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Rockford
Powertrain
Training Workshop
Process
Capability
and Cpk
Training materials, reference documents,
and functional SPC templates are
available free on the Rockford Powertrain
web site. Go to:
www.rockfordpowertrain.com/supplier
Process Capability
• Enables successful manufacturing and
sales
• Prevents scrap, sorting, rework
• Allows jobs to run well
• Has major impact on cost and schedule
“Process Capability”
is the ability of a process
to make a feature
within its tolerance.
Everything Varies
(and the variation can be seen if we measure precisely enough)
•
•
•
•
•
•
•
•
Heights
Weights
Lengths
Widths
Diameters
Wattage
Horsepower
Miles per Gallon
•
•
•
•
•
•
•
•
Pressure
Roughness
Strength
Conductivity
Loudness
Speed
Torque
Etc. etc. etc.
Eli Whitney in 1798
• Won a U.S. Military contract to supply 10,000 guns
• Reduced variation and created interchangeable parts for
assembly and service by:
– Installing powered factory machinery
– Using specialized fixtures, tools, jigs, templates, and end-stops
– Creating drawings, routings, operations & training
Manufacturing in the 21st Century
• International competition to provide defect-free products
at competitive cost
• Reducing variation and providing interchangeable parts
for assembly and service by:
– Using machine tools
– Using specialized fixtures, tools, jigs, templates, and end-stops
– Using drawings, routings, operations & training
Graphing the tolerance and a measurement
.512 .513 .514 .515 . 516 .517 .518 .519 .520 .521 .522 .523 .524 .525 .526
.527
.528
It’s useful to see the tolerance and the part measurement on a graph.
Suppose that:
Graphing the tolerance and a measurement
Specification
Limit MIN
.512 .513 .514 .515 . 516 .517 .518 .519 .520 .521 .522 .523 .524 .525 .526
.527
.528
It’s useful to see the tolerance and the part measurement on a graph.
Suppose that:
--the tolerance is .515”
Graphing the tolerance and a measurement
Specification
Limit MIN
Specification
Limit MAX
.512 .513 .514 .515 . 516 .517 .518 .519 .520 .521 .522 .523 .524 .525 .526
.527
.528
It’s useful to see the tolerance and the part measurement on a graph.
Suppose that:
--the tolerance is .515” to .525”
Graphing the tolerance and a measurement
Specification
Limit MIN
X
Specification
Limit MAX
.512 .513 .514 .515 . 516 .517 .518 .519 .520 .521 .522 .523 .524 .525 .526
.527
.528
It’s useful to see the tolerance and the part measurement on a graph.
Suppose that:
--the tolerance is .515” to .525”
--and an individual part is measured at .520”.
Graphing the tolerance and measurements
Specification
Limit MIN
X
X
X
X
Specification
Limit MAX
.512 .513 .514 .515 . 516 .517 .518 .519 .520 .521 .522 .523 .524 .525 .526
Suppose we made and measured several more
units, and they were all EXACTLY the same!
We wouldn’t have very many part problems!
.527
.528
Graphing the tolerance and measurements
Specification
Limit MIN
XX
XXX
XXXXX
XXXXXXX
Specification
Limit MAX
.512 .513 .514 .515 . 516 .517 .518 .519 .520 .521 .522 .523 .524 .525 .526
In the real world, units are NOT EXACTLY the same.
Everything VARIES.
The question isn’t IF units vary.
It’s how much, when, and why.
.527
.528
The “normal bell curve”
XXXX
XXX XXX
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XXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXX XXXXXX
XXXXXX XXXXXX
XXXXXXXXXXXXXX
XXXXXXXXXXXXXXXX
XXXXXXXXX XXXXXXXXX
XXXXXXXXXXX XXXXXXXXXXX
Widths, heights, depths, thicknesses, weights, speeds, strengths,
and many other types of measurements, when charted as a
histogram, often form the shape of a bell.*
A “perfect bell,” like a “perfect circle,” doesn’t occur in nature, but
many processes are close enough to make the bell curve useful.
(*A number of common industrial measurements, such as flatness and straightness, do NOT tend
to distribute in a bell shape; their proper statistical analysis is performed using models other than
the bell curve.)
What is a “standard deviation”?
XX
XXX
XXXX
XXXX
XXXXX
XXXXX
XXXXXX
XXXXXX
XXXXXXX
XXXXXXXX
XXXXXXXXX
XXXXXXXXXXX
Typical distance
from the center: -1
standard deviation
XX
XXX
Typical distance
XXXX
from the center: +1
XXXX
standard deviation
XXXXX
XXXXX
XXXXXX
XXXXXX
XXXXXXX
XXXXXXXX
XXXXXXXXX
XXXXXXXXXXX
If we measure the DISTANCE from the CENTER of the bell
to each individual measurement that makes up the bell curve,
we can find a TYPICAL DISTANCE.
The most commonly used statistic to estimate this distance is the
Standard Deviation (also called “Sigma”).
Because of the natural shape of the bell curve, the area of +1 to –1
standard deviations includes about 68% of the curve.
How much of the curve is included in how many standard
deviations?
-6
-5
-4
-3
-2
-1
0
+1 +2 +3 +4 +5 +6
From –1 to +1 is about 68% of the bell curve.
From –2 to +2 is about 95%
From –3 to +3 is about 99.73%
From –4 to +4 is about 99.99%
(NOTE: We usually show the bell from –3 to +3 to make it easier to draw, but in
concept, the “tails” of the bell get very thin and go on forever.)
What is Cpk? It is a measure of how well
a process is within a specification.
B
Specification
Limit
A
Cpk =
A divided by
B
Specification
Limit
Cpk = A divided by B
A = Distance from process mean to closest spec limit
B = 3 Standard Deviations (also called “3 Sigma”)
A bigger Cpk is better because fewer units will be beyond spec.
(A bigger “A” and a smaller “B” are better.)
“Process Capability” is the ability of a process
to fit its output within the tolerances.
A
B
Specification
Limit
Cpk =
A divided by
B
Specification
Limit
…a LARGER “A”
…and a SMALLER “B”
…means BETTER “Process Capability”
An Analogy
A
B
Specification
Limit
Cpk =
A divided by
B
Specification
Limit
Analogy:
The bell curve is your automobile.
The spec limits are the edges of your garage door.
If A = B, you are hitting the frame of your garage door with your car.
How can we make Cpk (A divided by B) better?
A
B
Specification
Limit
1.
2.
3.
4.
Cpk =
A divided by
B
Specification
Limit
Design the product so a wider tolerance is functional (“robust design”)
Choose equipment and methods for a good safety margin (“process capability”)
Correctly adjust, but only when needed (“control”)
Discover ways to narrow the natural variation (“improvement”)
What does a very good Cpk do for us?
A
B
Specification
Limit
Mean
This Cpk is
about 2.
Very good!
Specification
Limit
This process is producing good units with a good safety margin.
Note that when Cpk = 2, our process mean is 6 standard deviations from
the nearest spec, so we say it has “6 Sigma Capability.”
What does a problem Cpk look like?
A
This Cpk is just
slightly greater
than 1. Not good!
B
Specification
Limit
Specification
Limit
This process is in danger of producing some defects.
It is too close to the specification limits.
(Remember: the bell curve tail goes further than B…
…we only show the bell to 3-sigma to make it easier to draw.)
What does a very bad Cpk look like?
A
B
Specification
Limit
This Cpk is less
than 1. We desire
a minimum of 1.33
and ultimately we
want 2 or more.
Specification
Limit
A significant part of the “tail” is hanging out beyond the spec limits.
This process is producing scrap, rework, and customer rejects.
Notice that if distance “A” approaches zero…
…the Cpk would approach zero, and…
…the process would become 50% defective!
Free software is available to draw a histogram
and calculate average, standard deviation, and Cpk.
Located at: www.rockfordpowertrain.com/supplier
What “Six Sigma Philosophy” did Motorola
teach its suppliers in the 1980’s?
Specification
Limit
Specification
Limit
In the 1980’s, Motorola achieved dramatic quality improvements and won
the USA’s Malcolm Baldrige National Quality Award.
Motorola began seminars teaching its “Six Sigma Philosophy” to its
suppliers, and to other companies.
The following few slides depict some original messages from that time.
Robust Design – part of the original Six Sigma
Known
Existing
Process
New Product
Specification
Limit
New Product
Specification
Limit
The new design above has tolerances set “tight” to a known existing process, while
the one below has tolerances that allow “six sigma capability”.
Products have thousands of tolerances. They result from choices about shapes,
thicknesses, grades of materials, and grades of components. “Robust design” is
NOT about permitting “sloppiness.” It requires very smart engineering to allow
ample tolerances AND achieve satisfactory function.
Known
Existing
Process
New Product
Specification
Limit
New Product
Specification
Limit
Robust Design – part of the original Six Sigma
Known
Existing
Process
New Product
Specification
Limit
New Product
Specification
Limit
CAUTION:
Suppliers must negotiate the widening of tolerances BEFORE competitive bids,
quotations, and acceptance of orders. Competitive bids are commitments to meet
all existing tolerances. Failure to meet customer tolerances means failure to meet
contract requirements. Prevent breaches of contract.
Known
Existing
Process
New Product
Specification
Limit
New Product
Specification
Limit
Robust Processes – part of the original Six Sigma
New Process
choice “X”
New Product
Specification
Limit
New Product
Specification
Limit
The process above varies so much that it “fills” the design tolerance. The different
process below has good repeatability for “six sigma capability”.
It’s a false-economy to choose an allegedly lower-cost process that “uses up” all
tolerance. The resulting scrap, rework, rejections, recalls, damage to reputation,
crisis communications, and fire-fighting cancel out the alleged economy. “Robust
Process” requires skillful insight to choose ways to make defect-free product at the
lowest real cost.
New Process
choice “Y”
New Product
Specification
Limit
New Product
Specification
Limit
6 Sigma Philosophy – Not Just The Shop Floor
Getting every person “capable” and in “self control”
Achieving delivery and project deadlines
Meeting budgets & financial goals
Administrative tasks
Design work
Purchasing/sourcing
Special projects
Security and Safety
Health and Environmental
Legal compliance
Anything that can be
defined and measured
Getting every person “capable” and in “self control”
Defined & Understood
Requirements
The 3
Requisites
Of Self-Control
Ability to
Measure Results
Process
Capability and
Ability to Control
Summary:
•
•
•
To call a process “capable” typically requires at least a
Cpk of 1.33 (+ and - 4 standard deviations within
tolerance)
Many customers desire a Cpk of 2.0 (+ and - 6 standard
deviations within tolerance)
Organizations need:
1. Feasible designs
2. Capable processes
3. Process self-control
Conclusion:
Process Capability:
Yes:
No:
too wide
Yes:
No:
potentially capable
if re-centered
Yes:
No:
potentially capable
if re-centered
Review Question 1
What is “Process Capability?”
Review Question 2
How is the “process average” calculated
or estimated?
Review Question 3
What is a “Standard Deviation”?
(also known as a “sigma”)
Review Question 4
What is Cpk used for?
Review Question 5
Suppose that a feature tolerance is .750”/.760”,
and the process average is .759”,
and the process standard deviation is .002”
…is the process satisfactory and capable?
Review Question 6
Suppose that a torque tolerance is 25 foot
pounds minimum,
and the process average is 26 foot pounds,
and the process standard deviation is 3 foot
pounds…
…is the process capable?
Review Question 7
Suppose that a diameter tolerance is 8.010” to
8.060”,
and the process average is 8.041”,
and the process standard deviation is .002”…
…is the process capable?
Review Question 8
Fred is cutting an outside diameter on a lathe
and the diameter is easily adjustable.
The diameter tolerance is 5.050” to 5.090”,
the process average is 5.090”,
and the process standard deviation is .001”…
• What is the Cpk?
• What should Fred do with the process?
Review Question 9
Joe is boring an inside diameter on a lathe.
The diameter tolerance is 1.980” to 2.020”.
Joe has measured three random samples at
2.005”, 2.004”, and 2.006”.
• Estimate the process average.
• Estimate the standard deviation (best guess).
• Estimate whether the process can be
capable.
Review Question 10
TechCorp is demonstrating a new “high-precision”
grease dispenser machine.
TechCorp claims that they can “dispense grease all day
with an accuracy of plus or minus half an ounce.”
During the demo, ten samples of grease in a row were
dispensed (in ounces) as follows:
2.3, 2.0, 2.6, 3.0, 2.1, 2.7, 2.9, 2.5, 2.0, 2.4
• Based on the sample data, evaluate TechCorp’s
claim that they can “dispense grease all day with an
accuracy of plus or minus half an ounce.”
Quiz Question 1
True or False?
“Process Capability” can be defined as
the ability of a process
to make a feature
within its tolerance.
Quiz Question 2
True of False?
We can estimate the process average
by taking a set of sample measurements,
adding them up, and dividing by the
number of measurements.
Quiz Question 3
True or False?
A “Standard Deviation” can be thought of
as the “typical” distance of the
measurements from the average;
about 68% of the individuals will fall within
+ or – 1 standard deviation of a bell curve.
Quiz Question 4
True or False?
When using Cpk, the goal is to keep the
Cpk value as low as possible.
Quiz Question 5
True or False?
If the feature tolerance is .350”/.360”,
and the process average is .351”,
and the process standard deviation is .004”
…then the process should be called “capable.”
Quiz Question 6
True or False?
If a pressure tolerance is 250 PSI minimum,
and the process average is 260 PSI,
and the process standard deviation is 4 PSI,
…then the process is “capable.”
Quiz Question 7
True or False?
If a height tolerance is 7.010” to 7.060”,
and the process average is 7.042”,
and the process standard deviation is .002”…
…then the process is “capable.”
Quiz Question 8
True or False?
If Larry is cutting an O.D. and the diameter is
easily adjustable, the tolerance is 4.055” to
4.095”, the process average is 4.095”, and
the standard deviation is .001”…
…then Larry should be able to make the
process fully “capable” by adjusting the
process.
Quiz Question 9
True or False?
If Jill is boring an I.D. with a tolerance of 1.475”
to 1.525”, and has measured three samples
at 1.501”, 1.500”, and 1.499”…
…then the average of the samples is 1.501”,
the standard deviation is probably larger than
.010”, and the Cpk is probably zero.
Quiz Question 10
True or False?
If HiTechCo is demonstrating a new “high-precision”
surface coating machine, and claims that their
machine “can coat all day with an accuracy of plus or
minus .010 inches,” and during the demo the coating
thickness readings (in inches) were as follows:
.027, .028, .027, .029, .028, .029, .028, .029, .028, .027
…then the sample readings suggest that HiTechCo
might be telling the truth about being able to hold plus
or minus .010 inches.
Appendix
Cpk and PPM
(Parts Per Million Defective)
Cpk: Avoid confusion and pitfalls
•
•
•
DOES IT VARY? Cpk varies when sampled, because
it’s calculated from the average and the standard
deviation, both of which are estimated from samples.
CARROTS AND STICKS? Giving rewards or
reprimands based on minor, short-term fluctuations of
Cpk amounts to a lottery. Watch real trends.
MAKE A “PLANT AVERAGE” CPK? It’s unhelpful to
report a plant average Cpk of multiple characteristics
and products, because:
1. Cpk values depend on each chosen tolerance
2. An “okay average Cpk” could come from 50%
“good” and 50% “bad” numbers -- highly
misleading!
What is PPM (defect Parts Per Million)?
A
B
The defect PPM is
the area outside
spec limits
Specification
Limit
Specification
Limit
“PPM” is an estimate of the portion that is beyond the spec limit.
If we know the Cpk…
--we can look up the PPM “out of spec” in a statistics book table, or
--we can use software, such as Microsoft Excel, to calculate the PPM.
(REMEMBER that the “tail” of the bell goes out further than it is drawn.)
What is the “6-Sigma Philosophy”
“1.5-Sigma Shift”?
Unfavorable
process shift
of 1.5
standard
deviations
Specification
Limit
Specification
Limit
The “6 Sigma Philosophy” includes the premise that real-world processes move around
to some extent, and produce more defects than a static process. As an arbitrary
convention, this is represented as an “unfavorable shift” of 1.5 sigma in Parts Per
Million tables for Six Sigma programs. The intention is to plan conservatively.
(This means that the “PPM vs. Sigma” charts published for “6-Sigma Programs” show higher defect rates than the
similar but traditional “Z-tables” in statistical textbooks.)
The following page is a table showing the relationships
among the following:
• Cpk,
• “How Many Sigma Capability,”
• Parts Per Million according to traditional statistical tables
• Parts Per Million taking into account the “6-Sigma
Philosophy” of an unfavorable shift in the mean of 1.5
Sigma
Cpk, PPM, and "Six Sigma"
Cpk
(Defined as
distance from
process mean to
the nearest spec,
divided by 3
Standard
Deviations)
"How Many
Within Spec
Sigma
(Process Perfectly
Capability?"
Centered,
Distance of
Both Tails
Process Mean to
Considered)
Spec Limit in
Good Units Per
Standard Deviations
Million
PPM of the Bell
Curve
Out of Spec
(Process Perfectly
Centered,
Both Tails
Considered)
PPM of the Bell
Curve
Out of Spec
(Process Not
Centered,
Only One Tail
Considered)
PPM of the Bell
Curve
Out of Spec
The column AT
with Six-Sigma
LEFT equates to 1
Philosophy
defective out of how of 1.5 Standard
many total?
Deviation Penalty
for Anticipated
Unfavorable
Process Mean Drift
0
0
0
1,000,000
500,000
2
0.17
0.5
382,925
617,075
308,538
3
0.33
1
682,689
317,311
158,655
6
0.5
1.5
866,386
133,614
66,807
15
500,000
0.67
2
954,500
45,500
22,750
44
308,538
0.83
2.5
987,581
12,419
6,210
161
158,655
1
3
997,300
2,700
1,350
741
66,807
1.17
3.5
999,535
465
233
4,298
22,750
1.33
4
999,937
63
32
31,560
6,210
1.5
4.5
999,993.2
6.8
3.4
294,048
1,350
1.67
5
999,999.4
0.6
0.3
3,483,046
233
1.83
5.5
999,999.96
0.04
0.02
52,530,944
32
2
6
999,999.998
0.002
0.001
1,009,976,693
3.4
2.17
6.5
999,999.99992
0.00008
0.00004
24,778,276,273
0.3