Bb Wk1 330 Process Capability for Non-Normal Data

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Copyright © 2001-2005Six Sigma Academy International, LLC

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

Process Capability AnalysisFor Non-normal Data



Copyright © 2001-2005

Six Sigma Academy International, LLC


Process Capability Analysis

For Non-normal Data

Pg 1

The Breakthrough Strategy® And Process Capability

• Calculate baselineprocess capability

• Assess process

capability after

improvements are made

1. Select Output Characteristic

2. Define Performance Standards

3. Validate Measurement System

4. Establish Baseline Process Capability

5. Define Performance Objectives

6. Identify Variation Sources

7. Screen Potential Causes

8. Discover Variable Relationships

9. Establish Operating Tolerances

– Implement Improvements

10. Validate Measurement System

11. Determine Final Process Capability

12. Implement Process Controls







For Non-normal Data

Pg 2

Module Objectives

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

• Explain the need for reasonable normality when assessing process

capability for continuous data

• Assess data for normality

• Identify data distributions that can be readily transformed

• Transform a Non-normal Distribution into a Normal Distribution using

Box-Cox Power Transformation Capability Analysis

• Calculate the DPMO and Z of a transformed data set using MINITAB®’s

Six Sigma Process Report Module







For Non-normal Data

Pg 3

• Capability Analysis assumes that the data are from a near

normal distribution

- p(Y>USL), p(Y<LSL), and ZBench all utilize the Normal Distribution for

probability assessments

• Many processes do not produce data that follow a Normal Distribution

• Non-normal Capability Analysis allows the use of statistics that are

predicated upon a normal distribution for processes that do not produce

normally distributed data

• Provides process metrics for Benchmarking

• Easily accomplished in MINITAB

Why Learn Non-normalCapability Analysis? (1)

DPMO Conversion (continuous data) is based upon a

normal distribution.







For Non-normal Data

Pg 4

Capability Analysis

• Loses accuracy for non-normal sample data (inaccuracies can be either

high or low)

• If the data are highly skewed, the tail on one side will be very short, while

the tail on the other side will be very long

- Standard Deviation estimates (both long term and short term) can be

affected, usually they are inflated

- Using Z to estimate the probability of being outside the spec limit can

produce erroneous estimates

• If the data are not normal, but the distribution is fairly symmetric, effectsof non-normality are minimized

Why Learn Non-normalCapability Analysis? (2)

Capability Analysis usually assumes

normal data.







For Non-normal Data

Pg 5

• Moderate departures from normality, especially from distributions that are

symmetric may not pose much of an issue – Remember no data isperfectly normal and that the Z and DPMO values are only estimates

• Strong departures from normality can cause serious errors in the DPMO

and Z values

What If Non-normal Data IsEvaluated As Normal?






Introduction To Non-normalData Analysis







For Non-normal Data

Pg 7

Unfortunately not all process data follows a normal distribution.

Process Distributions May Vary

10

0

5

10

22.5 23 .5 2 4.5 2 5.5 26 .5 27 .5 2 8.5 29.5 30.5 31.5

0

5

2 2 23 24 25 26 27 2 8 29 3 0 3 1

Reasonably Normal Significant Right Skew

F r e q u e n c y

F r e q u e n c y

Normal Non-Normal

Developing accurate Z scores for non-normal data is a

fundamental necessity for benchmarking.







For Non-normal Data

Pg 8

Review: Testing For Non-normality

• Normal probability plots are a graphical technique to determine if a

distribution is normally distributed

• Using data file: Process Capability Non Normal.mpj

- Make worksheet “Review” active

- Stat Basic Stats Normality Test and evaluate “Normal” and “RS”

- Also graphically look at the data

- Graph Histogram Simple

with Fit

- Also Graph Probability Plot

[Distribution – Normal]Single,produces these graphs







For Non-normal Data

Pg 9

Normally distributed data will appear on the plot as a reasonably

straight line.

Interpreting The Normal ProbabilityPlot Data = “Normal”

Since p-value

> 0.05, we saythe data is

normal.







For Non-normal Data

Pg 10

Anderson-Darling Test

• If the p-value from the Anderson-Darling test < Alpha of .05, the data is

not normal per that test,

• However:

- The Anderson-Darling test is not robust to small sample sizes

- For samples less than 50 it is best to rely on the “Fat Pencil” test. If a

fat pencil can cover all of the points on the normal probability plot, the

data may safely be treated as normal.

- For large samples the Anderson-Darling can measure slight

departures from normality that will have little or no effect on the level of

analysis that we will be performing. Again use the “Fat Pencil” test todetermine reasonable normality.

- This is the only instance in this material where the graphical analysis is

considered the primary analysis and the p-value secondary







Pg 11

Interpreting The Normal ProbabilityPlot Data = “RS”

Fails both the

AD test and the

“Fat Pencil “test. This data is

non-normal.




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


Pg 12

Some Examples

• Make worksheet “Examples” active

• Run both a Histogram and a Normality Test on the following columns

- Fat Tail (acceptable) – Applying the assumption of normality here will slightly

underestimate the probability of a defect

- Thin Tail (not acceptable) – Applying the assumption of normality to a thin tail

distribution is a conservative approach as it overestimates the probability of adefect. Given symmetry and a reasonable histogram, we can accept the

assumption of normality. However, this thin tail distribution is NOT symmetric.

- Slight Skew (acceptable)

- Heavy Skew (not acceptable)

- Uniform – Worse case of fat tailed distribution (not acceptable)

- Bi-modal (not acceptable)

What do we do when we have data that is not reasonably normal?

We transform it (when we can).






Capability AndNon-normal Data






Pg 14

The Six Sigma team has a team member who insists upon knowing the fix

to the problem. He implements a change. After data is collected for theNEW PROCESS, (Worksheet “Case Study”) he runs a Capability Analysis.

The following summary is presented at the weekly team meeting:

Parameter Old Process New Process

Mean 1.54 1.24

StDev 1.42 1.03 (Smaller is better)

DPMOLT 7444 132 (Lower is better)

Case Study

Should the team implement the change as an improvement?






Pg 15

Torturing The Data A Little More

• Never make decisions solely based upon a tabulation of process

descriptive statistics. Always plot your data.• Graph Histogram

Simple with Fit

• Click “Multiple Graphs”

The histogram for the New Process looks pretty badly skewed – A

normality test should be run on the New Process data.






Pg 16

Normal Probability Plot Of New Process

Since p < 0.05 and it completely fails the “Fat Pencil” test,

this data IS NOT from a normal population.






Pg 17

Capability Analysis Of NewProcess Untransformed

Six Sigma Process Report for “New Process” with USL = 5 and

sub-group size = 1.






Pg 18

Let Us Transform The Data

• What is transformed data?

- We take non-normal data and perform a mathematical operation on the

data (including the spec limits) such that the new data is normal

• What are some common transformations (there are many)?

- Square root

- Natural Log

- Reciprocal (1/X)

• How do I choose a proper transform?

- The Box-Cox method will, in many cases, allow MINITAB to apply anappropriate transform

• We shall first demonstrate the Square Root and Natural Log Transforms






Pg 19

Normality Plots Of “New Process”

Data is in Worksheet

“Case Study.”

Better

Best






Pg 20

Histograms Of “New Process”

F r e q u e n c y

543210-1

30

20

10

01.500.750.00-0.75-1.50-2.25

16

12

8

4

0

2.01.61.20.80.4

24

18

12

6

0

New Process Natural Log

Square Root

New Process

-0.1092

StDev 0.8358

N 100

Square Root

Mean 1.030

StDev 0.4225

Mean

N 100

1.237

StDev 1.029

N 100

Natural Log

Mean

Histogram of New Process, Natural Log, Square RootNormal






Pg 21

Capability Comparisons

• The normality plots show that the Natural Log transform is better (more

normal) than the Sq.Root transform which is better than the raw data• Let us run a capability analysis for each condition using

- Six Sigma Process Report

- Sub-group size = 1

- USL as appropriate for each case – Must use transformed limits[See worksheet for USL values]

• Which Capability Value is the best estimate? Why?

As Measured Square Root Natural Log






Pg 22

Parameter Old Process New Process Actual

New Process

Mean 1.53 1.23

1.23*

StDev 1.86 1.03

1.03*DPMO 7444 132

20134**

*un-Transformed

** Transformed using the Natural Log

The Real Capability Comparison

Normal Process Capability is based upon the assumption

of normality, therefore, in this case, the data closest to

Normal (Natural Log transform) should be used.





The Box-Cox Transformation






Pg 24

What Is The Box-Cox Transformation?

• The Box-Cox Procedure for transforming data was developed by G.E.P.

Box and D.R. Cox. Essentially, it transforms the data by raising it to thepower l, where l is any number between -5 and 5

• It is an automated method within MINITAB to determine and directly

apply the proper transform

• Some values of l have specific meanings:

- Square root l = .5

- Natural log l = 0

- Reciprocall

= -1- Reciprocal Square Root λ = -0.5






Pg 25

Box-Cox Transformation – New Process (Stand Alone)

Make the worksheet Case Study.mtw active

Stat Control Charts Box-Cox TransformationNote: By clicking on the

Options box, you can store the

transformed data

C f






Pg 26

Box-Cox Transformation Results

• Vertical lines indicate 95% confidence intervals for Lambda

• Any Lambda within the interval is suitable• If the confidence interval contains 1 no transformation is required

• Choose a practical lambda if one exists (one that has meaning). In this case,

the suggested lambda is 0.0. We can apply a natural logarithm

We saw earlier

that the natural

log transformyielded a normal

distribution

B C T f ti






Pg 27

Six Sigma Process Report

Box-Cox TransformationMINITAB Six Sigma Process Report

MINITAB will automatically

transform the spec limits.

R lt






Pg 28

Results

Note MINITAB chose to use a Lambda = 0 Natural Log transform

Or iginal Data

1.500.750.00-0.75-1.50-2.25

USL

1009080706050403020101

100000010000

1001

0

Date:

Lower Spec:

Nominal:

O pportunity:

Reported by :

Project:

Department:

Process:

C haracteristic:Units:

Upper Spec: 1.60944 A ctual (LT)

Process Performance Process Demographics

A ctual (LT) Potential (ST)

Sigma(Z.Bench)

DPMO

2.05

20134.2

3.55

191.9

Process Benchmarks

Report 1: Executive SummaryUsing Box-Cox Transformation With Lambda = 0

Results are identical tomanual transform

using the Natural Log.

Li it ti O B C






Pg 29

Limitations On Box-Cox:Overcoming Error Messages

• MINITAB will warn you of two issues with error messages:

1.* ERROR * Specification limits must be positive when using theBox-Cox transformation with a non-positive lambda

• This occurs frequently with time (or other) data where the lower spec is Zero. Time data is frequently right skewed making it anotherwise good candidate for the Box-Cox

• Solution 1: Use a hard boundary of a slightly positive number

• Solution 2: Use a slightly positive lower spec limit

• Solution 3: Use a slightly positive Lambda when possible(providing the choice of Lambda is within its 95% CI – MINITAB)

2.* ERROR * All data must be positive when using theBox-Cox transformation

• Solution 1: Add a constant just large enough to shift thedistribution so that all data values are positive. Add the sameconstant to the upper and lower spec limits

• Solution 2: If all data (including specs) is negative, first multiply by(-1). Note: This will reverse the direction of any skew

Z A A H d Li it






Pg 30

• Your company has an EPA site permit to operate at a maximum of 2.5 ppm

emission level for the total plant. If the process emission is over 2.5 ppm, the sitewill be over the allowable limit and MUST shut down one or more processes to

keep the total site emissions under the maximum

• Your emission level is:

1.267 average; 0.543 StDev

• What percentage of time can the plant manager expect to have to shut a process

down due to process emissions?

• Make worksheet “EPA ” active

• NOTE THAT THE LOWER BOUNDARY IS ZERO; we cannot have less than

zero emissions

• Evaluate normality and capability of non-transformed data (significantly

right skewed)• Running Six Sigma Process Report with Specs as given (0 as hard LSL, 2.5 as

USL) and using Box-Cox results in an error message:

- “Specification limits must be positive when using the Box-Cox transformation

with a non-positive lambda.”

Zero As A Hard Limit

Z A A H d Li it






Pg 31

Zero As A Hard Limit

• How can we proceed to obtain a solution?

R l ti Gi E M






Pg 32

Resolution Given Error Message: Analysis With No Lower Spec

• Since we are only concerned with exceptions to the upper spec limit, we can

simply use no lower specification limit in our solution• Repeat the commands on the previous page, but do not enter a value for the

lower spec limit

Original Data

1.20.80.40.0-0.4-0.8

US L

757065605550454035302520151051

1000000

100000

10000

1000

100

10

1

Date:

Low er Spec:

Nominal:

O pportunity :

Reported by :

Project:

Department:

Process:

C haracteristic:

Units:

Upper Spec: 0 .916291 A ctual (LT)


A ctual (LT) Potential (ST )

Sigma(Z.Bench)

DPMO

1.87

30587.4

3.37

372.8

Process Benchmarks


R l ti Gi E M






Pg 33

Resolution Given Error Message:Use A Slightly Positive Lower Spec

• Enter 0.001 as the lower Spec

• Make sure “Optimal Lambda” is selected in the Options

Using A Slightl Positi e Spec Limit Res lts






Pg 34

Using A Slightly Positive Spec Limit: Results

Or iginal Data

1.20.0-1.2-2.4-3.6-4.8-6.0

LSL USL

757065605550454035302520151051

1000000

100000

10000

1000

100

10

1

Date:

Lower Spec: -6.90776

Nominal:

O pportunity :

Reported by :

Project:

Department:

Process:

Characteristic:

Units:

Upper Spec: 0.916291 A ctual (LT)



Sigma(Z.Bench)

DPMO

1.87

30587.4

3.37

372.8

Process Benchmarks


Using A Slightly Positive Lambda:






Pg 35

Using A Slightly Positive Lambda:Checking The Value Of Lambda

• Since MINITAB gave us a message that said that we had a Non-Positive

Lambda, let us find out exactly what MINITAB proposed to use

• Stat Control Charts Box-Cox Transformation

• Make sure “Optimal Lambda” is selected in the Options

95% Confidence Limits Of Lambda






Pg 36

95% Confidence Limits Of Lambda

Lambda

S t D e v

5.02.50.0-2.5-5.0

2.5

2.0

1.5

1.0

0.5

LowerCL UpperCL

Limit

Lambda

0.000000

(using 95.0% confidence)

Estimate -0.199885

LowerCL -0.823746

UpperC L 0.388547

Best Value

Box-Cox Plot of Stack Particulate PPM

MINITAB,

chose “zero”

as our optimal;

hence our problem.

We are given

a range (95%

CI) of from -

0.82 to +0.39.

Let us use the

small positive

value of

Lambda =

0.001

Rerun With Modified Choice Of Lambda






Pg 37

Rerun With Modified Choice Of Lambda

Six Sigma Process Report

Using A Slightly Positive Lambda:






Pg 38

Using A Slightly Positive Lambda:Results With Lambda = 0.001

Or iginal Data

1 . 0 0

0 8

0 . 9 7

3 0

0 . 9 4

5 2

0 . 9 1

7 4

0 . 8 8

9 6

0 . 8 6

1 8

0 . 8 3

4 0

0 . 8 0

6 2

LSL USL

757065605550454035302520151051

1000000

100000

10000

1000

100

10

1

Date:

Low er Spec: 0

Nominal:

O pportunity:

Reported by :

Project:

Department:

Process:

C haracteristic:

Units:Uppe r Spec: 1.00092

A ctual (LT)



Sigma

(Z.Bench)

DPMO

1.87

30557.6

3.37

372.2

Process Benchmarks

Report 1: Executive SummaryUsing Box-Cox Transformation With Lambda = 0.001

Summary Of Results:






Pg 39

Summary Of Results: All Three Approaches

Approach No Lower Spec

Slightly

Positive

Lower Spec

Slightly

Positive

Lambda

ZST 3.37 3.37 3.37

DPMOLT 30,587 30,587 30,557

• In this case, using no lower spec limit (and therefore no lower boundary

value) or a slightly positive lower spec designated as a boundary, will

yield the same results

• Using a slightly positive lambda is not the preferred approach in this case

Comments






Pg 40

Comments

• Right skewed data is relatively easy to transform using Box-Cox

• Standard transforms do not work well for left skewed or uniform data

• Sometimes combinations of functions are used

- Ln(1/x) or Ln(-1*x)

• With Bi-modal distributions that can be dissected to determine themeans and Standard Deviations of both subsets, the individual DPMOs

can be calculated and then summed in proportion to their presence in

the process

• Most importantly: Most data is normal enough to use as is

It will be rare when you cannot use either the “as is” data

or a simple Box-Cox transform. When you do, consult

with your Master Black Belt as Minitab has additional

functionality (i.e. Johnson Transform methods).

Key Learning Points






Pg 41

Key Learning Points

•

•

•

•

•

Objectives Review





Process Capability AnalysisFor Non-normal Data Pg 42

Objectives Review

The participant should be able to:

• Explain the need for reasonable normality when assessing process

capability for Continuous data

• Assess data for normality

• Identify data distributions that can be readily transformed

• Transform a non-normal distribution into a normal distribution using

Box-Cox Power Transformation Capability Analysis

• Calculate the DPMO and Z of a transformed data set using MINITAB’s

Six Sigma Process Report Module

Resources




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

Resources

“Montgomery, D.C., “Hypothesis Testing”, Design and Analysis of

Experiments, Fifth Edition, John Wiley and Sons, page 39, 2001.”




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Bb Wk1 330 Process Capability for Non-Normal Data