bb wk1 330 process capability for non-normal data
TRANSCRIPT
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Process Capability AnalysisFor Non-normal Data
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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
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Process Capability Analysis
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
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Process Capability Analysis
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.
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Process Capability Analysis
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.
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Process Capability Analysis
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?
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Introduction To Non-normalData Analysis
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Process Capability 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.
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Process Capability Analysis
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
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Process Capability Analysis
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.
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Process Capability Analysis
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
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Process Capability AnalysisFor Non-normal Data
Pg 11
Interpreting The Normal ProbabilityPlot Data = “RS”
Fails both the
AD test and the
“Fat Pencil “test. This data is
non-normal.
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Process Capability AnalysisFor Non-normal Data
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).
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Capability AndNon-normal Data
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Process Capability AnalysisFor Non-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?
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Process Capability AnalysisFor Non-normal Data
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.
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Process Capability AnalysisFor Non-normal 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.
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Process Capability AnalysisFor Non-normal Data
Pg 17
Capability Analysis Of NewProcess Untransformed
Six Sigma Process Report for “New Process” with USL = 5 and
sub-group size = 1.
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Process Capability AnalysisFor Non-normal Data
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
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Process Capability AnalysisFor Non-normal Data
Pg 19
Normality Plots Of “New Process”
Data is in Worksheet
“Case Study.”
Better
Best
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Process Capability AnalysisFor Non-normal Data
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
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Process Capability AnalysisFor Non-normal Data
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
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Process Capability AnalysisFor Non-normal Data
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.
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The Box-Cox Transformation
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Process Capability AnalysisFor Non-normal Data
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
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Process Capability AnalysisFor Non-normal Data
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
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Process Capability AnalysisFor Non-normal Data
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
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Process Capability AnalysisFor Non-normal Data
Pg 27
Six Sigma Process Report
Box-Cox TransformationMINITAB Six Sigma Process Report
MINITAB will automatically
transform the spec limits.
R lt
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Process Capability AnalysisFor Non-normal Data
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
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Process Capability AnalysisFor Non-normal Data
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
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Process Capability AnalysisFor Non-normal Data
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
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Process Capability AnalysisFor Non-normal Data
Pg 31
Zero As A Hard Limit
• How can we proceed to obtain a solution?
R l ti Gi E M
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Process Capability AnalysisFor Non-normal Data
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)
Process Performance Process Demographics
A ctual (LT) Potential (ST )
Sigma(Z.Bench)
DPMO
1.87
30587.4
3.37
372.8
Process Benchmarks
Report 1: Executive SummaryUsing Box-Cox Transformation With Lambda = 0
R l ti Gi E M
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Process Capability AnalysisFor Non-normal Data
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
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Process Capability AnalysisFor Non-normal Data
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)
Process Performance Process Demographics
A ctual (LT) Potential (ST)
Sigma(Z.Bench)
DPMO
1.87
30587.4
3.37
372.8
Process Benchmarks
Report 1: Executive SummaryUsing Box-Cox Transformation With Lambda = 0
Using A Slightly Positive Lambda:
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Process Capability AnalysisFor Non-normal Data
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
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Process Capability AnalysisFor Non-normal Data
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
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Process Capability AnalysisFor Non-normal Data
Pg 37
Rerun With Modified Choice Of Lambda
Six Sigma Process Report
Using A Slightly Positive Lambda:
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Process Capability AnalysisFor Non-normal Data
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)
Process Performance Process Demographics
A ctual (LT) Potential (ST)
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:
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Process Capability AnalysisFor Non-normal Data
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
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Process Capability AnalysisFor Non-normal Data
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
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Process Capability AnalysisFor Non-normal Data
Pg 41
Key Learning Points
•
•
•
•
•
Objectives Review
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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
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Resources
“Montgomery, D.C., “Hypothesis Testing”, Design and Analysis of
Experiments, Fifth Edition, John Wiley and Sons, page 39, 2001.”
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