A review / introduction of problem solving tools for achieving process improvement / control and waste reduction - this is available as an animated PowerPoint presentation

1. Making a Quality Product
Product
??????
What is required to make a product?
A REVIEW / INTRODUCTION OF PROBLEM SOLVING TOOLS FOR
ACHIEVING PROCESS CONTROL AND WASTE REDUCTION
please contact mrdrking@gmail.com for an animated PowerPoint presentation
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2. Making a Quality Product
Product
Raw
Material Processing
Cell
The process needs:
the raw materials ...
the equipment to produce the product ...
Is that all?
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3. Making a Quality Product
Process
Control
Process Control Chart
Product
Raw
Material Processing
Cell
The process also needs ... regulation or control using ...
A limited amount of the raw materials ...
- how much raw material can be processed at one time?
A limited range on the control factors ...
- temperature: how hot or cold?
- time: what duration?
Monitoring of materials and parameters ...
Is this enough to always make a good product?
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4. Making a Quality Product
Process
Control
Process Control Chart
Product
Raw
Material Processing
Cell
Sure! Why not?!
So start the process and make product.
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5. Making a Quality Product
Process
Control
Process Control Chart
Product
Raw
Material Processing
Cell
The customer expects uniformity.
Does all the product behave the same and conform to
the manufacturing specifications?
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6. Making a Quality Product
Process
Control
Process Control Chart
Product
Raw
Material Processing
Cell
Wait a second!
What’s this?
This product is different!
The customer won’t accept this part!
So this product gets trashed.
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7. Making a Quality Product
Process
Control
Process Control Chart
Product
Raw
Material Processing
Cell
And there is more trash,
and more ... $
and more ...
$
Hey, this is getting expen$ive!!
How can this be improved?
TRASH
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8. Making a Quality Product
Process
Control
Process Control Chart
Product
Raw
Material Processing
Cell
Tell the operator when bad Feedback $
product is made and to
watch the process better.
But the operator claims all
process parameters are
? $
being maintained!
What else can be done?
TRASH
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9. Making a Quality Product
Process
Control
Process Control Chart
Product
Raw
Material Processing
Cell
Feedback
Data
Find out what conditions produce SPC Chart
very good or bad product. Control
Inspection establishes data on the Charts
normal output of all product.
It would be easiest to monitor all output and look at what
conditions existed when a deviation from normal occurs.
Data is easily organized and interpreted with a Control Chart.
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10. Making a Quality Product
Process
Control
Process Control Chart
Product
Raw
Material Processing
Cell
Data has
several uses ... Feedback
Feedback Data
Control Charts produce SPC Chart
improvements by comparing Control
typical and unusual data
Charts
Design of Efficient experiments produce data that results in an
Experiments improved process yielding a better product
. (DOE)
Process Data is used to estimate the
ability of the process to produce
Capability conforming product
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11. Making a Quality Product
Process
Control
Process Control Chart
Product
Raw
Material Processing
Cell
Feedback
Feedback Data
Design of Experiments Engineering Analysis Control Charts
SPC Chart
Optimize Output Process Capability ID out-of-control events
Reduce Variation Cp > 2.0 TYPES
Factorial Design Cpk > 1.5 Variable (measurable)
Conventional & Taguchi Attribute (yes/no, on/off)
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12. Making a Quality Product
We will look at
Process
Control
How to identify an out-of-control
Process Control Chart
Product
Raw
process withProcessing
Material statistical process
Cell
control (SPC).
How to predict the amount of
Feedback Data
Feedback
non-conforming product from the
Design of Experiments Engineering Analysis Control Charts
process data. SPC Chart
Optimize Output Process Capability ID out-of-control events
Reduce Variation Cp > 2.0
How to improve the process by
Factorial Design
Conventional & Taguchi
Cpk > 1.5
TYPES
Variable (measurable)
conducting efficient experiments.
Attribute (yes/no, on/off)
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13. Making a Quality Product
SPC and DOE Reduce Variation in a Process
SPC Chart
Control Charts - reduce special (non-random) causes.
They are used by the operator as a feedback mechanism
to correct problems shown by the control chart.
Engineering Analysis - compares the process
capability to process tolerance. Scrap is reduced
when parts are processed through areas capable
of holding tolerance.
Design of Experiments - analyze the influence of factors
that cause variation. Factors are deliberately changed in an
controlled and organized fashion so that their effects can
be analyzed and then optimized to reduce output variation.
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14. Making a Quality Product
Here is an example of making and testing bullets to illustrate:
control charts
design of experiments
engineering analysis
The test of a well made bullet is to hit the target bull’s eye
•
This is what the customer and manufacturer wants!
Is this always produced?
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15. Making a Quality Product
Of course we can’t expect every bullet to be identical.
••
• ••
•
So we will look the process of making a bullet and show:
process control -
How are factors controlled in the manufacturing?
control charts -
Why do weed need control charts?
Show the measure of good performance.
Show when the process has poor performance.
engineering analysis -
Predict the amount of scrap.
design of experiments -
Show how to improve process performance.
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16. Making a Quality Product
So what are the input factors to be controlled in the manufacture. Let’s
assume only three factors require monitoring for process control.
A heavier weight projectile is slower so it hits the
target lower than a lighter and faster projectile, but
too little weight and the wind affects the path.
The path of a smaller diameter projectile is erratic
since the projectile wobbles, but too large and it
Projectile doesn't fit the barrel.
More powder weight makes the projectile faster
and less makes it slower.
Powder
No case factors influence bullet quality. Here, this
was chosen for convenience, but acquiring from
an approved vendor could reduce monitoring.
case
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17. Making a Quality Product
We need process control to monitor the input variables
The projectile has manufacturing limitations:
a maximum and minimum weight
a maximum and minimum diameter
Projectile
weight and diameter
The powder has manufacturing limitations:
a maximum and minimum weight
Powder weight
So let’s look at the process control or “rainbow”
charts for several of the most recent lots of bullets.
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18. Making a Quality Product
Process control monitors the input variables
Here are the “rainbow” charts for the lots 980701 through 980707
Operation Characteristic: WEIGHT of PROJECTILE
DATE 980701 980702 980703 980704 980705 980706 980707
TIME
MAX
PROJECTILE WEIGHT
MIN
INITIALS
OK
NOTES
Operation Characteristic: DIAMETER of PROJECTILE
DATE 980701 980702 980703 980704 980705 980706 980707
Projectile TIME
MAX
weight and diameter
PROJECTILE DIAMETER OK
MIN
INITIALS
NOTES
Operation Characteristic: WEIGHT of POWDER
DATE 980701 980702 980703 980704 980705 980706 980707
TIME
MAX
Powder weight MIN
POWDER WEIGHT OK
INITIALS
NOTES
Let’s look at the testing of these lots.
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19. Making a Quality Product
Control Charts monitor the output variables
To measure the quality of the product, a few of the bullets from lot must
be tested; this is called a sample. A sample is used because you can’t
use the entire lot in testing or there would be nothing left to sell.
The quality of the lot is determined by
the spread of the hole pattern
and
the distance the center of the spread is to the center of the bull’s eye .
• ••
•
• •
Here is the testing of lot 980701.
Let’s look closer at this pattern and put the results into a control chart.
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20. Making a Quality Product
So we will look the process of making a bullet and show:
Let’s look at
process control -
How are factors controlled in the manufacturing?
control charts -
Why do weed need control charts?
Show the measure of good performance.
Show when the process has poor performance.
DISCUSSION
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21. Making a Quality Product
Control Charts monitor the output variables
•
••
•• •
Test pattern of
lot 980701
The diameter of the blue circle around the pattern is 7 inches in
diameter. This circle represents the pattern spread and is a measure
of variation.
This distance from the center of the pattern to the center of the bull’s
eye is 6.5 inches. This is the a location measurement which compares
the output to the desired or true value.
A proper evaluation requires a variation and a location measurement.
Control charts plot both location and variation output measurements.
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22. Making a Quality Product
Variable Control Charts
Variable Control Chart (Average and Range)
Part Number Chart No.
Part Name (Product) Operation (Process) Specification Limits
Operator Machine Gage Unit of Measure Zero Equals
DATE
TIME
1
2
3
4
5
SUM
AVERAGE
RANGE
NOTES
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Control charts plot both location and variation output
measurements. On this control chart the location is
called the average and the variation is called the range.
Control charts also have boundaries called UCL and
LCL which stands for upper and lower control limits.
These boundaries represent values that a stable
process should not exceed. When the control
boundaries are exceeded, the operator needs look for
something that may be wrong with the process.
Let’s fill out the chart with the results from 980701.
Any change in people, equipment, materials, methods or environment to be noted on the reverse side; the notes will help to make corrections / improvements when indicated by the control chart.
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23. The chart is provided with previously established process control limits.
First fill in the information required intoQuality Product
Making a the header.
Part Number Chart No.
Part Name (Product)
Variable Control Chart (Average and Range)
Operation (Process)
123
Specification Limits
29
Big Bullet Final Test See Customer Spec
Operator Machine Gage Unit of Measure Zero Equals
Kim Tester #7 Tester #7 inch 0.0
DATE 01
TIME
1
2
3
4
5
SUM
AVERAGE 6.5 this is the distance of the pattern from the bull’s eye - the location of the sample data
RANGE 7.0 this is the diameter of the pattern - the variation of the sample data
NOTES
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
UCL 8
•
average
6
4
LCL 2
15
range
UCL
10
5 •
Let’s look at the tests for the remaining lots.
Any change in people, equipment, materials, methods or environment to be noted on the reverse side; the notes will help to make corrections / improvements when indicated by the control chart.
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24. Making a Quality Product
Oh good! We are just in time to see the tests of lots 980702 thruough 980707.
• • ••••
•
•••• •• •
•• •
•
that is 980702 to 980704
•
• • ••
• ••• ••• • •
• • • •
that is 980705 to 980707
Record the patterns of location and variation from the targets
and then plot them on the control chart.
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25. Making a Quality Product
980702 980703 980704 Fill in the table in with the
variation and location results.
• • ••••
•
•••• •• •
•• •
lot variation location
•
980702 6.5 2.5
6.5, 2.5 7.0, 3.0 5.0, 5.5
980703 7.0 3.0
• 980704 5.0 5.5
• • ••
• •• • • 980705 7.0 1.5
• •• • • •
•
7.0, 1.5 7.5, 1.0 •
13.5, 1.5 980706 7.5 1.0
980707 13.5 1.5
980705 980706 980707
Use this table to fill in
the control chart.
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26. Making a Quality Product
Part Number Chart No.
Variable Control Chart (Average and Range) 123 29
Fill in the information for lots
Part Name (Product)
Big Bullet Operation (Process)
Final Test 980702 to 980704. Spec
See Customer
Specification Limits
Operator Machine Gage Unit of Measure Zero Equals
Kim Tester #7 Tester #7 inch 0.0
DATE 01 02 03 04
TIME
1
2
3
4
5
SUM
AVERAGE 6.2 2.5 3.0 5.5
RANGE 7.0 6.5 7.0 5.0 Now plot the points on the average and range graphs
NOTES
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
UCL 8 lot variation location
•
average
6 980702 6.5 2.5
• 980703 7.0 3.0
4
• 980704 5.0 5.5
LCL 2
• 980705 7.0 1.5
980706 7.5 1.0
15
range
UCL 980707 13.5 1.5
• • • •
10
5
Let’s look at this before finishing.
Any change in people, equipment, materials, methods or environment to be noted on the reverse side; the notes will help to make corrections / improvements when indicated by the control chart.
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27. Making a Quality Product
Part Number Chart No.
Variable Control Chart (Average and Range) 123 29
Part Name (Product)
Big Bullet Operation (Process)
Final Testthe location and variation
All of See Customer Spec
Specification Limits
Operator Machine Gage Unit of Measure Zero Equals
Kim Tester #7 Tester #7 inch 0.0
DATE 01 02 03 04
data looks normal so the process
TIME
1
is behaving as expected.
2
3 None of the new values exceed
4
5
the dotted lines which are the
SUM control limits that signal when to
AVERAGE 6.2 2.5 3.0 5.5
RANGE 7.0 6.5 7.0 5.0 look for problems within the
NOTES
1 2 3 4 5 6 7 8 9 10 11 12 13 process.
14 15 16 17 18 19 20 21 22 23 24 25 26
UCL 8 lot variation location
•
average
6 980702 6.5 2.5
• 980703 7.0 3.0
4
• 980704 5.0 5.5
LCL 2 • 980705 7.0 1.5
980706 7.5 1.0
15
range
UCL 980707 13.5 1.5
10
5 • • • •
Let’s continue.
Any change in people, equipment, materials, methods or environment to be noted on the reverse side; the notes will help to make corrections / improvements when indicated by the control chart.
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28. Making a Quality Product
Part Number Chart No.
Variable Control Chart (Average and Range) 123
Ok, you know there is something 29
Part Name (Product)
Big Bullet Operation (Process)
Final Testwith the remaining data.Spec
wrong See Customer
Specification Limits
Operator Machine Gage Unit of Measure Zero Equals
Kim Tester #7 Tester #7 inch 0.0
DATE
TIME
01 02 03 04 05 06 07 Think about where the data
1
2
becomes unusual and what to do.
3
4
5
SUM
AVERAGE 6.2 2.5 3.0 5.0 1.5 2.0 2.5
RANGE 7.0 7.5 7.0 5.0 7.0 7.5 13
NOTES
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
UCL 8 lot variation location
•
average
6 980702 6.5 2.5
• 980703 7.0 3.0
4
980704 5.0 5.5
•
•
LCL 2 • •
980705 7.0 1.5
• 980706 7.5 1.0
15
•
range
UCL 980707 13.5 1.5
10
5 • • • • • •
Do you see a problem?
Any change in people, equipment, materials, methods or environment to be noted on the reverse side; the notes will help to make corrections / improvements when indicated by the control chart.
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29. Making a Quality Product
Part Number Chart No.
Variable Control Chart (Average and Range) 123
OK. There are some hints here! 29
Part Name (Product)
Big Bullet Operation (Process)
Final Test See Customer Spec
Specification Limits
Operator
Kim Machine
Tester #7 • DidTester #7
Gage you think the red location
inch 0.0 Unit of Measure Zero Equals
DATE 01 02 03 04 05 06 07 value was a problem?
TIME
1 • Did you think the blue variation
2
3 value was a problem?
4
SUM
5 • Are both a problem?
AVERAGE 6.2 2.5 3.0 5.0 1.5 2.0 2.5 • Maybe neither are a problem.
RANGE 7.0 7.5 7.0 5.0 7.0 7.5 13
NOTES
1 2 3 4 5 6 7 8 9 10 11 12 13 Do both values have to exceed a
14 15 16 17 18 19 20 21 22 23 24 25 26
UCL 8 limit at the same time to act?
•
average
6
• What do you think and why?
4
•
LCL 2 •
• • •
15 •
range
UCL
10 Take a minute to think.
5 • • • • • •
Any change in people, equipment, materials, methods or environment to be noted on the reverse side; the notes will help to make corrections / improvements when indicated by the control chart.
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30. So what do you think?
Variable Control Chart (Average and Range)
Part Name (Product)
Big Bullet Operation (Process)
Oh-Oh!?
Operator
DATE
TIME
Kim
01 02 03 04 05 06 07
Machine
Tester
Thinking The red dot, the blue dot,...
1
2 Both, neither,...
3
4
5 Maybe it’s a trick and
SUM
AVERAGE 6.2 2.5 3.0 5.0 1.5 2.0 2.5 it’s all the above.
RANGE 7.0 7.5 7.0 5.0 7.0 7.5 13
NOTES
1 2 3 4 5 6 7 8 9 10 11
UCL 8
•
average
6
•
4
• •
•
LCL 2
• •
15 •
range
UCL
10
5 • • • • • •
Any change in people, equipment, materials, methods or environment to be noted on the reverse
OK. Here’s the answer and why.

31. Making a Quality Product
Part Number Chart No.
Variable Control Chart (Average and Range)Certainly you would stop and look if 123 29
Part Name (Product)
Big Bullet Operation (Process)
Final Test Specification Limits
the location upper controlCustomer Spec See limit was
Operator Machine Gage
Kim Tester #7 exceeded. #7 That means Zerothe0.0
Tester Unit of Measure
inch Equals
hole
DATE 01 02 03 04 05 06 07 pattern has shifted a large distance
away from the bull’s eye and that is
TIME
1
2
3
bad.
4
5
But the location has exceeded the
SUM lower control limit (LCL).
AVERAGE 6.2 2.5 3.0 5.0 1.5 2.0 2.5
RANGE 7.0 7.5 7.0 5.0 7.0 7.5 13 That would mean that the hole pattern
NOTES
1 2 3 4 5 6 7 8 9 10 11 12 13 14 was close to the 20
15 16 17 18 19 bull’s 22 23 and that’s
21 eye 24 25 26
UCL 8 good. Why tell anyone if the process
is better than what is expected?
•
average
6 Well if the process got better perhaps
we can figure out why the process is
• better. So always look at what is
4 happening to the process when any
• control limit is exceeded.
LCL 2 •
• • • A special note. The variation limit
has not been exceeded at the same
time as the location value. This
15 •
range
UCL means that this may be a rare
10
• • • •
exception when a limit is exceeded
5 • • although the process is okay.
Any change in people, equipment, materials, methods or environment to be noted on the reverse side; the notes will help to make corrections / improvements when indicated by the control chart.
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32. Making a Quality Product
Part Number Chart No.
Variable Control Chart (Average and Range)Now if a variation and a location 123 29
Part Name (Product)
Big Bullet Operation (Process)
Final Test limit are exceeded at Spec
control Specification Limits
See Customer the
Operator Machine Gage Unit of Measure Zero Equals
Kim Tester #7 same Tester #7
time there is usually a real
inch 0.0
DATE 01 02 03 04 05 06 07 problem.
TIME
1
2
But the variation limit has been
3 exceeded by itself. Does this mean
4
5
there is a probelm?
SUM
AVERAGE 6.2 2.5 3.0 5.0 1.5 2.0 2.5 YES!
RANGE 7.0 7.5 7.0 5.0 7.0 7.5 13
NOTES A “well behaved” process will usually
1 2 3 4 5 6 7 8 9 10 11 12 13 14
have 16 17 18variation. When variation
15
stable 19 20 21 22 23 24 25 26
UCL 8 changes there is a good chance that
something has definitely influenced
•
average
6 the process.
•
4 When any control limit is exceeded,
assume there is a problem and look
•
• • •
for a source that influences the
LCL 2
• variation and/or the location value.
15 •
range
UCL
10
5 • • • • • • How are these problems identified?
Any change in people, equipment, materials, methods or environment to be noted on the reverse side; the notes will help to make corrections / improvements when indicated by the control chart.
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33. Control Charts and Probability
SPC Chart
It would be valuable to know when a
process is producing parts that meet a
desirable outcome (like high reliability or
yield) and if it was not producing, why
not?
Control charts are used to visually show
when a process is producing parts within
specification and when is it not
Come on snake producing parts within specification.
eyes!
We want to build parts that would be
identical, but we know all parts are not
the same. The parts vary.
Probability relates the possibility of
meeting and not meeting a desirable
outcome.
The discussion of control charts requires
some understanding of probability.
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34. Control Charts and Probability
SPC Chart
Just as in gambling we cannot predict
what will be the outcome of an event
before it happens,
for instance rolling a two with a pair of
dice,
we can know how frequently we should
expect that event to occur.
Come on snake
eyes!
When we make an item we can also
predict how frequently the part should be
out of some desirable range. When the
frequency gets too high then we should
look for the source that causes the part
to vary too much so it is unacceptable.
We can pictorially represent the shape of
how frequently events occur.
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35. Probability
Come on snake
eyes!
What is the probability of rolling a “one” with one die?
A = the number of ways an event can happen
B = the number of way an event fails to happen
A + B = the total number of all possibilities
Probability is calculated by dividing A by the sum of A and B
A 1
Probability = =
A+B 1+5 1 way to
Probability = 16.6% get a one
5 ways fail to get
a one
What is the probability of a “head” with a coin toss?
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36. Probability
Come on snake
eyes!
What is the probability of a “head” on a coin toss?
A = the number of ways an event can happen
B = the number of way an event fails to happen
A + B = the total number of all possibilities
Probability is calculated by dividing A by the sum of A and B
A 1
TAILS Probability = =
A+B 1+1
1 way to fail to Probability = 50% 1 way to
get a head get a head
What is the probability of tossing two coins and both are “heads”?
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37. Probability
Come on snake
eyes!
What is the probability of tossing two coins and
both are “heads”?
What are all the HH HT
combinations?
TH TT
HT
A 1
TH TT Probability = =
A+B 1+3 1 way to get
3 ways to fail to
get two heads two heads
Probability = 25%
What is the probability of tossing coins five
consecutive times and getting “heads”?
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38. Probability
Come on snake
eyes!
What is the probability of tossing coins
five consecutive times and getting “heads”?
HHHHH
HHHHT
HHHTH
HHHTT
HHTHH
What are all the combinations? HHTHT
HHTTH
HHTTT
31 ways to fail to get HTHHH
HTHHT
five heads HTHTH
HTHTT
HTTHH
1 way to get five heads HTTHT
HTTTH
HTTTT
THHHH
THHHT
THHTH
A 1 THHTT
THTHH
Probability = = THTHT
THTTH
0 1 1
A+B 1 + 31 THTTT
TTHHH
1 5 5 TTHHT
2 10 10 Probability = 3.125% TTHTH
TTHTT
3 10 10
12 TTTHH
4 5 5 TTTHT
5 1 1 Note how some outcomes are more
10 TTTTH
TTTTT
likely and some are less likely and how this
8
influences the shape of the distribution.
6
4
What is the probability of rolling a
“two” with a pair of dice?
2
0
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0 1 2 3 4 5

39. Probability
Come on snake
eyes! What is the probability of rolling a
“two” with a pair of dice??
What are all the outcomes from 2 dice?
36 total combinations 1st die 2nd die
1 way to get a two 6 1,2,3,4,5,6
35 ways to fail to get a two 5 1,2,3,4,5,6
4 1,2,3,4,5,6
3 1,2,3,4,5,6
A 1 2 1,2,3,4,5,6
Probability = = 1 1,2,3,4,5,6
A+B 1 + 35
Probability = 2.78%
The graphical
2 1 1 11
3presentation
2 2 12
4
5
3
4
3
4
31
41
8 The developing shape is similar to the
8
6 5 5 51
6
4
“Normal Distribution Curve”.
6
4
7 6 6 61
8 5 5 62 2 2
9 4 4 63 0 0
10 3 3 64 2 3 4 5 6 7 8 9 10 11 12
11 2 2 65
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40. Making a Quality Product
o oo
o o
o
o
o
o
o
o
o
This is precise but This is accurate but
not accurate. not precise.
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41. Probability
The Characteristics of a Normal Distribution Curve
When we make an item the location
(mean/average) is not a zero value as
shown here. There is an actual length
or weight or whatever is important
mean
enough to be measured.
All items do not have the same value; this
is the variation.
The shape of the curve results from the
fact that most items will have a value
at the peak of the curve and other
items will have other values, but these
will occur less frequently.
0
-15 -10 -5 0 5 10 15
variation
maybe a histogram of parts being measured would help more
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42. Probability
The Characteristics of a Normal Distribution Curve
location
X The Normal Distribution Curve has a
mean = 0 location and a variation value which
describes the entire shape of the
curve.
Literally these are the essential variables of
the mathematical equation
The location value is called the mean.
The variation value is called the
standard deviation.
0
-15 -10 -5 0 5 10 15
variation
S uo
standard deviation = +/- 5
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43. Probability
Note how changes in location and variation affect the characteristics of a
Normal Distribution Curve
Horizontally the graphs show changes in variation
The standard deviation is, from left to right, 3, 5, and 9
3 5 9 As the standard
deviation gets
0 bigger, the curves
gets wider and
lower.
0
3
-8
The change in
location moves -8
the curve left 3
and right
5
5
Vertically the graphs show changes in location
The mean is, from top to bottom, 0, -8, and 5
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44. Probability
The Characteristics of a Normal Distribution Curve
100% of all possibilities are within the curve!
+/- S INSIDE OUTSIDE
1 1 1 68.25% 32.75%
2 2 2 95.44% 4.56%
3 3 3 99.73% ???%
This describes the possibilities
of obtaining an outcome for any
process that is totally random
0
3 2 1 +/- S 1 2 3
axis marked in units of std. dev.
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45. SPLAT

46. MASH!

47. MATH!

48. Probability
The Characteristics of a Normal Distribution Curve
How are the location (X) and variation (S) values determined?
Gather a sample from the group to be evaluated.
Measure the response (length, time, pressure, ...).
Calculate the mean, X, by adding all the measured
values and divide by the number of measurements
added together.
find X of 5 measurements: 2, 4, 5, 8, 9
(2+4+5+8+9) = 28
S = ?? 28 / 5 = 5.6
Calculate the standard deviation, S, by summing
the square obtained from subtracting each
measured value from the average, divide this sum
by the number of measurements minus 1, and then
0
take the square root of that number.
X = ?? find S of same 5 measurements 2, 4, 5, 8, 9
(5.6-2)2+(5.6-4)2+(5.6-5)2+(5.6-8)2+(5.6-9)2 = 33.2
33.2 / (5-1) = 8.3
(8.3)1/2 = 2.88
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49. Probability
The Characteristics of a Normal Distribution Curve
What is variation and of what is it composed?
Variation is composed of common and special sources.
Common cause of variation - is the stable random pattern caused by
natural or inherent conditions of a process. Performance is predictable
and is a state of statistical control. This is the type of variation handled
by probability and depicted with the Normal Distribution Curve.
Special cause of variation - is a source of variation that is intermittent,
unpredictable unstable; sometimes called assignable causes. This is
tool wear, a balance missing a weight, a misread gage.

50. Process Capability
Cpk = X - nearest limit
3s
Cpk = 1
says the manufacturing tolerance is equal to 6
sigma and is evenly centered about the process
capability
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51. Process Capability
You are a car salesperson.
You want to sell your customer a new SUV (Sports Utility Vehicle).
Assume the width of the car represents the capability of the process (that’s what
your selling) and the width of the garage door represents the customer’s
specifications (they are limited to what can be bought)
To get the SUV through the door is
easiest when the door is much wider
than the car.
It is easiest to meet requirements
when the customer’s specification is
big compared to what the process
delivers.
Customer Specification Process Capability

52. Process Capability
Comparison of Cp (Process Capability) and Cpk (where the Process
Capability is k
centered with respect to the specifications)
Process
Process Capability Customer Specification Disruption
Cp < 1 Cp = 1 Cp > 1
Cpk < Cp Cpk = Cp Cpk < Cp Cpk << Cp
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53. Process Capability
when Cp > or = 1 then it it starts to get easier to get the car
through the garage door
to get a calculation of process capability
remove all assignable causes - this is done with the control
chart
once all random events achieved in the process
get x bar and std dev
calculate Cp and Cpk
calculate process yield
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54. Process Capability
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
This is called a “z” table. 0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
The table is used to find the 0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
probability that events will 0.3
0.4
0.3821
0.3446
0.3783
0.3409
0.3745
0.3372
0.3707
0.3336
0.3669
0.3300
0.3632
0.3264
0.3594
0.3228
0.3557
0.3192
0.3520
0.3156
0.3483
0.3121
occur. 0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
In the next slide we will look 0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
up 1.81 because we want to 1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
know what is the possibility 1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
of an event occurring 1.81 1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681
standard deviations away 1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
from the mean. 1.6
1.7
0.0548
0.0446
0.0537
0.0436
0.0526
0.0427
0.0516
0.0418
0.0505
0.0409
0.0495
0.0401
0.0485
0.0392
0.0475
0.0384
0.0465
0.0375
0.0455
0.0367
1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0329 0.0314 0.0307 0.0301 0.0294
This illustrates how to look 1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233
2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
up 1.81 and see that it 2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
represents 0.0351 or 3.51% 2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
probability an event will
2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
occur. 2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0013 0.0015 0.0015 0.0014 0.0014
Careful when using these; the table
3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
can be single or double tailed. This 3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
one is single tailed; the probability is 3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
for one tail. 3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
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55. Process Capability
Using the Z Table
(Using a single tailed table)
Z tables are used to determine the percent probability of an event in the
tail of a distribution (a variation in an input or an output variable).
This is a look up table for the % probability between two events, the
mean (x bar) and another event, the distance between them given in
standard deviation units.
1.81 S = 0.0351
Normalized Gaussian
120
or 3.51% of all
100 events within one tail
80 at 1.81 standard
60
40
deviations units and
20 beyond
0
-4 -3.6 -3.2 -2.8 -2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4
std dev units
Remember that these predictions work if the distribution is Normal / Gaussian. If the data is not
Normal then use control charts to find the assignable causes.

56. X = 1.7 Process Capability
S = 0.010
Calculate the out-of-spec parts for this process.
UCL = 1.715 Calculate the control limits in standard deviation units
LCL = 1.670 UCL = (1.715 - 1.7) / 0.010 = (0.015) / 0.010 = 1.50
LCL = (1.670 - 1.7) / 0.010 = (0.030) / 0.010 = 3.00
look up the “z” fraction beyond the points 1.50 and 3.00
Z1.50 = 0.0668 and Z3.00 = 0.00135 or
add together and make it a percent: 6.815% out-of-spec
Calculate the UCL & LCL in z
0.0
0.00
0.5000
0.01
0.4960
units found in a standard 0.1
0.2
0.3
0.4602
0.4207
0.3821
0.4562
0.4168
0.3783
Normal Distribution table 0.4
0.5
0.6
0.3446
0.3085
0.2743
0.3409
0.3050
0.2709
0.7 0.2420 0.2389
1.4 0.0808 0.8 0.2119 0.2090
0.9 0.1841 0.1814
the second curve is 1.5 0.0668 0.0 1.0 0.1587 0.1562
1.6 0.0548 0 1.1 0.1357 0.1335
centered on zero by 1.2
1.3
1.4
0.1151
0.0968
0.0808
0.1131
0.0951
0.0793
subtracting the 2.9 0.0019 1.5
1.6
1.7
0.0668
0.0548
0.0446
0.0655
0.0537
0.0436
3.0 0.0013 0.
average value 3.1 0.0010
1.8
1.9
0.0359
0.0287
0.0351
0.0281
2.0 0.0228 0.0222
2.1 0.0179 0.0174
2.2 0.0139 0.0136
2.3 0.0107 0.0104
the third curve has 2.4
2.5
0.0082
0.0062
0.0080
0.0060
2.6 0.0047 0.0045
the variation scaled 2.7
2.8
0.0035
0.0026
0.0034
0.0025
2.9 0.0019 0.0018
in whole units of 3.0
3.1
0.0013
0.0010
0.0013
0.0009
3.2 0.0007 0.0007
standard deviation 3.3
3.4
0.0005
0.0003
0.0005
0.0003
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57. Process Capability
Measurement adds variation
Adjust machines to get x bar in the center of UCL and LCL so Cpk
becomes as large as possible
What happen when Cpk produces yields of 0.98, 0.95, and 0.92?
(0.90) * (0.95) * (0.92) = 0.7866
When variation improves, get smaller, then yields improve.
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58. Process Capability
Assume a product tolerance of 1.00 +/- 0.01
1.00 x = 1.003 Cp = 0.02 / 0.0009486 = 0.35
1.02 s = 0.009486 Cpk = (1.10 - 1.003) / (3 * 0.009486)
1.00 LCL = 0.99 = 0.007 / 0.028458
0.99 UCL = 1.01 = 0.245
1.01
Z statistics
0.99
(1.003 - 0.99) / 0.009486 = 1.37
1.00
1.37 = 0.0853 or 8.53%
1.01
1.00 (1.003 - 1.01) / 0.009486 = 0.7379
0 0.7379 (about .74) = .2296 or 22.96%
1.01
total of 31.46% Out of Tolerance
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59. Process Capability
Assume a product tolerance of 2.5 + / - 0.05
2.49 x = 2.509 Cp = 0.10 / (6 * 0.02331) = 0.71
2.50 s = 0.02331 Cpk = (2.509 - 2.55) / (3 * 0.02331)
2.54 LCL = 2.45 = 0.041 / 0.06993
2.50 UCL = 2.55 = 0.586
2.47
Z statistics
2.49
(2.509 - 2.55) / 0.02331= 1.76
2.51
1.76 = 0.0392 or 3.92%
2.52
2.54 (2.509 - 2.45) / 0.02331= 2.53
0
2.53 2.53 = 0.0057 or 0.57%
total of 3.44% Out of Tolerance
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60. Control Charts
All Data
Yes / No, Good / Bad, Pass / Fail Measurable
Attribute
Variable
Defects Defectives
X bar & R individual &
Unlimited limited moving x bar
C u p np mixed sample short run
size production
fixed variable variable fixed
sample sample sample sample
size size size size Best to use variable
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61. Control Charts
P CHART
When variable data cannot be obtained
When charting fraction rejected as non-conforming
When screening multiple characteristics for potential control charts
When tracking the quality level of a process before (how? By counting the number of defective items from a
sample and then plotting the percent defective)
Conditions
to be of help: there should be some rejects in each observed sample
the higher the quality level, the larger the sample size needs to be, since needs rejects. For example, 20% of a
product is rejectable......................................................................................................
needed. However, a sample of 1000 will give a ......................................................................................
sample if 0.1% of the product is rejectable
UCL - pbar + ( 3(pbar (1-pbar)/n)1/2 .... LCL
want a normal dist
UCL 7 LCL are calc
calc pbar on n = 20 ( also based on a small
LCL and UCL) sample size; LCL
0
usually = 0; if LCL = -3s
then part is bad <
0.0135%
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