1 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Process Capability (Cp / Cpk / Pp / Ppk)
Global Training MaterialGlobal Training Material
Creator : Global Mechanics Process Manager
Function : Mechanics
Approver : Gary Bradley / Global Process Team
Document ID : DMT00018-EN
Version / Status : / Approved
Location : Notes : \\…\ NMP \ DOCMANR4 \ PCP \ PC Process Library DocMan
Change History :
Issue Date Handled By Comments
21st Dec’01 Jim Christy & Søren Lundsfryd Approved for Global Use
NOTE – All comments and improvements should be addressed to the creator of this document.
2 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Contents
Section Heading / Description Page
1 Variation, Tolerances and Dimensional Control 4
2 Population, Sample and Normal Distribution 15
3 Cp and Cpk Concept 28
4 Use of the NMP Data Collection Spreadsheet 44
5 Confidence of Cpk 52
3 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Process Capability
- Evaluating Manufacturing Variation
Acknowledgements
• Benny Matthiassen (NMP CMT, Copenhagen, Denmark)
• Frank Adler (NMP Alliance, Dallas, USA)
• Joni Laakso (NMP R&D, Salo, Finland)
• Jim Christy (NMP SRC, Southwood, UK)
4 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Section 1
Variation, Tolerances and
Dimensional Control
5 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Two Types of Product Characteristics
Variable: A characteristic measured in physical units,
. millimetres, volts, amps, decibel and seconds.
ON
OFF
Attribute: A characteristic that by comparison to some
standard is judged “good” or “bad”, . free from
scratches (visual quality).
In this
train
ing w
e dea
l with
varia
bles o
nly
6 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
The Sources of Process/System Variation
Methods
Operators
Customer Satisfaction
Material
Environment
Equipment
Process
7 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Two Types of Processes
• All processes have:
–Natural (random) variability
=> due to common causes
• Stable Process:
A process in which variation in
outcomes arises only from
common causes
• Unstable Process:
A process in which variation
is a result of both common
and special causes
USL
LSL
nominal value
Defect
USL
LSL
nominal value
–Unnatural variability
=> due to special causes
8 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Shewhart (1931)Shewhart (1931)
The Two Causes of Variation
• Common Causes:
–Causes that are implemented in the process
due to the design of the process, and affect
all outcomes of the process
–Identifying these types of causes requires
methods such as Design of Experiment
(DOE), etc.
• Special Causes:
–Causes that are not present in the process all
the time and do not affect all outcomes, but
arise because of specific circumstances
–Special causes can be identified using
Statistical Process Control (SPC)
USL
LSL
Nominal
value
Defect
USL
LSL
nominal
value
9 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Tolerances
LSL (lower specification limit)
10,7
USL (upper specification limit)
10,9
Acceptable
part
Rejected
PartRejected
Product
Nominal
10,80,1
Rejected
Part
A tolerance is a
allowed maximum
variation of a
dimension.
10 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Measurement Report
In most cases we measure only one part per cavity for
measurement report
11 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Example of Capability Analysis Data
• For some critical dimensions we need to measure more than 1
part
• For capability data we usually measure 5 pcs 2
times/hour=100 pcs (but sampling plan needs to be made on
the basis of production quantity, run duration and cycle time)
12 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Process Capability - What is it?
• Process Capability is a measure of the inherent
capability of a manufacturing process to be able to
consistently produce components that meet the
required design specifications
• Process Capability is designated by Cp and Cpk
• Process Performance is a measure of the performance of
a process to be able to consistently produce
components that meet the required design specifications.
Process Performance includes special causes of variation
not present in Process Capability
• Process Performance is designated Pp and Ppk
13 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Why Make Process Capability Studies
LSL (lower specification limit)
10,7
USL (upper specification limit)
10,9
Nominal
10,80,1
This part is within
spec. The tool
would be
approved if only
this part was
measured
These parts
are out of
spec and
could be
approved if
only one good
part was
measured
A process
capability
study would
reveal that the
tool should
not be
accepted
When a dimension needs to be
kept properly within spec, we
must study the process capability
…. but still this is no guarantee for
the actual performance of the
process as it is only an initial
capability study
14 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
E1 E2 E3 E4 E5
The Nokia Process Verification Process
Black diamonds to
be fixed by E3
(often a change of
a white diamond)
Proposal for black
diamonds to be
discussed with
Supplier.
Max: 105,85Ongoing Process Control (SPC)
Tolerances
applied to
drawing
T
yp
e
1
F
un
ct
io
na
l
C
ha
ra
ct
er
is
tic
s
- 1 part/cavity
measured for
measurement report
White
diamonds(s)
to be agreed
White
diamonds(s)
to be
discussed
with supplier
10 parts/cavity
measured for
measurement report
Capability study: Requirement: Cp and Cpk >
by E3.
Quantities to be agreed with supplier. Minimum 5
parts pr 1/2 hour in 10 hours measured for each
cavity = 100 parts. Can vary depending on tool
capacity, . stamped parts (see DMY00019-EN)
15 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Section 2.
Population, Sample and
Normal Distribution
16 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
The Bell Shaped (Normal) Distribution
• Symmetrical shape with
a peak in the middle of
the range of the data.
• Indicates that the input
variables (X's) to the
process are randomly
influenced.
17 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
“Population Parameters”
= Population mean
= Population standard deviation
Population versus Sample
Population
• An entire group of objects
that have been made or will
be made containing a
characteristic of interest
Sample
• The group of objects actually
measured in a statistical
study
• A sample is usually a subset
of the population of interest
Po
pu
la
tio
n
Sam
ple
“Sample Statistics”
x = Sample mean
s = Sample standard deviation
18 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
The Normal Distribution
19 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
What Measurements Can Be Used to
Describe a Process or System ?
Example: x1 = 5 x2 = 7 x3 = 4 x4 = 2 x5 = 6
• mean (average) or describes the location of the
distribution
• µ (mü), a measure of central tendency, is the mean or
average of all values in the population. When only a
sample of the population is being described, mean is
more properly denoted as (x-bar) :
20 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Example: x1 = 5 x2 = 7 x3 = 4 x4 = 2 x5 = 6
• The most simple measure of variability is the range. The
range of a sample is defined by as the difference between
the largest and the smallest observation from samples in a
sub-group, . 5 consecutive parts from the manufacturing
process.
What Measurements Can Be Used to
Describe Process variation?
21 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
• sST - often notated as or sigma, is another measure of
dispersion or variability and stands for “short-term standard
deviation”, which measures the variability of a process or
system using “rational” sub-grouping.
where is the range of
subgroup j, N the number of subgroups, and d2* depends
on the number N of subgroups and the size n of a
subgroup (see next slide)
What Measurements Can Be Used to
Describe Process variation?
22 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
d2* values for SST
Where: N = no. of sub-groups, n = no. of samples in each sub-group
dd22**
dd22
Typical:
N=20 &
n=5
23 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
•
•
•
•
•
•
• xx33
• xx22• xx11
• xx1010
xx
__
tt
Example:
What Measurements Can Be Used to
Describe Process variation?
24 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
The Difference Between SST and sLT !!
mean
Time
Dimension
Short term Standard Deviation
Long term Standard Deviation
Subgroup size n = 5
Number of subgroups N = 7
TRE
ND
Subgroup
No. 1
25 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
The difference between the standard deviations sLT and sST gives an
indication of how much better one can do when using appropriate
production control, like Statistical Process Control (SPC).
Short-term standard deviation :
Long-term standard deviation ::
The difference between sST and sLT
26 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
The difference between sST and sLT
•The difference between sLT and sST is only in the way that the
standard deviation is calculated
•sLT is always the same or larger than sST
•If sLT equals sST, then the process control over the longer- term is
the same as the short-term, and the process would not benefit
from SPC
•If sLT is larger than sST, then the process has lost control over the
longer- term, and the process would benefit from SPC
•The reliability of sLT is improved if the data is taken over a longer
period of time. Alternatively sLT can be calculated on several
occasions separated by time and the results compared to see
whether sLT is stable
27 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Exercise 1: Sample Distributions
1. In Excel file "Data exercise " you find 100
measurements being the result of a capability study.
The specification for the dimension is 15,16,01
2. How well does the sample population fit the
specification, . should we expect any parts outside
spec?
3. Mention possible consequences of having a part
outside spec .
4. Mention possible causes of variation for parts.
5. Calculate the sample mean and sample standard
deviation for the 100 measurements. Use the average
and stdev functions Excel.
28 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Section 3.
Cp and Cpk Concept
29 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Defining Cp and Pp
Sample mean
Process
variation
6*s
USL-LSL
LSL USL
Nominal dim
The tolerance area divided by the total process
variation, irrespective of process centring.
30 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Defining Cpk and Ppk
Sample mean
Process
variation
3s
Process
variation
3s
Mean - LSL USL-Mean
LSL USL
Nominal dim
Cpk and Ppk Indexes account also for process centring.
31 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
What is the Difference Between Cp and Cpk?
• The Cp index only accounts for
process variability
• The Cpk Index accounts for process
variability and centering of the process
mean to the design nominal
• Therefore, Cp Cpk
• NOTE: Same applies also for Pp and
Ppk
Cp = Cpk
(both low)
LSL USL
Mean = Nominal
Reject partsReject parts
Cp high, Cpk low
Process should be optimized!
Nominal
LSL
Mean
USL
Reject parts
32 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
What Do These Indexes Tell Us ??
• Simple numerical values to describe the quality of the process >> The
higher the number the better
•Requirement for Cp and Cpk is min.
•Recommendation for Pp and Ppk is min.
• This leaves us some space for the variation, . a safety margin
• Are we able to improve our process by using SPC?
• If index is low, following things should be given a thought:
• Is the product design OK?
• Are tolerance limits set correctly?
• Too tight?
• Is the process capable of producing good quality products?
Process variation? DOE required?
• Is the measuring system capable? (See Gage R&R)
33 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Cpk - With a 2-sigma safety margin
- 3 sST + 3 sST
LCL UCL
LSL USL
Mean value
= Nominal value or Target
•Requirement for Cp and Cpk is min. is a ratio
of = 5/3 or 10/6.
6 * standard deviation
10 * standard deviation
2 * standard deviation
2 * standard deviation
34 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
•Cpk < the process NOT CAPABLE
Acceptability of Cpk Index
•Cpk >= the process is CAPABLE
•Cpk >= the process has reached Six Sigma level
35 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
What Do These Indexes Tell Us ??
• If Cp = Cpk,
• If Pp = Ppk,
• If Cpk < Cp,
• If Ppk < Pp,
• If Cp = Pp,
• If Cpk = Ppk,
• If Pp < Cp,
• If Ppk < Cpk,
… then process is affected by special causes.
Investigate X-bar/R-chart for out-of-control
conditions. SPC may be effective
… then process is not affected by special causes
during the study run. SPC would not be
effective in this case
… then process perfectly centred
… then process not centred (check process
mean against design nominal)
36 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Cp and Cpk Indices and Defects
(both tails of the normal distribution)(both tails of the normal distribution)
Pp=Ppk=1,33
63 ppm defects = 0,006%
Cp=Cpk=1,67
0,6 ppm defects = 0,00006%
Note: Ppm reject rates calculated from Cp & Cpk are based on the short term variation which may not
represent the long term reject rate
37 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
The Effects of Cpk and Cp on FFR
38 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Exercise 2: Cp and Cpk
• Calculate Cp and Cpk for the 100 measurements in the file "Data
exercise "
• Determine the approximate Cp and Cpk for the 4 sample
populations on the following page
• Should actions be made to improve these processes. If yes,
which?
39 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Estimate Cp and Cpk?
The width of the normal distributions shown include ±3*s
LSL USL
A)
LSL USL
B)
LSL USL
C)
USLLSL
D)
40 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Estimate Cp and Cpk? - A)
LSL USL
A)
Mean and nominal
USL - LSL
6*s
USL - MeanMean - LSL
3*s
41 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Estimate Cp and Cpk? - B)
LSL USL
B)
Nominal
Mean
USL - LSL
6*s
USL - MeanMean - LSL
3*s
42 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Estimate Cp and Cpk? - C)
LSL USL
C)
Nominal
Mean
USL - LSL
6*s
USL - MeanMean - LSL
3*s
43 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Estimate Cp and Cpk? - D
USLLSL
D)
Nominal
Mean
USL - LSL
6*s
USL - MeanMean - LSL
3*s
44 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Section 4.
Use of the NMP Data
Collection Spreadsheet
45 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Example of how to Collect Data
1. Run in and stabilise process
2. Note the main parameters for
reference
3. When the process is stable run
the tool for 10 hours
3. Take 5 parts out from each cavity
every half hour and mark them
with time, date and cavity. Total
20 sets of 5 parts from each
cavity must be made, or
according to agreement.
4. After the last sample lot note
the main process
parameters for reference
5. Measure and record the
main functional
characteristics (white
diamonds)
6. Fill data into the NMP data
collection spreadsheet
7. Analyse!
See DM
Y 0001
9-EN
Classif
ication
and M
arking
of Fun
ctional
Charac
teristic
s
Time
Dimension
Subgroup size n = 5
Number of subgroups N = 20
0,5 hours between samples taken
Note: For clarity, only 6
subgroups are shown
46 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Data Collection Sheet ()
47 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Data Collection Sheet ()
48 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Graphical Presentation: Histogram
• What kind of distribution?
Location versus tolerance area
Width (deviation)
• Example : Cp Pp
• Cpk Ppk
49 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Graphical Presentation: X-bar and R-Chart
X-Bar Chart
R-Chart
50 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Graphical Presentation - Time
Series Plot
Something happened here !!!
51 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Exercise 3: Cpk Data Spreadsheet
• Open spreadsheet "Data exercice ". Dim 13
is identical to the data from the previous
exercises.
• Verify the results from the previous exercises
for dimension 13.
• Analyse the remaining data sets an comment
the process, should any actions be made?
Remember to create and look at the charts.
52 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Section 5.
Confidence of Cpk
53 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Confidence of Cpk
• Cpk values are not definite numbers as they are based on
relatively small samples of a population.
• The 95% confidence interval determines the interval which
includes the true Cpk value with a probability of 95%, .
"there is a probability of 5% that Cpk is either lower or higher"
than this confidence interval.
95%
confidence
interval
Actual cpk
Cpk upper
confidence
limit
Cpk lower
confidence
limit
54 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Confidence of Cpk
Small sample sizes gives
wide confidence intervals
55 © NOKIA 2001 21-Dec-2001 / Jim Christy Company Confidential
Cpk Confidence Limits with a sample size of
100 and a nominal Cpk of
