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9. Machine and Process Capability
Quality Management in the Bosch Group | Technical Statistics
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3. Edition, 01.07.2004
2. Edition 29.07.1991
1. Edition 11.04.1990
The minimum requirements given in this manual for capability and performance indices are
valid at the time of publication (edition date). In case of conflict, the requirements of QSP0402
are binding and take precedence over this manual.
2004 Robert Bosch GmbH
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Machine and Process Capability
Table of Contents
Page
1. Introduction ............................................................................................................................... 4
2. Terms......................................................................................................................................... 4
3. Flow Chart for Machine and Process Capability Study ............................................................ 6
4. Machine Capability Study......................................................................................................... 7
4.1 A Machine Capability Study in Detail ............................................................................ 8
4.2 Data Evaluation ............................................................................................................... 9
4.2.1 Study of Temporal Stability.................................................................................. 9
4.2.2 Standard Method................................................................................................... 9
4.2.3 Manual Calculation Procedure............................................................................ 11
4.3 Machine Capability Study with Reduced Expense........................................................ 12
5. Process Capability Study......................................................................................................... 14
5.1 Procedure ....................................................................................................................... 14
5.2 Data Evaluation (Standard Method).............................................................................. 14
5.2.1 Studying the Process Stability (Analysis of Variance and F Test) ..................... 14
5.2.2 Studying the Statistical Distribution ................................................................... 15
5.2.3 Calculating Process Capability Indices............................................................... 15
5.3 Data Evaluation (Manual Calculation Procedure)......................................................... 16
5.3.1 Studying the Process Stability ............................................................................ 16
5.3.2 Studying the Statistical Distribution ................................................................... 16
5.3.3 Calculating Process Capability Indices............................................................... 16
6. Interpretation of Capability Indices......................................................................................... 18
6.1 Relation between Capability Index and Fraction Nonconforming ................................ 186.2 Effect of the Sample Size .............................................................................................. 19
6.3 Effect of the Measurement System................................................................................ 19
7. Capability Indices for Qualitative Characteristics................................................................... 20
8. Report of Capability or Performance Indices.......................................................................... 20
9. Methods for Calculating Capability Indices............................................................................ 21
9.1 Method M1 .................................................................................................................... 21
9.2 Method M2 .................................................................................................................... 22
9.3 Method M3 (Range Method)......................................................................................... 22
9.4 Method M4 (Quantile Method) ..................................................................................... 23
10. Distribution Models .............................................................................................................. 24
10.1 Distributions from the Johnson Family ....................................................................... 24
10.2 Extended Normal Distribution..................................................................................... 25
11. Examples ............................................................................................................................... 26
12. Capability Indices for Two-Dimensional Characteristics ..................................................... 30
13. Forms..................................................................................................................................... 31
14. Abbreviations ........................................................................................................................ 37
15. References ............................................................................................................................. 39
Index............................................................................................................................................ 40
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1. Introduction
Suitable methods must be applied for monitoring, and where applicable, measurement of
processes. These methods shall demonstrate the ability of the processes to achieve planned
results. When planned results are not achieved, correction and corrective action shall be taken,
as appropriate, to ensure conformity of the product. (see [2])
Examples of characteristics to assess the process performance or capability are or include the
following:
Capability indices
Response time
Cycle time or throughput
Reliability and safety
Rate of yield
Effectiveness and efficiency
Use of suitable technology
Avoidance and reduction of waste
Costs
2. Terms
Process
This document deals exclusively with production and assembly processes. A process is under-
stood as a series of activities or procedures in which raw materials or pre-machined parts or
components are further processed to generate a finished product.
The definition in the standard [1] is as follows: Set of interrelated or interacting activities
which transforms inputs into outputs.
Capability Studies
A process capability study (Process analysis, see [3]) is performed for a new or changed
production process (including assembly) in order to verify the (preliminary) process capability
or performance and to obtain additional inputs for controlling the process (see [3]).
References [10] and [11] distinguish between long-term and short-term studies. In a short-term
study (e.g., machine capability study), characteristics of products manufactured in one con-
tinuous production run are evaluated. A long-term study evaluates parts manufactured over a
longer time-span which is representative of the variation encountered in series production.
Capability and Performance Indices
Quantitative measures for evaluating capability include the machine and process capability orprocess performance indices (see [4]). These must achieve or surpass the specified minimum
values.
The minimum requirements given in this manual for capability and performance indices are
valid at the time of publication (edition date). In case of conflict, the requirements of QSP0402
are binding and take precedence over this manual. Higher minimum requirements for process
capability or performance may exist for special characteristics, or may be specified internally on
a product-by-product basis.
Machine Capability Study
The machine capability study is a short-term study with the sole aim of discovering the
machine-specific effects on the production process.
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Process Capability Study
The process capability study is a longer-term study. In addition to variation arising from the
machine, all other external factors that influence the production process over a longer operating
time must be taken into account.
Stable Process
A stable (in statistical control) process is only subject to random influences. In particular, the
location and variation of the product characteristic are stable over time. (see [4])
Quality-Capable Process
A process is quality-capable when it can meet all the specified requirements without exception.
Capability Indices Cmk, Cpk and Performance Index Ppk
In accordance with the QS-9000 reference documents [10] and [11], the term pkC must only be
used for a stable process. A process is stable if the following synonymous statements apply to it:
Mean and variance are constant. No systematic variations of the mean such as trend, batch-to-batch variation, etc., occur.
There is no significant difference between sample variation and and total variation.
Every sample represents the location and variation of the total process.
If the process is not stable, one speaks of process performance, and the index is called the
process performance index, pkP . This applies to all processes with systematic variation of the
mean such as trend or batch-to-batch variation (see Chapter 3). It is, therefore, the process
behavior which determines whether the index is named pkC or pkP .
In a machine capability study (initial process study or short term study see [10]), the index
is always called mkC , except where different customer requirements are specified. mkC is
understood to be an index for a short-term capability study in terms of [10] and [11].
Only when sufficient data has been collected over a longer term (e.g., as the result of a process
capability study, pre-production run with at least 125 values or evaluation of several control
charts) it is possible to calculate and distinguish between pkC and pkP on the basis of the
process behavior.
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3. Flow Chart for Machine and Process Capability Study
Process stable?
yes
Data recording
Normally distributed?
yes
no
Normal distribution Assign distribution mod
Cp/Cpk Cp/Cpk
Process without
systematic variation
of the mean
Cm/Cmk
Assign distribution model
Calculation of the indices
Duration of study?
Short-term study Long-term study
no
yes
Normally distributed?
Cm/Cmk
Cmk Machine capab
Cpk Process capabPpk Process perfor
k = katayori (japanese term
Normal distribution
(Machine capability) (Process capability/
performance)
Section
4.2.2 or
4.2.3
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4. Machine Capability Study
A machine capability study concentrates exclusively on the characteristics of the machine, i.e.,
to the extent possible, the influence or effects of variables external to the machine (noise
factors) are minimized. Some examples of variation sources are:
Man - Personnel- Shift changes
Machine - Speed- Feedrate- Tools- Cycle times- Coolant flow rate and temperature- Pressures- Current (in the case of welding equipment)- Power (in the case of laser welding)- Change status (in the case of optimization measures)
Material - Semi-finished products, rough parts or blanks from different lots ormanufacturers
Method - Run-in (warm-up) time of the machining facility before sampling- Differing pre-machining or production flow
Environment
(mother nature)
- Room temperature (temperature changes during production of thesample)
- Relative humidity, atmospheric pressure- Vibration acting upon the machining facility- Location of the machining facility in the building (storey)
- Unusual events
It is expected that only the machine's inherent sources of variation will affect the product and its
characteristics if these possible influences are kept constant. In cases where this is not possible,
the changes in the external influencing factors should be documented in the record of test
results. This information can be used as the basis for optimization measures if the capability
specifications are not met.
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4.1 A Machine Capability Study in Detail
A machine capability study cannot be performed in the absence of capable measurement or test
processes (see also 6.3 and [15]).
Note: for information on reducing the sample size, see Section 4.3.
Manufacturing of a representative number (minimum:
50, if possible: 100) of parts in a continuous,
uninterrupted production run. Deviations must be
documented.
Measurement of the parts characteristic(s) and
documentation of the results in accordance with the
production sequence.
Statistical evaluation:
- qualitative evaluation of temporal stability
- study of the distribution of these values
- calculation of capability indices
Preparation of the machining device (pre-production run)
so that the measured values are in the middle of the
tolerance zone as far as possible. For characteristics
limited to one side, choose the best possible setting with
reference to the limiting value (or target value).
Start
Machine is capable
Minimum requirement met?Problem analysis;
make improvements
no
yes
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4.2 Data Evaluation
4.2.1 Study of Temporal Stability
On the basis of the single value chart, a qualitative evaluation is now performed to determine
whether the measured values are stable over time.
Are systematic variations visible in the time-series?
Are the individual values concentrated in the vicinity of the set target value?
Do all individual values lie within a zone corresponding to approximately 60% of thetolerance range?
The following are specific signs that a process is not stable:
There are single, inexplicable outliers
There are inexplicable steps or a trend
Most of the individual values are above or below the target value
If the characteristic is limited to two sides:
Most of the individual values are close to both limit values.
If the series appears chaotic and is not plausible, the cause(s) for this behavior must be
investigated and eliminated. The capability study then must be repeated.
4.2.2 Standard Procedure
The standard procedure for evaluationg capability, described below, should be used to calculate
the machine capability indices. However, this method can only be used if a distribution model
has already been determined. This method can only be used with special statistics software - in
some cases, the best fitting distribution model is automatically selected. Otherwise, an
evaluation based on the manual calculation procedure can be used (Section 4.2.3).
Study of the Statistical Distribution
Knowledge of the production procedure and the type of tolerance often aid in selecting a
theoretical distribution which is appropriate for describing the empirical distribution. For
example, if there is an equal probability of a characteristic's values deviating upwards and
downwards from the nominal value (positive or negative deviation), one can expect the
characteristic to be approximately normally distributed. However, this is not always the case.
In contrast, characteristics which are naturally limited on one side typically are represented by
skewed, asymmetrical distributions. For example, concentricity and roughness are non-negativeby definition. In such a case, zero acts as a natural lower limit.
If a characteristic has two natural limits (a lower value, below which the characteristic can not
fall, and an upper value, above which the characteristic can not rise), the characteristic can be
approximated by a rectangular distribution.
It must be emphasized that a process characteristic may or may not behave in accordance with
these rules. In some cases, major deviations may be observed (for more information, see Section
5.2.2).
If a statistical software program is used, the user is faced with the problem of selecting an
appropriate distribution, i.e., one that represents the random sample on hand.
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Within the framework of a machine capability study, a statistical test is used to distinguish only
roughly between
normal distribution and
other distributions
If the characteristic values are not normally distributed, a mathematical procedure the
Johnson Transformationcan be used to select the most suitable distribution from a range
of possible distributions. If this automatic adjustment is not available, probability plots or
statistical goodness-of-fit tests can be used to aid in distribution selection.
For information on evaluating short-term studies, see also the flow chart in Chapter 3.
Calculating Machine Capability Indices
The quantile method is the preferred way to calculate machine capability indices (see method
M4 in Section 9.4). The capability indices mC and kmC are calculated as follows:
00135.099865.0
mQQ
TC
=
Cmk= minimum value of
00135.099865.0 Qx~
LSLx~;
x~Q
x~USL
Unlike kmC , mC accounts only for the spread but not the location of the distribution relative to
the tolerance zone (see Figure on the following page).
The machine is capable if 67.1C mk (for information on minimum requirements, see also Chapter
2.)
If characteristics are limited to one side (by USL and zero alone, or just by LSL), the formula
related to the given specification limit applies, i.e., only kmC is calculated.
Methods M1, M2 and M3 shown in Chapter 9 also can be used.
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3.6 s
5 s
Cm = 1.67
Cmk = 1.2
Cm = 1.67Cmk = 1.67
USLLSL
Comparison of mC and kmC
4.2.3 Manual Calculation Procedure
If no special software is available, mC and kmC also can be calculated as follows. Themean x and the overall standard deviation totals are calculated from the n measured values
ix :
=
=n
1i
ixn
1x ( )
=
=n
1i
2
itotal xx1n
1s
Then:
total
ms6
TC = with T = USL - LSL
Cmk= minimum value of
totaltotal s3
LSLxand
s3
xUSL
If characteristics are limited to one side (by USL and zero alone, or just by LSL), the formula
related to the given limit applies.
Since no information is available here on the distribution model and totals is used, this method
leads to comparatively small results.
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4.3 Machine Capability Study with Reduced Expense
As specified in Section 4.1, at least fifty ( 50n= ) parts should be manufactured for a
machine capability study, but use of one hundered ( 100n= ) parts is preferred. In practice,
capability studies often incur high costs due to expensive measurements. In such cases, the
following, two-stage procedure may be used to minimize cost:
1. Of the 50 parts produced consecutively, begin the study by measuring only every
second part, i.e., parts 2, 4, 6, ..., 50. This step yields 25 measured values per
characteristic. The machine is considered capable if the capability index calculated from
the 25 values is 0.2C mk .
2. If 0.2C67.1 mk
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In special cases, it may be unavoidable to reduce the sample size even further (regardless of the
capability requirement). This may be the case if the measurement procedure is very expensive
or the test is destructive.
Naturally, the smaller the sample size, the less accurate the conclusions (larger confidence
interval of the characteristic calculated from the sample). The quality assurance office must be
consulted before the sample size is reduced.
In such cases, the machine or process parameters should be given priority instead of the product
parameters. This also applies to the problem of qualitative product characteristics dealt with in
Chapter 7.
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5. Process Capability Study
The process capability study is a long-term study that is conducted over an extended operating
time and includes sources of variation external to a machine. These sources are typically
summarized under the headings of Man, Machine, Material, Method and Environment.
5.1 Procedure
A process capability study includes the following steps:
Select parts from series production in rational samples (not sorted); at least 25 subgroupsshould be evaluated. The preferred sample size is n = 5. Overall, at least 125 parts should be
examinded.
Measure part characteristics and record the results along with production sequence.
Statistical evaluation of the data: Evaluate temporal stability and statistical distribution.Calculate capability indices.
Note: In special cases, use of fewer than 125 parts may be unavoidable due to time or cost of
making the necessary measurements, or if the test is destructive. Smaller sample sizes lead to
larger confidence intervals of the characteristic(s) being studied. In turn, this reduces the
accuracy of the conclusions that may be drawn from the data. The quality assurance office must
be consulted before the sample size is reduced.
5.2 Data Evaluation (Standard Method)
The following evaluation steps 5.2.1 to 5.2.3 must also be performed in the same way if the
process capability or performance is to be calculated on the basis of the data from quality
control charts.
5.2.1 Studying the Process Stability (Analysis of Variance and F Test)
First of all, it is specified whether measured values are temporally stable or not. If statistical
software is being used, this information can be gained using the analysis of variance (ANOVA).
For this purpose, the total variation (variance) of all single values is divided into two parts: a
variation Within subgroup of groups of five, for example, and a variation Between
subgroups of means from group to group. An F test is then used to check whether the variation
Between subgroups is significantly larger than the variation Within subgroup or not.
What is this information used for? Ultimately, a capability index must be given as the result of
the process capability study, whether or not a trend or batch-to-batch variation is detected in the
process. The result of the F test now allows at the very least a rough distinction to be made as to
whether a process model with a stable mean or an extended normal distribution can be used as a
process model.
More critical however are cases where the control limits for the standard deviations are
exceeded. This indicates that the process variation is not stable, that the process behavior cannot
be explained statistically and hence, that the process is not in statistical control.
It is necessary to study and eliminate the causes of this chaotic behavior and to repeat the
capability study.
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5.2.2 Studying the Statistical Distribution
The observed (measured) characteristic values are interpreted as realizations of a statistical
random variable. An expression such as Determining the distribution shape probably gives the
impression that the measured values hide a specific distribution which is not known initially but
which can be found objectively by using statistical methods.
In reality, one can merely select a distribution model and test it statistically to see if this very
distribution is at the very least almost compatible with the observed data. All other conclusions
made on the strength of the model (e.g. normal distribution) stand and fall with the validity of
the model.
Which distribution models can be used for descriptive purposes?
In the standard [8], some distribution models which are suitable for describing real production
processes are displayed qualitatively. Qualitative here simply means that all that is shown is
how the resulting distribution arises from a Momentary distribution with varying location and
variation parameters and whether the distribution is uni-modal or multi-modal.
If a statistical software program is used, the user is faced with the problem of selecting an
appropriate distribution, i.e., one that represents the random sample on hand. During the process
capability study, statistical tests are used to distinguish only roughly between
normal distribution,
extended normal distribution,
other distributions
The normal distribution or the extended normal distribution acts as the standard distribution.
Using a mathematical procedure called the Johnson Transformation, it is possible to select the
most suitable distribution from the range of other distributions here. The parameters of thedistribution selected are adjusted as well as possible to the data set to be evaluated. If this
automatic distribution adjustment is not available, probability plots or statistical goodness-of-fit
tests can be used to aid in distribution selection.
5.2.3 Calculating Process Capability Indices
The quantile method is recommended as the standard method for calculating machine capability
indices. See Chapter 9 for a discussion of advantages and disadvantages of this and alternative
methods. Manual calculation methods are described in Section 5.3.3.
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5.3 Data Evaluation (Manual Calculation Procedure)
5.3.1 Studying the Process Stability
Test Using the Distribution of the Means
Here, one uses the fact that the means of sufficiently large random samples (approximately n =
4 or higher) are approximately distributed normally. This is not affected by the distribution of
the individual values and is a result of the central limit theorem of statistics.
For the stability test, control limits based on the normal distribution are calculated for the
means and the standard deviations of e.g. groups of five.
If the control limits for the means are exceeded, this shows that the process is not stable and that
its position has changed systematically (see form in Chapter 13).
Test Using the Standard Deviation of the Means
This stability test variant is illustrated in the flow chart in Section 5.3.3.
5.3.2 Studying the Statistical Distribution
A qualitative evaluation can be made and a distribution assigned using graphic representations
such as an individual value plot, a histogram, probability plots etc. Calculating the shape
parameters (skewness, kurtosis) or performing distribution tests can act as quantitative methods.
5.3.3 Calculating Process Capability Indices
If no special software is available, capability and performance values can also be calculated as
follows.
Normal distribution: method M1, M2 (see Sections 9.1 and 9.2)
Random distribution:From the n measured values ix , the total mean x and the overall standard deviation totals
are calculated:
=
=n
1i
ixn
1x ( )
=
=n
1i
2
itotal xx1n
1s
and with T = USL - LSL finally
total
p
s6
TP
= kpP = minimum value of
totaltotal s3
LSLx;
s3
xUSL
If characteristics are limited to one side (by USL and zero alone, or just by LSL), the relevant
formula depending on the given limit applies. As no information is available here on the
distribution model and totals is therefore used, this method can lead to comparatively smaller
results.
Extended normal distribution: See Section 10.2 and flow chart on the following page.
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Manual Calculation Procedure for Extended Normal Distribution
If the queryn
xan
s4.1
s
> receives a yes answer, a systematic variation of the mean has
occurred (trend, batch-to-batch variation). ( )=
==m
1j
2
jxx xx1m
1s is the standard
deviation of the means.
xam = mean of the 3 largest means ix
nim = mean of the 3 smallest means ix
MM (moving mean, trend) is the leeway which the systematic variation of the mean demands
(see also Section 10.2). For and parameter an, see Section 9.1.
Characteristiclimited to only one
side?
noyes
Only USL given?
yes
no
( )
( )
minmax
min3min2min1min
max3max2max1max
MM
xxx31
xxx3
1
=
++=
++=
=
3
USLP
max
pk
=
3
LSLP
min
pk
=
=
3
LSL;
3
USLminP
6
MMTP
minmax
pk
p
x2
s,s,x ==
yes
Process is stable;
no systematic variation
of the mean
no
n
xan
s4.1s
>
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6. Interpretation of Capability Indices
The following section contains information which everyone who performs or evaluates machine
or process capability studies should be familiar with.
The aim of a machine or process capability study is to reach a conclusion about the process
behavior (in control or not?) and an as yet non-existent parent populationnamely the totalityof parts to be manufactured in the future on the basis of the random sample results. This is
called an indirect (or inductive) statistical inference. Such an inference can only be reached if
the distribution of the parent population is already known. On the basis of the (representative)
random sample, the parameters of this distribution are all that have to be estimated.
In reality however, the situation is different. After the sample values have been recorded,
nothing is known about temporal stability, the distribution of the values, the distribution
parameters or their time behavior. All this must be evaluated on the basis of the low number of
single values available.
6.1 Relation Between Capability Index and Fraction Nonconforming
Most of the literature on process capability shows that there is a direct relation between a
calculated pkC value and a fraction nonconforming, e.g.: 33.1Cpk= corresponds to 32 ppm
(one-sided). This relation is based on the normal distribution model. If the real characteristic
distribution deviates from the normal distribution, different fractions nonconforming normally
arise (see Chapter 10).
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
99.99994
%
99.9937 %
99.73 %
95.45 %
68.27 %
1350 ppm
32 ppm
0.3 ppm
2.275%
15.865 %
0.3 ppm
32 ppm
1350 ppm
2.275%
15.865 %
Normal distribution: percentages within the ranges 1s, 2s, 3s, 4s, 5s, as well as
fractions nonconforming at top and bottom.
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6.2 Effect of the Sample Size
The sample size has a major effect on the quality of statistical conclusions. This is reflected in
the size of the confidence intervals for estimated distribution parameters such as mean and
standard deviation. Seen statistically, a machine or process capability index is also a random
variable which can vary from sample to sample even if the process remains unchanged.
In particular, the smaller the random sample, the more critical the allocation of a distributionmodel (based on the Johnson algorithm for example). A suitable distribution model is selected
and adjusted depending on the skewness and kurtosis. However, these variables react very
sensitively to extreme values if the random sample is small. Therefore, making a minor change
to a low number of individual values can lead to a change-over when the distribution model is
selected and a corresponding change to the capability index.
6.3 Effect of the Measurement System
The measuring device and measuring procedure used to measure the parts in the random sample
are very important for evaluating the process later on. If the measuring device is not accurateenough or if the measuring procedure is unsuitable, the tolerance for the production process is
reduced unnecessarily. A large %GRR value or a small gC value impair the machine and
process capability indices.
Please note also that a capable measuring device is no use if the parts are dirty, non-tempered,
deformed or have excessive shape deviations during the test.
Information, examples and calculation procedures for calculating the Capability of measure-
ment and test processes are given in [15].
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7. Capability Indices for Qualitative Characteristics
Capability indices such as pC and pkC can only be calculated using normal formulae if the
characteristic is measurable. However, there are some processes where the characteristic is not
measurable. During the printed circuit board component assembly or soldering processes, for
example, defects can occur which are merely counted but not measured in the true sense of theword. Such defects may include component assembly defects (incorrect or missing component
or wrong direction) and soldering defects (cold soldering point, short-circuit, missing
connection). In order to calculate a capability index in such a case, the following option is
suggested in the literature:
The ration
kp= is considered as a theoretical fraction nonconforming of a normal distribution.
In this ratio, k stands for the number of defects in a random sample, and n stands for the sample
size. If p1u designates the quantile of the standard normal distribution to the probability value
p1 , then3
uC
p1
pk
= is the capability index. This procedure corresponds to method M2 in
Section 9.2 and conforms with [8].
8. Report of Capability or Performance Indices
In order to achieve as much transparency as possible in calculating and circulating capability
indices (reporting), the following information should always be available (see [8]):
Example:
Process capability index Cp = 1.75
Process capability index Cpk = 1.47
Calculation method M4
Number of values taken as a basis 200
Optional:
- Sampling interval- Time and duration of data recording- Distribution model (reason)- Measuring system- Technical framework conditions
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9. Methods for Calculating Capability Indices
In this chapter, all values are called pC or pkC to make things easier.
In concrete applications, it is the statistical distribution (process model) which determines
whether the capability indices pC or pkC , or the process performance indices pP or pkP are
specified. The calculation method has no effect on this.
9.1 Method M1
This method can only be used for normal distribution.
=
6
LSLUSLCp pkC = minimum value of
3
LSL;
3
USL
Estimating the process average:
=
==m
1j
jxm1x Total mean (mean of the sample means)
=
=n
1i
ij xn
1x Mean of a sample with size n (e.g. n = 5)
Estimating the standard deviation of the process
( )
=
=
=
=
==
==n
1i
2
jjijm
1j
j
n
m
1j
2j
22
xx1n
1swhere
sm
1swhere
a
s
sm
1swheres
nd
R= where
=
=m
1j
jRm
1R
totals= where = =
=m
1j
n
1i
2ijtotal )xx(
1nm
1s
n 2 3 4 5 6 7 8 9 10
na 0.798 0.886 0.921 0.940 0.952 0.959 0.965 0.969 0.973
nd 1.128 1.693 2.059 2.326 2.534 2.704 2.847 2.970 3.078
%99PA=
Advantages:
pkC can also be calculated manually.
Disadvantages
The value of the calculated index varies slightly with the formula used to estimate thestandard deviation.
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9.2 Method M2
If Lp is the fraction nonconforming at the lower specification limit LSL and
Up the fraction nonconforming at the upper specification limit USL,
the capability index is
pkC = minimum value of
3
u;
3
uUL p1p1
For information on applying this method for qualitative characteristics, see also Chapter 7.
9.3 Method M3 (Range Method)
minmax
pxx
LSLUSLC
= and pkC = minimum value of
minmax x
LSL;
x
USL
Calculation of as for M1 or =
==m
1j
jx~
m
1x~ (mean of the sample medians)
Advantages:
Always works
It is not necessary to select an approximating distribution
pkC can easily be calculated manually as well
Disadvantages:
The result depends on n
This method does not use all sample values. Outliers have a major effect on the result. Thismethod is therefore not recommended.
Note: this method covers the 60% rule in accordance with QS-Info 2/1996.
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9.4 Method M4 (Quantile Method)
The width of the range, which corresponds to 99.73% of the distribution of the population, is
defined as the process spread. The limits of this range are called 0.135% quantile = 00135.0Q
and 99.865% quantile = 99865.0Q .
0.135% of the values of the population are to be found both below 00135.0Q and above 99865.0Q .
The hat on the Q shows that this is an estimated value calculated from a random sample.
00135.099865.0
pQQ
LSLUSLC
=
pkC = minimum value of
00135.099865.0 Qx~
LSLx~;
x~Q
x~USL
If characteristics are limited to one side (by USL and zero alone, or just by LSL), the relevantformula depending on the given limiting value applies.
Schematic representation
of the method.
In this example of a
normal distribution, the
median x~ is the same as
the mean x , and
because of
s3x~Q 99865.0 = , the
result for pkC is the
same as the result
achieved with the
formula
s3
xUSLCpk
=
Advantages:
This method works for all empirical distributions which one can expect to meet in practice.
Disadvantages
It is necessary to select an approximating distribution.
The result depends on this distribution.
This procedure can only be used with the assistance of a computer (in case of a normaldistribution, still graphically using a probability plot).
USLLSLx~USL
x~ 99865.0Q
x~Q 99865.0
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10. Distribution Models
10.1 Distributions from the Johnson Family
There are several theoretical distribution models. A wide range of distributions can be covered
using the Johnson distributions, e.g. Lognormal distribution.
Unbounded distributions (system of unbounded distributions; see [17])
Bounded distributions (system of bounded distributions; see [17])
Unlike the bell-shaped normal distribution, these functions reach zero. They have contact with
the x-axis. Outside these contact points, they adopt the value zero. For this reason, theoretical
fractions nonconforming with respect to a limiting value LSL or USL can also be exactly zero in
these cases.
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10.2 Extended Normal Distribution
The extended normal distribution arises when a normally-distributed process exhibits an
additional variation of the average position (MM = moving mean). See also example 4 in
Chapter 11.
There are several methods of calculating the MM:
a) Variance-analytical calculation of the variance of the means, followed by calculation ofMM (standard in QS-STAT).
b) Calculation on the basis of the variance of the sample means as long as these are normally
distributed (e.g. x15.5MM = )
c) Calculation of MM as the difference between an upper and lower process location (see flow
chart in Section 5.3.3): minmax MM =
MM
maxmin
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11. Examples
Example 1
Machine capability study, characteristic limited to two sides
Evaluation with quantile method M4
Feature: disk height in mm
n = 50
9.980
9.985
9.990
9.995
10.000
[mm]
0 10 20 30 40 50
USG
OSG
x_
-3 s
+3 s
Normal distribution
988.9x~ = LSL = 9.98 USL = 10.0
8.1Qx~
LSLx~C
00135.0
mk =
=
1.2QQ
LSLUSLC
00135.099865.0
m =
=
USL
LSL
0
5
10
15
20
25
rela
tiveHufigkeit
9.980 9.985 9.990 9.995 10.000
Scheibenhhe [mm] NV
USG OSGx~-3 s +3 s
USLLSL
Relativefrequency
Disk height [mm] ND
Value no.
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Example 2
Machine capability study, characteristic limited to one side on the top
Evaluation with quantile method M4
Characteristic: roughness Rz in m
n = 50
Distribution in accordance with
Johnson transformation
4,1x~ = 0.4USL=
8.0x~Q
x~USLC
99865.0
mk =
=
In this example, there is no point incalculating mC .
The machine is not capable.
Relativefrequency
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Rz[m
]
0 10 20 30 40 50
Wert Nr.
OSG
x_
Up3
Op3
00135,0Q
USL
99865,0Q
Value no.
00135.0Q
99865.0Q
0
10
20
30
40
re
lativeHufigkeit
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
Rz [m] Johnson SB
OSGx~Up3 Op300135,0Q 99865,0
QUSL00135.0
Q 99865.0Q
Relativefrequency
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Example 3
Machine capability study, characteristic limited to two sides
Evaluation with quantile method M4
Feature: housing length in mm
n = 50
Distribution in accordance with
Johnson transformation
033.54x~ = 0.54LSL= 1.54USL=
74.1
Q
x~
LSLx~C
00135.0
mk =
=
49.2QQ
LSLUSLC
00135.099865.0
m =
=
Note: Because of the Johnson transformation, a bounded distribution is allocated to the
empirical distribution in this case. The theoretical fraction nonconforming for both LSL and
USL is zero.
54.00
54.01
54.02
54.03
54.04
54.05
54.06
54.07
54.08
54.09
54.10
[mm]
0 10 20 30 40 50
Wert Nr.
USG
OSG
x_
Up3
Op3
00135,0Q
99865,0Q
USL
LSL
Value no.
00135.0Q
99865.0Q
0
4
8
12
16
20
relativ
eHufigkeit
54.00 54.02 54.04 54.06 54.08 54.10
[mm] Johnson SB
USG OSGx~Up3 Op3
USLLSL
00135,0Q 99865,0Q
Relativefrequency
00135.0Q 99865.0Q
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Example 4
Evaluation of long-term process capability, characteristic limited to one side on the top
Characteristic: cylindricity in m
n = 775
Extended normal distribution
8.1x~ = USL = 4.0
44.1x~Q
x~USLP
99865.0
pk =
=
In this example, there is no point incalculating Pp.
Long-term data was evaluated here. As the process exhibits systematic variation of the mean, itis not stable within the meaning of [4]. The process performance index Ppkis therefore given.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Zylinderform
0 100 200 300 400 500 600 700
OSG
x_
Up3
Op3
00135,0Q
99865,0Q
USL
Cylindricity
00135.0Q
99865.0Q
0
4
8
12
16
20
relativeHufigkeit
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Zylinderform NV()
OSGx~Up3 Op300135,0Q 99865,0Q USL
Relativefrequency
Cylindricity ND()
00135.0Q 99865.0Q
Value no.
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12. Capability Indices for Two-Dimensional Characteristics
The position of a bore is one example of a two-dimensional characteristic. The position in aplane is clearly determined by indicating two x and y coordinates relative to the origin (pointwith the (0, 0) coordinates). The position tolerance can be given by means of a circle with radius
T where the center is identical to the target position.
In the following, it is assumed that the measured positions )y,x( ii are subject to a two-
dimensional normal distribution, i.e. each component is distributed normally. These positionscan be displayed as points in the x-y diagram. Using suitable software, a two-dimensionalnormal distribution is adjusted to these positions and an elliptical random variation range
calculated. This range includes the p1 percentage of the parent population and lies
completely within the tolerance circle.
Just as in method M2 for one-dimensional characteristics, the process capability index is then
given as3
uC
p1
pk
= , where p1u is the (one-sided right) threshold value (quantile) to the
fraction nonconforming p of the (one-dimensional) standard normal distribution.
If the measured points are moved in such a way that their center )y,x( is the same as the center
of the tolerance circle, the elliptical random variation range (still completely within the
tolerance circle) is larger. The relevant process capability index pC is given as3
uC
p1
p
= .
Notes:The tolerance circle can be distorted into an ellipse in a rectangular screen even when the xand y-scale are selected immediately. If the positions are given in polar coordinates (radius andangle), they must be turned into Cartesian coordinates. The sketched procedure can be appliedto any bivariate characteristic (e.g. unbalance) and can be generalized for characteristics with p
components using the p-variate normal distribution.
The 4 s-ellipse (99.9968% range)touches the tolerance circle,Cpk = 1.33.
When the cloud of points is centered,the 7.6 s-ellipse touches the tolerancecircle,
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
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13. Forms
Machine and process capability studies are normally evaluated using special computerprograms. The forms listed here are therefore given purely as aids for collecting data manuallyand using manual calculation procedures.
Evaluation form for machine capability study
Evaluation form for process capability study
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Normal distribution
Order: 047011
Quality Assurance Sheet no.: 1 von 1
Machine No.: Designation: Year: Workshop:
314084 Milling machine 1998 W33007
Part: Disk Tool: xyz Evaluation performed by: Schmidt
Material: Steel Meas. system: Digital Gage Date: 13/10/01
Nominal value 10.00 mm Standard: Gage block Machine cycle time: 2 / min
Tolerance: 0.20 mm Operation: Milling Duration of random sampling: 100 min
Upper limit: 10.20 mm Start of random sampling: 8:50 PM
Lower limit: 10.00 mm End of random sampling: 10:30 PM
Process and ambient parameters:
Batch A
Cutting speed vx, rpm nx
Tool 3
Machine temperature 27.3 C
Air temperature 24.5 C
Punching machine not operating
Location: Building 17/2
Machine was not switched off during breakfast from 8:30 pm to 8:45 pm
Note: Enter measured values overleaf
Evaluation: (Take values from overleaf):
10.116 mm 0.016
10.120 mm 0.016
10.112 mm 0.024
Smallest value
Cmk is valid!
Stability test (if "no", process unstable):
Stability limits for mean values: Stability limit for standard deviations:
10.137 0.034
10.095
Are and X Yes X Yes
within UCL and LCL? No No
RobertBoschGmbH
reservesallrightsevenintheeventofindustrialprope
rtyrights.
Wereserveallrightsofdisposalsuchascopy
ingandpassingontothirdparties.
1.75
2.08
2.42
Evaluation Sheet
for Machine
Capability Analysis
H. u. K.
Manufacturer:
0.116
0.0160
0.200
0.0960
0.084
0.0480
=
=
total
ms6
TC
=x
=maxx
=mi nx
=totals
=s
=maxs
maxx mi nx
=
=
=
=
=
total
mks3
LSLxC
=
=
total
mks3
xUSLC
?UCLsIssmax
=+= s3.1xUCL
== s3.1xLCL
== s1.2UCLs
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Individual value chart
Table values in mm Deviation from
m ----> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1
x1 0.13 0.11 0.10 0.09 0.11 0.13 0.10 0.14 0.13 0.12
x2 0.10 0.14 0.12 0.12 0.10 0.12 0.11 0.10 0.15 0.10
x3 0.11 0.10 0.11 0.13 0.13 0.15 0.14 0.13 0.12 0.12
x4 0.12 0.12 0.13 0.10 0.10 0.10 0.12 0.12 0.09 0.11x5 0.11 0.13 0.11 0.14 0.12 0.09 0.10 0.11 0.10 0.13
0.114 0.120 0.114 0.116 0.112 0.118 0.114 0.120 0.118 0.116
s 0.011 0.016 0.011 0.021 0.013 0.024 0.017 0.016 0.024 0.011
Evaluation:
Robert Bosch GmbH reserves all rights even in the event of industrial property rights. We reserve all rights of disposal such as copying and passing
0.116 0.120
10.00
0.00.0160.112
s
x
x
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
1 2 3 4 5 6 7 8 9 10 11 12 13 14 1
=x =maxx =minx =maxs=s
0.04 -
0.03 -
0.02 -
0.01 -
0.14 -
0.13 -
0.12 -
0.11 -
0.10 -
0.09 -
0.20 -
0.10 -
0 -
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Normal distribution
Order: 047011
Quality Assurance Sheet no.: 1 von 1
Machine No.: Designation: Year: Workshop:
20113 BZ20 1998 W 1391
Part: Housing Tool: Milling cutter Evaluation performed by: Rl.
Material: Al Meas. system: Trimos Date: 13/05/03
Nominal value 54.10 mm Standard: Machine cycle time: 3 / min
Tolerance: 0.40 mm Operation: Mill end face Duration of random sampling: 3 h
Upper limit: 54.30 mm Start of random sampling: 6:45 PM
Lower limit: 53.90 mm End of random sampling: 9:45 PM
Process and ambient parameters:
Blank from supplier R. & B.
Cutting speed vx, rpm nx
Feed sx
Milling cutter 2 HSS
Machine temperature 30.1 C (run-in time 15 min)
Air temperature 25.7 C
Location Building 17/2
Note: Enter measured values overleaf
Evaluation (Take values from overleaf):
54.114 mm
54.123 mm 0.039 0.0415
54.101 mm 0.045
Smallest value
Cpk is valid!
Stability test (if "no", process unstable):
Stability limits for mean values: Stability limits for standard deviations:
54.165 0.082
54.063
Are and X Yes X Yes
within UCL and LCL? No No
RobertBoschGmbH
reservesallrightsevenintheeventofindustrialprope
rtyrights.
Wereserveallrightsofdisposalsuchascopy
ingandpassingontothirdparties.
0.400
0.2489
0.214
0.12451.72
1.61
1.490.186
0.1245
Evaluation Sheet
for Process
Capability Analysis
Steinel
Manufacturer:
=x
=maxx
=mi nx
=totals
=s
=maxs
maxx mi nx
=
=
6
TCp
=
=
=
==
94.0
s
=
=
3
xUSLCpk
=
=
3
LSLxCpk
?UCLsIs smax
=+= s3.1xUCL
== s3.1xLCL
== s1.2UCLs
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Individual value chart
Table values in mm Deviation from
m ----> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1
x1 0.06 0.07 0.11 0.13 0.06 0.10 0.15 0.10 0.15 0.08
x2 0.09 0.11 0.17 0.11 0.17 0.05 0.08 0.14 0.05 0.15
x3 0.08 0.14 0.06 0.09 0.14 0.16 0.12 0.07 0.14 0.14
x4 0.11 0.18 0.12 0.16 0.07 0.14 0.10 0.17 0.09 0.10x5 0.17 0.10 0.09 0.10 0.11 0.10 0.07 0.12 0.16 0.15
0.101 0.120 0.112 0.117 0.112 0.109 0.104 0.121 0.118 0.123
s 0.042 0.041 0.042 0.029 0.044 0.042 0.033 0.037 0.045 0.031
Evaluation:
Robert Bosch GmbH reserves all rights even in the event of industrial property rights. We reserve all rights of disposal such as copying and passing
0.114 0.123
54.00
0.00.0390.101
s
x
x
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
1 2 3 4 5 6 7 8 9 10 11 12 13 14 1
=x =maxx =minx =maxs=s
0.10 -
0.05 -
0.00 -
0.20 -
0.15 -
0.10 -
0.05 -
0.00 -
0.20 -
0.10 -
0 -
-
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14. Abbreviations
na Factor for calculating from the mean standard deviation s
gC Capability index of a measurement process without taking account
of the systematic deviation
mC , kmC Machine capability index
pC , kpC Process capability index
nd Factor for calculating from the mean range R
LCL Lower control limit (lower limit for the stability test)
LSL Lower specification limit
m Number of groups of five or number of single samples
MM Moving mean; measure for the systematic variation of the mean
max Mean of the three largest means
min Mean of the three smallest means
n Number of values per column or number of values in a single sample
AP Confidence level
pP , kpP Process performance index
%GRR Overall variation of a measurement processreferred to the tolerance of the characteristic
Lp Fraction nonconforming at the lower specification limit LSL
Up Fraction nonconforming at the upper specification limit USL
00135.0Q Estimated value for the 0.135% quantile of a characteristic distribution
99865.0Q Estimated value for the 99.865% quantile of a characteristic distribution
R Range of a set of numbers
jR Range of the jthsample
R Mean of ranges
s Empirical standard deviation
2
s Mean variance; mean of squared standard deviations
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maxs Largest single value from a set of standard deviations
ns Standard deviation of a random sample of n single values
totals Standard deviation of all single values
xs Standard deviation of the means
s Mean standard deviation from m random samples of equal size
T Tolerance of a characteristic
p1u Quantile of the standard normal distribution to value 1-p
UCL Upper control limit (upper limit for the stability test)
USL Upper specification limit
ix i -th single value in a random sample
jix i -th single value in the j-th random sample
maxx Largest single value in a set of numbers (maximum)
minx Smallest single value in a set of numbers (minimum)
x Arithmetic mean
jx jtharithmetic mean
x Mean of means
x~ Median
Estimated value for the standard deviation of the parent population
Sum
Reference to other commonly-used abbreviations:
Booklet No. 9 DIN 55319 ISO/DIS
21747
QS-STAT
Lower specification limit LSL L L LSL
Upper specification limit USL U U USL
Lower Quantile00135.0Q 00135.0Q %135.0X 3obQ
Upper Quantile 99865.0Q 99865.0Q %865.99X 3unQ
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15. References
[1] EN ISO 9000:2000 Quality management systemsFundamentals and vocabulary
[2] EN ISO 9001:2000 Quality management systemsRequirements
[3] ISO/TS 16949 Quality management systems Particular requirements for theapplication of ISO 9001:2000 for automotive production and relevant service partorganizations
[4] ISO/DIS 21747:2002 Process Performance and Capability Indices
[5] ISO/DIS 3534-2 StatisticsVocabulary and Symbols
DIN 55350 Begriffe der Qualittssicherung und Statistik
[6] DIN 55350-11 Begriffe des Qualittsmanagements[7] DIN 55350-33 Begriffe der statistischen Prozesslenkung (SPC)
[8] DIN 55319 Qualittsfhigkeitskenngren
[9] Chrysler, Ford, GM: QS-9000, Quality System Requirements, 1995
[10] Chrysler, Ford, GM: Statistical Process Control, Reference Manual, 1995
[11] Chrysler, Ford, GM: Production Part Approval Process, PPAP, 1999
Bosch, Booklet Series: Quality Assurance in the Bosch Group, Technical Statistics[12] No. 1, Basic Concepts of Technical Statistics, Variable Characteristics[13] No. 3, Evaluation of Measurement Series[14] No. 7, Statistical Process Control[15] No. 10, Capability of Measurement and Test Processes
[16] Dietrich/Schulze: Statistical Procedures for Machine and Process Qualification, 2003,Hanser-Verlag
[17] Elderton and Johnson, Systems of Frequency Curves, 1969, Cambridge Univ. Press
[18] Davis R. Bothe: Measuring Process Capability, 1997, McGraw-Hill
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Index
Page
5M ............................................................................................................................................... 14Capability indices.......................................................................................................................... 5Capability or performance indices ................................................................................................ 4Distribution models..................................................................................................................... 15Extended distribution ............................................................................................................ 17, 25Fraction nonconforming.............................................................................................................. 18Indirect inference ........................................................................................................................ 18Johnson distribution family......................................................................................................... 24Kurtosis ....................................................................................................................................... 19Machine capability study .............................................................................................................. 4Manual calculation procedure ......................................................................................... 11, 16, 17
Measurement system................................................................................................................... 19Process capability
Indices Cp, Cpk....................................................................................................................... 15Process capability study ................................................................................................................ 5Process performance ..................................................................................................................... 5Qualitative characteristics ........................................................................................................... 20Quantile method.............................................................................................................. 10, 15, 23Range method.............................................................................................................................. 22Sample size.................................................................................................................................. 19Skewed distribution....................................................................................................................... 9Skewness..................................................................................................................................... 19Stability ......................................................................................................................................... 9
Stability test................................................................................................................................. 16Two-dimensional characteristics................................................................................................. 30
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Robert Bosch GmbH
C/QMM
Postfach 30 02 20
D-70442 Stuttgart
GermanyPhone +49 711 811-4 47 88
Fax +49 711 811-2 31 26
www.bosch.com