etc.

• So thus, there isn‟t a specific number of data points that make data set

“long term”

What if we could take out the effects of long term shift and drift? The

variation remaining would only be short term, or what MINITAB refers to

as “within.” To do that, we need to understand Rational sub-grouping.

What Is A Rational Sub group?






Rational sub-grouping separates variation due to inputs that influence

the process in the short term from those that influence the process in thelong term.

In other words, a Rational sub-group is a group of data selected where you

have tested to determine the variability within the group is smaller than

the variability between groups. This allows estimation of pure short term

variability and the long term drift of the process.

2

Total =

2

Within +

2

Between

What Is A Rational Sub-group?

Variation due to Xs

changing in the

short term

Variation due to Xs

changing in the

long term

Let‟s see how this looks graphically.

How Sub grouping Works






How Sub-grouping Works

Sub-grouping allows for the mathematical separation of between

and within sub-group variation.

10

5

0

5040302010

• Long term

or Between

sub-group

Variation

• Short term

or Within

sub-groupVariation

Sub-group 1

Sub-group 2

Sub-group 3

Sub-group 4

Sub-group 5

How Do I Form Rational Sub groups?






How Do I Form Rational Sub-groups?

• As we collect our data, we should “data tag” it to facilitate grouping

• We can organize our data into groups where the variation is most likely

common cause, for example:

- Machines

- Plant

- Invoice types- Department

- Operator

- Shift

In summary, if we choose our sub-groups well and include only common

cause variation in them, we get an estimate of short term capability and

insight into what we could work on to minimize the long term shift and drift.

Now an example…

10

5

0

5040302010

Dept A

Dept B

Dept C

Dept D

Dept E

Definitions: Process Capability (C )






Definitions: Process Capability (Cp)In Real Life

Consider the task of parking a car in your garage. If the garage door

opening is larger than the width of the car, it is possible to park thecar inside.

Definitions: C






Definitions: Cp

• A capability index is a single value that expresses the ability of a

process to meet its requirements – There are several of these metrics;one of the most used is Cp

• Cp is the ratio of the tolerance width to the process width

• Bigger is better

VOP

VOC

6

LSLUSL

pC TOLERANCE

WIDTH OF DISTRIBUTION

Definitions: C In Real Life






Definitions: Cpk In Real Life

Sometimes, the car COULD make it, but alignment prevents it.

Definitions: C






• To assess the process relative to the USL, or drivers‟ side, use

this formula

• On the other side, compare the process to the LSL

3

USL

C pu

3 LSLC pl

Definitions: Cpk

Definitions: C






• To express the performance of the process as a whole, we consider

the limit closest to the process mean (the minimum of Cpu and Cpl )This is Cpk

• Cp can also be calculated from Cpu and Cpl – It is the average of these two values (recall that Cp does not take the process center

into account)

2),(avg

pl pu

pl pu p

C C

C C C

+

),min( pl pu pk C C C

Definitions: Cpk

Relationship Between C And C




Six Sigma Academy International, LLC All rights reserved; for use only in compliance with SSA license. Process Capability Pg 43

C

C

p

pk

1

1

C

C

p

pk

1

0

CC

p

pk

1

1

LSL USL Some details...

Relationship Between Cp And Cpk

• Cp is positive – It is the ratio of two

positive numbers

• Cpk can be positive, zero

or negative

• When Cpk is zero, yield is 50%

• When Cpk is negative, yield is less

than 50%

• If the process is centered,

Cpk = Cp

LSL USL

LSL USL

Short Term Capability Indices





LSL

Short Term Process Width

Design Width – Voice Of Customer

USL

T

+ 3s- 3s

Short Term Capability Indices

Min{Cpu, Cpl}

Cp =

Cpk =

Cpu =

Cpl =

(USL – LSL)

6*σshort term

(USL – µ)

3*σshort term

(µ – LSL)

3*σshort term

Long Term Capability Indices For





+ 3s

USLLSL

Process Width – Voice Of Process

Design Width – Voice Of Customer

T

- 3s

Long Term Capability Indices For Process Performance

Pp =

Ppk =

Ppu =

Ppl =

Min{Ppu, Ppl}

(USL – LSL)6*σlong term

(USL – µ)

3*σlong term

(µ – LSL)

3*σlong term

C k Relationship To Z





Cpk Relationship To Z

Point of Interest =USL

Mean

ZUSL = ?

Z =

Point of

Interest

Process Standard

Deviation

– Process

Mean

1

Cpk = 1 corresponds to 3 Sigma performance.

13

3

3

Z

3σ

USLCpk

OK I‟m Ready





OK, I m Ready Let‟s Calculate Capability

• Before collecting data from a process for capability calculations, there

are some things we need to consider

- What type of data is available? (Attribute or variable)

- Is the process stable and in control?

- Is the process output normally distributed?

Let‟s discuss each of the above questions.

Is The Process Stable And In Control?





Is The Process Stable And In Control?

0

0.2

0.4

0.6

0.8

1

1.2

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

?

?

?

?

?

?

?

? ?

3 . . .how it will vary in

the future.

1Based on past experience . . .

2 . . .we can

predict, within

limits . . .

If a process is in control, we can use the data to predict it‟s behavior in

the future, within limits. How do we assess it‟s state of control?

Variation And Control – Review





Variation And Control Review

• All processes can exhibit two types of variation

- Common Cause

• Completely random, natural variation of a process. Common cause

variation arises out the process, or out of the way the process or

organized and updated.

- Special Cause

• Non-random process variation. It is the result of an event, an

action, or a series of events or actions. It is localized in nature.

A process that is in control is free of Special Cause Variation.

How do we separate Special Cause from Common Cause Variation?

Separating Common And Special Causes






Separating Common And Special CausesThe Control Chart

10

9

8

7

6

5

4

3

2

1

0

0 5 10 15 20

Upper Control

Limit

Lower Control

Limit

Mean

Observation number

O b s e r v

a t i o n v a l u e

Region of Common

Cause Variation

Region of Special Cause Variation

Region of Special Cause Variation

NOTE: The data within the

“Region of Common Cause

Variation” is also evaluated

for runs, trends or patterns

that are indicators of Special Cause Variation

(even though no points fall

outside the control limits).

The Control Chart provides a simple means of

identifying Special Cause Variation… We‟ll return to this

later to fully discuss Control Charts and their application

to both process monitoring and process improvement.

MINITAB Capability Example







MINITAB Capability ExampleStability Check

Open File: Process Capability.mpj data column, “Distance 25”

Stat Control Charts Variables Chart for Individuals I-MR








MINITAB Capability ExampleStability Check

Observation

I n d i v i d u a l V a l u e

12110997857361493725131

11.0

10.5

10.0

9.5

9.0

_ X=9.769

UC L=10.542

LCL=8.997

Observation

M o v i n

g R a n g e

12110997857361493725131

1.00

0.75

0.50

0.25

0.00

__ MR=0.290

UC L=0.949

LCL=0

1

11

I-MR Chart of Distance 25

Is The Process







Is The ProcessOutput Normally Distributed?

• Capability Analyses should be done with a distribution type that best

fits the data

• Most Capability Analyses assumes that the data are from a

Normal Distribution

• If we incorrectly assume our data follows a Normal Distribution, our

performance metrics (DPMO, Z, Cp, Cpk, Pp and Ppk) will be incorrectand thus misleading

• Thus, we need to perform a check of normality – Non-normal

Capability Analysis is addressed in the next module

• Recall from the Basic Statistics Module, the Anderson-Darling test

for normality








MINITAB Capability ExampleNormality Check

Distance 25

P e r c e n t

11.010.510.09.59.0

99.9

99

95

90

80

7060504030

20

10

5

1

0.1

Mean

0.421

9.769

StDev 0.3029

N 125

AD 0.370P-Value

Probability Plot of Distance 25Normal

Stat Basic Stats Normality Test

Normally distributed data will appear on the plot as a straight line

If the p-value from the Anderson-Darling test <

alpha of .05, the data is not normal









Stat Quality Tools Capability Analysis Normal

Enter

• Single column: „Distance 25‟.

• Sub-group size: „Sub-group 25‟ or 5

• LSL: 9

• USL: 11

Note that we‟re grouping

the five consecutive shots

taken by a given operator

MINITAB Capability Output







Capab y Ou pu

10.810.510.29.99.69.39.0

LSL Target USL

Process Data

S ampleN 125

StDev (Within) 0.243594

StDev (Ov erall) 0.303479

LSL 9

Target 10

USL 11

Sample Mean 9.76944

Potential (Within) Capability

CC pk 1.37O v erall C apability

Pp 1.10

P PL 0.85

P P U 1.35

Ppk

C p

0.85

C pm 0.87

1.37

C P L 1.05

C P U 1.68

C pk 1.05

O bserv ed Performance

PPM<LSL 0.00

PPM>USL 0.00

PPMTotal 0 .00

Exp. Within Performance

PP M<LSL 792.39

PPM > U S L 0.22

PPMTotal 792.61

Exp. O v erall Performance

PP M<LSL 5616.09

PPM>USL 25. 08

PPMTotal 5641.18

Within

Overall

Process Capability of Distance 25

0.00 ppm Observed…

Simply a count of the data points

outside the spec limits converted to

PPM

Using the within sub-group variation only

(red distribution),

792 ppm fall outside the spec limits

Using the overall

variation (black

dotted distribution,

5640 ppm fall outside

the spec limits

ProcessData

Capability

estimates with

the betweensub-group

variation

removed

Capability

estimates withno sub-grouping

Alternatively, We Can Display







y, p yThe Z Values Directly

In the Options menu of

Stat Quality Tools Capability Analysis Normal

10.810.510.29.99.69.39.0

LSL Target USL

Process Data

S ampleN 125

StDev (Within) 0.243594

StDev (Ov erall) 0.303479

LSL 9

Target 10

USL 11

Sample Mean 9.76944

Potential (Within) Capability

C C pk 1.37

O v erall C apability

Z.Bench 2.53

Z.LS L 2.54

Z.USL 4.05

Ppk

Z.Bench

0.85

C pm 0.87

3.16

Z.LS L 3.16

Z.USL 5.05

Cpk 1.05

O bserv ed Performance

PPM<LSL 0.00

PPM>USL 0.00

PPMTotal 0.00

Exp. Within Performance

PPM<LSL 792.39

PPM>USL 0.22

PPMTotal 792.61

Exp. O v erall Performance

PPM<LSL 5616.09

PPM>USL 25.08

PPMTotal 5641.18

Within

Overall

Process Capability of Distance 25

Process Capability Sixpack







p y p

What about checking for a stable, in-control process? We can check the

state of control, normality and get capability metrics all in one displayStat Quality Tools Capability Sixpack Normal

S a m

p l e M e a n

252321191715131197531

10.00

9.75

9.50

_ _ X=9.7694

UCL=10.0963

LCL=9.4426

S a m p l e R a n g e

252321191715131197531

1.0

0.5

0.0

_ R=0.567

UCL=1.198

LCL=0

Sample

V a l u e s

252015105

11

10

9

10.810.510.29.99.69.39.0

10.510.09.59.0

Within

Overall

Specs

Within

StDev 0.243594

C p 1.37

C pk 1.05

CC pk 1.37

Overall

StDev 0.303479

Pp 1.1

P pk 0.85

Cpm 0.87

1

Process Capability Sixpack of Distance 25

Xbar Char t

R Chart

Last 25 Subgroups

Capability Histogram

Normal Prob Plot AD: 0.370, P: 0 .421

Capability Plot

OK, All Set







,Still A Few Unanswered Questions

• What if I don‟t have long term data? What do I do?

• What will MINITAB do with short term data?

• Can I estimate long term capability from short term data?

• What if I don‟t have rational sub-groups?

All I Have Is Short Term Data And No







Rational Sub-groups… Now What?

• MINITAB wants long term data!

• If your data is short term and there are no obvious sub-groups, you canestimate short term capability by entering a 1 for sub-group size

• When doing this the overall capability, Pp, Ppk, is then considered to be

an estimate of the short term capability

• What is the within called?

- In this case, the within capability is based on the moving range and

thus is an estimate of the VERY short term capability

• Can I estimate the long term capability from my short term estimate?

- You can roughly estimate your long term capability from your short

term estimate, but be cautious – You don‟t know how much your process shifts and drifts

- This estimate is as follows:

Z shift = 1.5 is common industry benchmark

ZLT = ZST – Z shift

Summary For Variable Data







y

• Process Capability is assessed by comparing the process distribution to

the customer specifications (VOC vs. VOP)• For Continuous Data, we calculate Cp, Cpk, Pp, Ppk

• Rational sub-grouping can be used to identify improvement opportunities

• MINITAB “likes” long term data – If you have actual long term

performance and it will estimate your short term performance based onyour chosen sub-groups

• MINITAB will ALWAYS return short term and long term capabilities

regardless of the data you supply it – You must interpret results based on

your knowledge of the data

• If you have only short term data, you can roughly estimate

long term capability






Attribute Capability Analysis:MINITAB Six SigmaProduct Report Module

Case Study 1







yStep 1: The Business Case (Identifying CTS‟s)

The Business Case

• The Duffers Golf Association (DGA) is interested in issuing a credit cardwith their logo on it – They have been soliciting various banks to

determine how well these banks can handle their account

• LDB has made claims that it can satisfy all of the DGA needs –

Based on this limited information the DGA has been evaluating

LDB‟s performance

• The DGA is evaluating several branches from the LDB‟s

many locations

We will evaluate the capability of the Left Dogleg Bank (LDB) to provide a

credit card service

Case Study 1







yBackground Information

Internal to LDB and for the purposes of this Case Study, issuing a credit

card is a four step process that entails:1. Data entry for credit card application

2. Background credit checks

3. Customer setup in the system

4. Issue the card

Case Study 1







Background Information

Within each step of the process, there are more detailed steps that allow

the applications to be converted into a credit card. Listed below are someof the detailed steps:

1.Data entry for credit card application

• Enter various customer demographic information

• Identify bank account numbers

2.Background credit checks• Verify savings

• Debt analysis

3.Customer setup in the system

• Demographic information

• Establish credit limits

4.Issue the card

• Create the card

• Mail

Flow Chart Of Our Case Study







For Each Branch

Data Entry for

Credit

Application

• Demographics• Acct. #‟s

• Salary History

Credit Checks

• Verification of

Savings

• Ver. of Salary• Credit Report

• Debt analysis

Set person up

in system

• Demographics

• Credit Limits

Supplies

Issue Card

• Create Card

• Mate to mailer info

• Mail• Phone Verification

Mail

ClerkINSPECTOR

DGA Member

Received card

and activates

DGA Member

Customer AppliesLeft Dog-Leg Bank

DGA Corporate

evaluates satisfaction

of members

Detailed Flow Chart







(A Subset Of A Credit Card Process)

Start

Receive

Customer

Application

Selectappropriate

Credit

Bureau and

Request info

ReconcileCredit report

with

Applicant file

Does

the report

reconcile?

Was

Applicant

info entered

correctly?

End

Establish

person in

System

End

Reject Applicant

Walk to 3rd

floor and

check

against

Customer

application

Does

applicant meetrequirements?

Yes

No

Yes

No End

Call Credit

Bureau

Yes

Correct info

in Applicant

file

No

X's

- Applicant file

- List of Bureaus

- Computer System

Y's

- Selected

Bureau

- A Requested

Credit Report

X's

- Applicant file

- Requested

Credit Report

Y's

- Comparison of

Report and file

X's

- Comparison of

Report and file

Y's

- Reconciled

Report

Y's

- Identified

Gap

- Applicant

file

X's

- Identified Gap

- Applicant file

- Customer Application

Y's

- Checked Gap

against Application

X's- Reconciled

Report

- List of

Requirements

X's

- Identified Gap

- Applicant file

- Customer Application

Y's

- Corrected file

Step 2: Define Performance Standard







The Spec

• If you don‟t know what makes something good or bad, how can youmeasure the problem in terms of defects (DPMO)?

• If the definition of what makes a defect (the specification) is not in the

same units of measure as the selected CTS from Step 1, you have the

wrong spec

• The definition needs to be crystal clear and brief

• Does your customer agree with your spec?

• For our case a defect is any step (operation,) on any card, with anything

missing or incorrect – Each card has 16 Steps

What is a “good” Y? What is a “bad” Y?

Step 3: Validate Measurement System







Can I See?

• If you can‟t measure something, how do you know where you are,where you have been or where you are going?

• If your Measuring System is incapable, STOP and FIX IT

before proceeding

• Note: MSA must be done on any and all defects you wish to count(watch out if you have a visual inspection of six characteristics)

MUST be done before evaluating capability

Is it really Y?

Step 4: Establish Baseline Capability







Process Capability: A measurement of the capability of the process to

deliver product or services that meet the needs of the customer as definedby the product/service specifications – The most common

Six Sigma measurements of process capability are DPMO and ZBench

Baseline

• I know what needs to be fixed (Step1), what defines it as good or bad

(Step 2) and I can accurately measure it (Step 3) so:

- Now I can say how my process is performing in terms of long and

short term Z and DPMO

- I improve from here

• Common tools: Six Sigma “Product Report” for Attribute Data

What is your ability to make “good” Ys?

Product Capability Report Setup(A ib D )







(Attribute Data)

Total number of

Units (25 cards)

Number of

Opportunities for

defect in each unit

Number of

Defects at the

opportunitylevel

Descriptor:

In this case the

name of thebranch

Product Capability Report Dialog Box(Att ib t D t )







(Attribute Data)

MINITAB Command: Six Sigma Product Report

Product Capability Report Output(Att ib t D t )







Eagle 72500.0 1.500 2.957 0.29993229 25 16 * 29 25 400 1.16

Total 72500.0 1.500 2.95729 400

C omponent DPMO Z.Shift Z.ST YTPDefs

O bs

Units

O bs

Unit

perOpps

Cmplx Defs

A dj

Units

A dj

Opps

Total A dj

DPU

Report 7: Product Performance

(Attribute Data)

Data we entered

Total Opportunities

DPU = Defects Per Unit.

This is the average number

of Defects for every Credit

Card

Z score Short Term (calculated

from the DPMO Long Term andshifted 1.5)

DPMO = Defects Per Million Opportunities.

This is the average number of Defects for

each opportunity multiplied by a million.

Note as MINITAB assumes that Attribute

Data is collected “Long Term”, this is

assumed to be Long Term DPMO = “What

the Customer Feels”

Product Capability Report Additional Output(Att ib t D t )







(Attribute Data)

Z.Bench (Short-Term)

Z . S

h i f t

76543210

3

2

1

0

Technology

Zone of Average

Control

Typical

Zone of

Performance

Six Sigma

Report 8B: Product Benchmarks

Z.Bench (Short-Term)

D P M O

6543210

1000000

100000

10000

1000

100

10

1

Report 8A: Product Benchmarks

A Few Discussion Points







• Process should be stable – This implies that the process is not changing

over time creating significantly different defect rates- May be checked with daily Run Chart of defect rate or by use of an

Attribute Control Chart (more on this later)

- What should we do if our process is not stable?

• We must stabilize first otherwise we do not even know what our baseline is

• As was stated earlier, MINITAB assumes that due to the large amount of

data generally collected for Attribute Data Analysis, the collected data is

assumed to be long term – What the customer actually feels

Some Other Uses Of The ProductC bilit R t







Capability Report

• What if we wished to drill down further to see what department was

making the most errors so that we could target our fix?- See following slides to view a drill down of the Eagle Branch Data

by department

Drill Down







• CAUTION: The prior slides showed us getting the Process Capability at

the branch and company levels• In a real project, you will frequently have sufficient data to measure the

capability at a department level (four departments) or even the operation

level (all 16 sub operations in the total process)

• This “more targeted” evaluation may lead to strong hints as to where to

look for improvements

• As an example we will look at the same “Eagle Branch” data broken down

by the four departments – This time we will use credit cards

as units and each process step (operation) as an opportunity.

Capability Of Eagle By Department







Opps

Defs Units Unit Cmplx Defs Units Opps DPU DPMO Z.Shift

A dj

Z.ST YTP

Data Entry 8 25 4 * 8 25 100

O bs

0.32 80000.0 1.500 2.905 0.726149

C redit C heck 15 25 6 *

O bs

15 25 150 0.60 100000.0 1.500 2.782 0.548812

Set Up A ct 1

per

25 2 * 1 25 50 0.04 20000.0 1.500 3.554

A dj

0.960789

Issue C ard 5 25 4 * 5 25 100 0.20

A dj

50000.0 1.500 3.145 0.818731

Total 29 400 72500.0 1.500 2.957

Total

C omponent

Report 7: Product Performance






Variable Capability Analysis:MINITAB Six SigmaProduct Report Module

Case StudyCatapult Distance







Catapult Distance

Let us revisit the Catapult File: 2270MB00a_Process Capability.mpj

As before we will evaluate capability of hitting a target of 10 inches with aLSL of 9 and a USL of 11

• Open MINITAB file

• Run Six Sigma Process Report

• Fill in as per next slide

Process Report Dialog Box







• For an initial look at the data we will

use the default which is the first tworeports. These will help us verify that

we have collected enough data and

assess the stability of

the process.

• Report two also contains information

to assess the capability of theprocess but we will check data

adequacy and stability first.

• Report 3 provides descriptive

statistics but we should check

stability first.

• Reports 4 – 6 are useful

diagnostic reports for

troubleshooting problems.

Report 1 – Summary







10.810.510.29.99.69.39.0

LSL USL

24222018161412108642

1.0000E+06

10000.000

100.000

1.000

0.000

Date:

Low er S pec: 9Nominal:

O pportunity :

Reported by :

Project:

Department:

Process:

C haracteristic:

Units:

U pper S pec: 11

A ctual (LT) Potential (ST)

Process Performance Process Demographics

A ctual (LT) Potential (ST)

Sigma

(Z.Bench)

DPMO

2.53

5641.2

3.94

40.4

Process Benchmarks

Report 1: Executive Summary

Actual process:

• Not perfectly centred• Long term variation

Process at entitlement:

• Perfectly centred

• Short term variation

Cumulative estimates of

DPMO:

• Short term

• Long term

Critical Metrics:

• ZST

• DPMOLT

Report 2







252321191715131197531

10.00

9.75

9.50

_ _ X=9.7694

UCL=10.0963

LCL=9.4426

Subgroup

252321191715131197531

0.4

0.2

0.0

_ S=0.2290

UCL=0.4783

LCL=0

ST

4.11 4.05

Z.LSL 4.11 2.54

Z.Bench 3.94 2.53

Z.Shift 1.41

LT

1.41P.USL 0.0000202 0.0000251

P.LSL 0.0000202 0.0056161

P.Total 0.0000404 0.0056412

Mean

Yield 100.00 99.44

DPMO 40.4 5641.2

Cp 1.37 *

Cpk

10

1.05 *

CC pk 1.37 *

Pp * 1.10

Ppk *

9.76944

0.85

StDev 0.244 0.303

Z.USL

119

10.73089.26922

119

10.67998.859

Xba r and S C har t

Process Tolerance

Specifications

Potential (ST) Capability

Process Tolerance

Specifications

A ctual (L T ) C apabilit y

Data Source:

Time Span:

Data Trace:

Capability Indices

Report 2: Process Capability for Distance 25

Process statistics and

capability metrics:

• Short term• Long term

X bar S Chart – Stability

Capability plots:

• Short term

• Long term

Interpretation Of Data







• Short term mean is the center

of the process specs• Short term Standard Deviation is

a measure of the Pooled or

“average” within

sub-group variation

• Short term ZBench is the “short

term” Sigma value of the process.

How many Standard Deviations

can fit between the short term

mean and the spec if all defects

are on one side?

• Long term mean is actual mean of

the collected data• Long term Standard Deviation is a

measure of the overall variation of

the process

– What the customer feels

• Long term ZBench is the “long term”

Sigma value of the process. How

many Standard Deviations can fit

between the long term mean and the

spec if all defects are on one side?

Z shift is the shift and drift difference between

Short Term ZBench and Long Term ZBench.

Interpretation(Cont‟d)







(Cont d)

• P. USL = Probability of a defect above the Upper Spec

• P. LSL = Probability of a defect below the Lower Spec

• P. Total = Total probability of a defect whether above the Upper Spec

or below the Lower Spec

• Yield = % good

• PPM = DPMO

• Cp and Cpk = short term indices

• Pp and Ppk = long term indices

Summary Takeaways







• Long term process is centered on mean of actual data and has a

variation associated with all the data – This is what the customer feels• Short term process is centered artificially at the center of the tolerance

and has a variation that is related to the Pooled (“average”) Standard

Deviation of a sub-group – This is the best we could do, if we centered

our process and held everything as constant as possible

• Six Sigma projects should always report, as a minimum, DPMO long

term (what customer feels) and ZBench short term (Entitlement – An

estimate of what we should be able to do with our present process)

Steps In Establishing Process Capability







• Pick out variable for study (Y)

• Set the process to run in “standard” mode

• Record values for key input variables

• ID potential variables for rational sub-grouping (variable data only)

• Run the process (no tweaking!)

• Take notes

• Record key output variable values

• Calculate Capability based on data type

- Variable – Review normality, stability, control then calculate capability using

Capability Analysis and Capability Sixpack- Attribute – Calculate capability using Six Sigma Product Report

• If sub-grouping was done successfully, review key input variables that changed

between groups and determine plan to control (variable data only)

Calculating Capability







All processes can be described by the Z

score, making process comparisons easy.

Attribute Data Variables Data

Six Sigma Product ReportCapability Analysis and Six-pack

and/or Six Sigma Process Report

Sigma (Z)

DPMO DPMO

Key Learning Points







•

•

•

•

•

Objectives Review







The participant should be able to:

• Explain where Capability Analyses fit into the12 Step Breakthrough Strategy

• Compute capability statistics with Variable Data

• Compute capability statistics with Attribute Data

• Discuss the concepts of stability as a prerequisite for Capability Analysis

• Utilize MINITAB for Capability Studies



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Breakthrough Change StrategySM, Breakthrough DesignSM, Breakthrough DiagnosisSM, Breakthrough ExecutionSM, Breakthrough Sigma LeanSM, Breakthrough Six SigmaSM,

Breakthrough Software DesignSM, FASTARTSM, Six Sigma Gold BeltTM, SOLVING YOUR BUSINESS PROBLEMS FOR THE LAST TIMESM. Six Sigma is a federally

registered trademark of Motorola, Inc. MINITAB is a federally registered trademark of Minitab, Inc. SigmaFlow is a federally registered trademark of Compass Partners, Inc.

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