February 2005
VW
101 30
Contents
Page
1
2
3
3.1
3.1.1
3.1.2
3.1.3
3.2
3.2.1
3.2.2
3.3
3.4
4
4.1
4.2
4.3
4.4
4.4.1
4.4.2
4.4.3
4.4.4
4.4.5
4.4.6
4.4.7
4.4.8
4.5
4.6
4.7
4.8
5
6
7
8
Form FE 41 - 01.03
Numerical notation according to ISO practice (see VW 01000).
The English translation is believed to be accurate.In case of discrepancies the German version shall govern.
Konzernnorm
Descriptors: machine capability investigation, capability index, quality capability, machine
capability
Purpose and scope............................................................................................................ 2
Principle of the machine capability investigation ................................................................ 3
Theoretical principles......................................................................................................... 4
Distribution models............................................................................................................ 4
Normal distribution............................................................................................................. 4
Absolute value distribution, type I ...................................................................................... 5
Type II absolute value distribution (Rayleigh distribution) .................................................. 7
Determination of capability ................................................................................................ 8
Determination of capability for defined distribution models ................................................ 9
Determination of capability for undefined distribution models .......................................... 16
Limit values of machine capability ................................................................................... 18
Statistical tests ................................................................................................................ 20
Carrying out a machine capability investigation ............................................................... 21
Test equipment use ......................................................................................................... 24
Sampling ......................................................................................................................... 24
Special regulation for restricted MFI ................................................................................ 25
Data analysis................................................................................................................... 25
Selection of the expected distribution model.................................................................... 25
Test for outliers................................................................................................................ 26
Take outliers out of the calculation of statistics ................................................................ 26
Test for change of the production location ....................................................................... 26
Test for deviation from the specified distribution model ................................................... 26
Evaluation according to normal distribution ..................................................................... 26
Evaluation according to specified model.......................................................................... 26
Distribution-free evaluation .............................................................................................. 27
Documentation ................................................................................................................ 27
Results evaluation ........................................................................................................... 28
Machine optimization....................................................................................................... 29
Handling incapable machines.......................................................................................... 29
Examples ........................................................................................................................ 30
Referenced standards ..................................................................................................... 33
Referenced literature ....................................................................................................... 33
Keyword index................................................................................................................. 34
Page 1 of 36
Fachverantwortung/Responsibility
Normung/Standards (EZTD, 1733)
K-QS-1/2
Bestenreiner
Neupert
Tel.: -79298
Tel.: +49-5361-9-26513
Sobanski
Confidential. All rights reserved. No part of this document may be transmitted or reproduced without prior permission of a Standards Department of the Volkswagen Group.
Parties to a contract can only obtain this standard via the B2B supplier platform “www.vwgroupsuppy.com”.
 VOLKSWAGEN AG
QUELLE: NOLIS
Machine Capability Investigation for
Measurable Characteristics
Norm vor Anwendung auf Aktualität prüfen / Check standard for current issue prior to usage.
Klass.-Nr. 11 00 5
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VW 101 30: 2005-02
Introduction
An evaluation of the machine capability regarding observable measurable production
characteristics is an important prerequisite for fulfillment of the specified quality requirements.
However, for many practical cases of the capability investigation, to date no standards or uniform
Group regulations exist, so in identical cases completely different capability evaluations could
result. This standard was developed in order to make it possible to carry out the capability
investigation under uniform rules for all practical cases and thus ensure the comparability of the
results in the Volkswagen Group.
This standard contains all of the theoretical principles, as a self-contained whole, that are needed
for use and understanding. Only the statistical tests that are already described in detail in
standards or standard works in the literature of statistics will be indicated with references.
To carry out a machine capability investigation according to this standard, a computer program is
required in which the algorithms described are implemented. If such a computer program is
available, the user can basically concentrate on the rules described in Section 4 and look up
theoretical principles if necessary. The most important sub-sections in it are:
─
4.2 Sampling
─
4.5 Documentation
─
4.6 Results evaluation
Section 5 also lists examples that can be used to help with the results evaluation.
1
Purpose and scope
The goal of a machine capability investigation is a documented evaluation of whether the machine
to be tested makes possible secure production of a characteristic considered within defined limit
values. Ideally, in this process, only machine-related influences should have an effect on the
production process.
How and under what requirements machine capability investigations are to be carried out is the
object of this standard. It is applicable to any continuous1) (measurable) production characteristics.
1) In the following, the designation “measurable characteristics” is used for this characteristic type instead of the designation
recommended by DIN “continuous characteristics”, since this has become established at Volkswagen.
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2
Principle of the machine capability investigation
Basically different values of a considered characteristic result during the production of the same
type of parts with the machine to be tested due to random influences. These characteristic values2)
disperse around a location caused by systematic influences, depending on the production quality.
Therefore there is a test of how well the distribution of the characteristic values fits into the
tolerance zone defined by the designer (Figure 1). The evaluation of this is expressed by the
capability indexes cm and cmk (c as in capability), whereby only the production dispersion is taken
into consideration in the cm value and the process location is additionally taken into consideration in
the cmk value. These statistics must be at least as great as the defined limit values in order to meet
the requirement for a capable machine.
To determine the capability indexes with respect to the considered characteristic, an adequately
large random sample of produced parts (generally n = 50) is taken in direct sequence. So that
essentially only the machine influence is recorded, the samples shall be taken under conditions
that are as ideal as possible regarding the influence categories material, human, method and
environment. From this random sample, the location µ and dispersion range limits x0.135% and
x99.865% are estimated for the population of the characteristics values (theoretically infinite quantity)
and compared to the tolerance zone [Gu, Go] (Figure 1). In this case, the dispersion range limits
are specified in such a way that the percentage of characteristic values outside the dispersion
range is pe = 0.135% on both sides. In addition, there is a test of whether the distribution of the
characteristic values corresponds to an expected regularity.
Tolerance
Toleranz
Defined
process
dispersion range
definierte
Prozeßstreubreite
Tolerance
Frequency
Frequency
Häufigkeit
Defined process dispersion
range
pe = 0,135%
0
Gu
2
pe = 0,135%
x 0,135%
4
µ
6
8
Characteristic
value
Characteristic
Merkmalswert
value
x 99,865%
10
Go
12
Figure 1 - Example of a distribution of characteristic values within a defined tolerance zone
2) The term characteristic value should not be confused with the term measured value, since in contrast to the first, the latter contains an
uncertainty.
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3
Theoretical principles
3.1
Distribution models
The distribution of characteristic values can be described by a distribution model for most types of
production characteristics. This means that most production characteristics with two-sided
tolerance zones, e.g. length dimensions, diameters and torques can be based on a normal
distribution.
In contrast, the variation behavior of production characteristics with upper limited tolerance zone
only can be described by absolute value distributions of type I or type II. For example, the type I
absolute value distribution can be used as the basis for the characteristic types parallelism and
asymmetry and the type II absolute value distribution can be used for the characteristic types
position and coaxiality.
3.1.1
Normal distribution
The function of the probability density (density function for short) of a normal distribution that
is shown graphically in Figure 2 is:
fN (x ) =
1
σ ⋅ 2π
⋅e
1  x −µ 
− 

2 σ 
2
(1.1)
Probability density
Wahrscheinlichkeitsdichte
with the parameters mean value µ and standard deviation σ, that identify the location and width
of a distribution, whereby the square of the standard deviation σ2 is identified as variance.
σ
Wendepunkte
Turning
points
pe = 0,135%
pe = 0,135%
-4
µ-4σ
-3
µ-3σ
-2
µ-2σ
-1
µ-σ
0µ
1
µ+σ
2
µ+2σ
3
µ+3σ
4
µ+4σ
Merkmalswertvalue
Characteristic
Figure 2 – Function of the probability density of a normal distribution*)
*)
Illustrations, graphics, photographs and flow charts were adopted from the original German standard, and the numerical notation
may therefore differ from the English practice. A comma corresponds to a decimal point, and a period or a blank is used
as the thousands separator.
Page 5
VW 101 30: 2005-02
The portion of the area below the graph in Figure 2 within a considered time interval can be
interpreted as probability. The probability of finding a characteristic value x in a population that is at
most as high as a considered limit value xg is thus specified by the integral function (distribution
function). For normal distribution, this is
xg
FN (xg ) =
∫ fN (x ) ⋅ dx
(1.2)
−∞
where
∞
∫ fN (x ) ⋅ dx = 1
(1.3)
−∞
and fN (x ) ≥ 0 for all values x.
With the transformation
u=
x−µ
(1.4)
σ
(1.1) produces the probability density of the standardized normal distribution
ϕ (u ) =
1
2π
⋅e
1
− u2
2
(1.5)
and the distribution function
ug
Φ (u g ) = ∫ ϕ (u ) ⋅ du
(1.6)
−∞
with the standard deviation σ =1
3.1.2
Absolute value distribution, type I
The type I absolute value distribution results by convolution of the density function of a normal
distribution at the zero point, where the function values on the left of the convolution are added to
those on the right.
The density function and the distribution function of the type I absolute value distribution are thus
1  x + µN 
 − 1  x − µ N 

− 
 2  σ N 

1
2 σ
fB1 (x ) =
⋅ e
+e  N 
σ N ⋅ 2π 

2
 xg − µN
FB1 (x g ) = Φ 
 σN

 x + µN
 +Φ g

 σ
N



 −1


2





for x ≥ 0
(1.7)
(1.8)
where
µ N:
mean value of the original normal distribution that identifies a systematic zero point shift
σN:
standard deviation of the original normal distribution
Φ:
distribution function of the standardized normal distribution
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Figure 3 shows density functions that result from convolution of the density of the normal
distribution at different zero point shifts
µN = 0
µN = 2σN
µN = 3σN
Probability density
Wahrscheinlichkeitsdichte
µN = 1σN
0
1σN
2σN
3σN
4σN
Merkmalswert
Characteristic value
5σN
6σN
Figure 3 - Density function of the type I absolute value distribution with different zero point
shifts
Mean value and variance of the type I absolute value distribution are:
 µ 
 − µN
µ = µ N ⋅ Φ  N  − Φ 
 σN
 σN 
 2 ⋅ σ N
  +
⋅e

2
π

1 µ
−  N
2σN




2
σ 2 = σ N2 + µ N2 − µ 2
(1.9)
(1.10)
For the case of a zero point shift µ N = 0, the following result from (1.9) and (1.10)
µ=
2 ⋅σ N
2π


σ 2 = 1 −
2 2
 ⋅σN
π
(1.11)
(1.12)
As Figure 3 shows, with increasing zero point shift, the type I absolute value distribution
approaches a normal distribution. Thus for the case
µ
≥3
σ
(1.13)
the type I absolute value distribution can be replaced with a normal distribution with good
approximation.
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3.1.3
Type II absolute value distribution (Rayleigh distribution)
The type II absolute value distribution results from the vectorial values of the orthogonal
components x and y of a two-dimensional normal distribution, where equal standard deviations are
assumed for the components. This case is present for many production characteristics in the form
of radial deviations r from a considered point or a considered axis.
The density function and the distribution function of the type II absolute value distribution are
generally
f B 2 (r ) =
r
2π ⋅ σ N2
⋅e
1  z 2 +r 2
− 
2  σ N2




2π
⋅ ∫e
z ⋅r ⋅cos α
σ N2
dα
for r ≥ 0
(1.14)
0
rg
FB 2 (rg ) = ∫ fB 2 (r ) ⋅ dr
(1.15)
0
where
σN:
Standard deviation of the orthogonal components x and y, from which the radial deviation r
of a reference point or reference axis results
z:
Eccentricity: distance between coordinate origin and frequency midpoint
Figure 4 displays density functions of the type II absolute value distribution that result in units of σN
for different eccentricities.
z = 1σN
z = 2σN
z = 3σN
Wahrscheinlichkeitsdichte
Probability density
z=0
0
1σN
2σN
3σN
4σN
Characteristic
Merkmalswert
value
5σN
6σN
Figure 4 - Density functions of the type II absolute value distribution with different
eccentricities
Mean value and variance of the type II absolute value distribution are
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∞
µ = ∫ fB 2 (r ) ⋅ r ⋅ dr
(1.16)
0
σ 2 = 2σ N2 + z 2 − µ 2
(1.17)
For an eccentricity z = 0, (1.14) and (1.15) density function and distribution function of the Weibull
distribution, with the shape parameter value 2 yield
fB 2 (r ) =
r
σ N2
1 r
− 
2σN




1  rg
− 
2 σN




⋅e
FB 2 (rg ) = 1 − e
2
(1.18)
2
(1.19)
and from them, in turn, mean value and variance
µ = σN ⋅


σ 2 = 2 −
π
(1.20)
2
π
 ⋅σN
2
2
(1.21)
As Figure 4 shows, with increasing eccentricity, the type II absolute value distribution approaches a
normal distribution. Thus for the case
µ
≥6
σ
(1.22)
the type II absolute value distribution can be replaced with a normal distribution with good
approximation.
3.2
Determination of capability
The capability indexes cm and cmk indicate how well the production results comply with the
tolerance zone of a considered characteristic. In this process, only the production dispersion is
taken into consideration in the cm value. The production location is taken into consideration by the
cmk value. Because of this, on the one hand, there can be an expression of what value is possible
in an ideal production location and, on the other, a comparison of the two values makes it possible
to express how much the production location deviates from the specified value. The higher the
capability indexes determined, the better the production.
There are different evaluation formulas, which have to be selected appropriately in the individual
case, for determining the capability indexes. Since the determination of the capability indexes can
only be carried out using random sample values, the results only represent estimates of the values
for the population which are to be determined and are thus identified by a roof symbol3).
3) The identification of the estimated capability indexes by a roof symbol is only important for understanding the theory, so they can be
dispensed with during the evaluations in practice.
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3.2.1
Determination of capability for defined distribution models
3.2.1.1
Capability indexes
For a production characteristic which is to be examined and whose random sample values are not
in contradiction to a theoretically expected distribution model, the capability indexes are estimated
in a manner appropriate to the respective case (see also examples 1 and 2 in Section 5) according
to the following formulas:
Capability indexes for characteristics with two-sided tolerance zones (according to DIN 55319,
measuring method M4), e.g. for length dimension:
G o − Gu
x̂ 99 ,865% − x̂ 0 ,135%
ĉ m =
 Go − µˆ
µˆ − Gu 
ĉmk = min 
;

 x̂99 ,865% − µˆ µˆ − x̂0 ,135% 
(2.1)
(2.2)
Capability indexes for characteristics with upper limited tolerance zone and natural lower limit value
zero, e.g. for radial runout deviation:
ĉ m =
Go
x̂ 99 ,865% − x̂ 0 ,135%
ĉ mk =
Go − µˆ
x̂ 99 ,865% − µˆ
(2.3)
(2.4)
Capability indexes for characteristic with lower limited tolerance zone only, e.g. for tensile strength:
ĉ mk =
µˆ − Gu
(2.5)
µˆ − x̂ 0 ,135 %
where
Go, Gu : Upper and lower limiting value, respectively
µ̂ :
Estimated mean value
x̂ 0 ,135 % , x̂ 99 ,865% : Estimated values for dispersion range limits (quantiles, below which the
specified percentage p of measured values lie)
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3.2.1.2
Estimation of the statistics
The statistics mean value µ and standard deviation σ of a population can be estimated,
independently of the distribution model, from the measured values of a random sample according
to expectation, by
1 ne
µ̂ = x =
⋅ ∑ xi
n e i =1
σ̂
2
(2.6)
ne
1
( x i − x )2
=s =
∑
n e − 1 i =1
2
(2.7)
where
n e = n − n A : effective random sample size
(2.8)
n : defined random sample size
n A : number of outliers
x i : ith characteristic value
In the case of data to be evaluated in the form of a frequency distribution of classified
measured values, e.g. from manual recordings (tally) in a class subdivision of the value range, the
statistics µ and σ can be estimated by
µ̂ = x =
σ̂ = s =
1 K
⋅ ∑ ak ⋅ x k
n e k =1
(2.9)
K
1
2
⋅ ∑ a k ⋅ (x k − x )
n e − 1 k =1
(2.10)
where
x k : mean value of the kth class
a k : absolute frequency of the measured values in the kth class (without outliers)
K : maximum number of measured value classes
3.2.1.3
Estimation of the dispersion range limits
The dispersion range limits depend on the distribution model and are estimated as follows:
Dispersion range limits for normal distribution:
If the normal distribution model is the fitting distribution model, the values µ̂
and σ̂ determined
according to (2.6) and (2.7) or (2.9) and (2.10) respectively result as estimated values for the
dispersion range limits
x̂ 99 ,865% 
 = µˆ ± 3σˆ
x̂ 0 ,135% 
(2.11)
which, used in turn in formulas (2.1) and (2.2) result in the classical formulas for calculating the
capability indexes (see also example 1 in Section 5).
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Dispersion range limits of the type I absolute value distribution:
To determine the dispersion range limits for a type I absolute value distribution, first the statistics µ
and σ are estimated according to formulas (2.6) and (2.7) or (2.9) and (2.10) respectively.
For the case µˆ σˆ < 3 , the estimated statistics µ and σ are used to estimate the parameter values
µ N and σN to be determined of the type I absolute value distribution to be fitted, in the following
manner:
Equation (1.9) yields the function
 µˆ
µˆ
= ψ B1  N
σˆ N
 σˆ N
 µˆ N
 =
 σˆ N
  µˆ
⋅ Φ  N
  σˆ N

 − µˆ N
 − Φ 

 σˆ N

2
  +
⋅e

2π

1  µˆ
−  N
2  σˆ N




2
(2.12)
With equation (1.10), this yields the following function
 µˆ
µˆ
= ϑB1  N
σˆ
 σˆ N

 =

 µˆ N
 σˆ N
ψ B1 
 µˆ
1 +  N
 σˆ N



2


 µˆ
 − ψ B1  N


 σˆ N

(2.13)

 


2
Equations (1.11) and (1.12) result in the condition
µˆ
2
≥
= 1,3236
σˆ
π −2
(2.14)
Under the condition (2.14), the parameter values to be determined of the type I absolute value
distribution can be estimated using
σˆ N = σˆ ⋅
 µˆ 
1+ 
 σˆ 
2

 µˆ  
1 +  ξ B1   
 σˆ  

2
 µˆ 
µˆ N = ξ B1   ⋅ σˆ N
 σˆ 
(2.15)
(2.16)
where
 µˆ 
ξ B1   : inverse function of (2.13)
 σˆ 
In the case where the ratio µˆ σˆ is lower than the limit value 1.3236 from condition (2.14) because
of random deviations of the random sample statistics, the ratio µˆ σˆ is set to this limit value, at
which the following parameter values result:
µˆ N = 0 and according to formula (1.12)
σˆ N =
π
π −2
⋅ σˆ = 1,659 ⋅ σˆ
(2.17)
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The connection between the parameter values µN and σN of the type I absolute value distribution
and the statistics µ and σ is shown graphically in Figure 5, related to σ
Relative
parameter
relativerdistribution
Verteilungsparameter
3,5
3,0
µN / σ
2,5
2,0
1,5
σN / σ
1,0
0,5
0,0
1,2
1,4
1,6
1,8
2,0
2,2
2,4
2,6
2,8
3,0
Relative
location
relative
Lage µ / σ
Figure 5 - Relative parameter values of the type I absolute value distribution
in relationship to the relative location*)
For the fitted type I absolute value distribution, the dispersion range limits can then be determined
numerically and their relationships with the relative location are shown in Figure 6.
RelativeStreubereichsgrenze
scatter range limit
relative
7
x99,865% / σ
6
5
4
3
2
1
x0,135% / σ
0
1,2
1,4
1,6
1,8
2,0
2,2
2,4
2,6
2,8
3,0
relative
Lage µ / σ
Relative
location
Figure 6 - Relative dispersion range limits of the type I absolute value distribution in
relationship to the relative location*)
*)
Illustrations, graphics, photographs and flow charts were adopted from the original German standard, and the numerical notation
may therefore differ from the English practice. A comma corresponds to a decimal point, and a period or a blank is used
as the thousands separator.
Page 13
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For direct determination of the statistics µ N and σN from the statistics µ and σ, for
1,3236 ≤ µˆ σˆ < 3 the following approximation can also be used as an inverse function of (2.13)
with adequate precision (error related to σ less than 0.01)
 µˆ 
 µˆ

 ≅ 1,64 ⋅  − 1,3236 
 σˆ 
 σˆ

0 ,286
ξ B1 
 µˆ

+ 0 ,634 ⋅  − 1,3236 
 σˆ

1,05
(2.18)
In addition, the determination of the dispersion range limits for 1,3236 ≤ µˆ σˆ < 3 is carried out
directly with the use of the following approximation (error related to σ less than 0.02)
x̂ 99 ,865 % 

 ≅ σˆ
x̂ 0 ,135 % 
0 ,67

 µˆ

 µˆ

+ 2 ,505 ⋅  − 1,3236  + 5 ,3171
− 2 ,47 ⋅  − 1,3236 

 σˆ

 σˆ

⋅
4
 µˆ


0 ,018 ⋅  − 1,3236  + 0 ,0028


 σˆ

(2.19)
For µˆ σˆ ≥ 3 the calculation of the dispersion range limits is carried out according to formula
(2.11).
Dispersion range limits of the type II absolute value distribution:
To determine the dispersion range limits for a type II absolute value distribution (see also example
2 in Section 5) first the statistics µ and σ are estimated according to formulas (2.6) and (2.7) or
(2.9) and (2.10) respectively. For the case µˆ σˆ < 6 , the estimated statistics µ and σ are then used
to estimate the parameter values z and σN to be determined of the type II absolute value
distribution to be fitted, in the following manner:
Equations (1.14) and (1.16) yield the function
 ẑ
µˆ
= ψ B 2 
σˆ N
 σˆ N

1
 =
⋅e
π
2

1  ẑ
− 
2  σˆ N




2
∞
1
2π
ẑ
− v
⋅ ∫  v 2 ⋅ e 2 ⋅ ∫ e σ N

0
0
2
⋅v ⋅cos α

dα  ⋅ dv


(2.20)
where
v=
r
σˆ N
(2.21)
With equation (1.17), this yields the following function
 ẑ
µˆ
= ϑ B 2 
σˆ
 σˆ N

 =

 ẑ
 σˆ N
ψ B 2 
 ẑ
2 + 
 σˆ N
2





 ẑ
 − ψ B 2 


 σˆ N


 


2
(2.22)
Equations (1.20) and (1.21) result in the condition
µˆ
π
≥
= 1,9131
σˆ
4 −π
(2.23)
Page 14
VW 101 30: 2005-02
Under the condition (2.23), the parameter values to be determined of the type II absolute value
distribution can be estimated using
σˆ N = σˆ ⋅
 µˆ 
1+ 
 σˆ 
2

 µˆ  
2 +  ξ B 2   
 σˆ  

(2.24)
2
 µˆ 
ẑ = ξ B 2   ⋅ σˆ N
 σˆ 
(2.25)
where
 µˆ 
ξ B 2   : inverse function of (2.22)
 σˆ 
In the case where the ratio µˆ σˆ is lower than the limit value 1.9131 from condition (2.23) because
of random deviations of the random sample statistics, the ratio µˆ σˆ is set to this limit value, at
which the following parameter values result:
ẑ = 0 and according to formula (1.21)
σˆ N =
2
⋅ σˆ = 1,526 ⋅ σˆ
4 −π
(2.26)
The connection between the parameter values z and σN of the type II absolute value distribution
and the statistics µ and σ is shown graphically in Figure 7, related to σ.
Relative
relativerdistribution
Verteilungsparameter
parameter
7
6
5
z/σ
4
3
2
σN / σ
1
0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
5,0
5,5
6,0
relative
Lage µ / σ
Relative
location
Figure 7 - Relative parameter values of the type II absolute value distribution
in relationship to the relative location*)
*)
Illustrations, graphics, photographs and flow charts were adopted from the original German standard, and the numerical notation
may therefore differ from the English practice. A comma corresponds to a decimal point, and a period or a blank is used
as the thousands separator.
Page 15
VW 101 30: 2005-02
For the fitted type II absolute value distribution, the dispersion range limits can then be determined
numerically and their relationships with the relative location are shown in Figure 8
Relative
scatter range limit
relative Streubereichsgrenze
9
8
x99,865% / σ
7
6
5
4
3
2
x0,135% / σ
1
0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
5,0
5,5
6,0
relative
Lage µ / σ
Relative
location
Figure 8 - Relative dispersion range limits of the type II absolute value distribution in
relationship to the relative location*)
For direct determination of the statistics z and σN from the statistics µ and σ, for 1,9131 ≤ µˆ σˆ < 6
the following approximation can also be used as an inverse function of (2.22) with adequate
precision (error related to σ less than 0.02)
 µˆ 
 µˆ

ξ B 2   ≅ 2 ,1 ⋅  − 1,9131
 σˆ 
 σˆ

0 ,313
 µˆ

+ 0 ,466 ⋅  − 1,9131 
 σˆ

1,22
(2.27)
In addition, the determination of the dispersion range limits for 1,9131 ≤ µˆ σˆ < 6 is carried out
directly with the use of the following approximation (error related to σ less than 0.03)
0 ,62
0 ,9

µˆ
µˆ




x̂ 99 ,865% 
+ 2 ,075 ⋅  − 1,9131  + 5 ,5485
 − 1,64 ⋅  − 1,9131 
σˆ




 σˆ

 ≅ σˆ ⋅ 
0 ,9
  µˆ


 µˆ



x̂ 0 ,135% 
2 ,6 ⋅ exp  −  − 1,7031  ⋅ 0 ,8  + 1,425 ⋅  − 1,9131  − 2 ,1206


 σˆ

  σˆ


(2.28)
For µˆ σˆ ≥ 6 the calculation of the dispersion range limits is carried out according to formula
(2.11).
*)
Illustrations, graphics, photographs and flow charts were adopted from the original German standard, and the numerical notation
may therefore differ from the English practice. A comma corresponds to a decimal point, and a period or a blank is used
as the thousands separator.
Page 16
VW 101 30: 2005-02
3.2.2
Determination of capability for undefined distribution models
If no fitting distribution model can be assigned to a production characteristic or the measured
values from the random samples contradict the assumed distribution model, a distribution-free
estimate of the capability indexes is carried out according to the range method, in the following
modified form under consideration of the random sample size4) (see also example 3 in Section 5):
Capability indexes for characteristics with two-sided tolerance zones:
ĉm =
Go − Gu
x̂o − x̂u
G − x̂50% x̂50% − Gu 
ĉmk = min  o
;

x̂
−
x̂
x̂50% − x̂u 
50 %
 o
(2.29)
(2.30)
Capability indexes for characteristics with upper limited tolerance zone and natural lower limit zero:
ĉm =
Go
x̂o − x̂u
ĉmk =
Go − x̂50%
x̂o − x̂50%
(2.31)
(2.32)
Capability indexes for characteristics with lower limited tolerance zone only:
ĉmk =
x̂50% − Gu
x̂50% − x̂u
(2.33)
where
x̂o , x̂u : estimated values of the upper and lower dispersion range limits
x̂50% : estimated value of the 50% quantile
In the case of single values
x̂50% = ~
x
(2.34)
where
~
x : median value, the value that lies in the middle of an ordered sequence of measurements
The dispersion range limits are estimated with
x̂ o 
R
 = xc ± k ⋅
x̂ u 
2
where
4) In this case, it is simply a matter of a somewhat different representation of the same method of calculation as in the previous
Volkswagen operating equipment specification BV 1.40 for characteristics without normal distribution.
(2.35)
Page 17
VW 101 30: 2005-02
xc =
x max + x min
2
(2.36)
R = x max − x min : range
(2.37)
x max , x min : maximum and minimum measured value, respectively, of the effective total random
sample
By the correction factor
k=
6
dn
(2.38)
the effective random sample size ne is taken into consideration, where
d n : expected value of the w distribution5)
The value dn is given in Table 1 for a few random sample sizes ne
Table 1 - Expected value of the w distribution in relationship to ne
ne
dn
20
3,74
25
3,93
30
4,09
35
4,21
40
4,32
45
4,42
50
4,50
For random sample sizes that are greater than 20, the expected values of the w distribution can be
determined according to the following approximation formula:
d n ≅ 1,748 ⋅ (ln(n e ))
0 ,693
(2.39)
In the case of a frequency distribution of classified measured values
x̂o , x̂u are the upper limit, lower limit, respectively, for the top/bottom occupied class
and
x̂ 50% = x uk
n − Auk
n
+ 2
⋅ ∆x for Auk < ≤ Auk + a k
2
ak
(2.40)
where
x uk : lower limit of the kth class
∆x : class width
a k : absolute frequency of the measured values in the kth class
Auk : absolute cumulative frequency of the measured values to the lower limit of the kth class
5)Strictly speaking, a normally distributed population of the individual values is a prerequisite for the expected value of the w distribution.
However, this prerequisite is not taken into consideration due to the lack of a suitable method for the distribution-free calculation of
the capability indexes.
Page 18
VW 101 30: 2005-02
3.3
Limit values of machine capability
To obtain the machine capability for a considered characteristic, the capability indexes determined
must fulfill the following requirement with respect to the specified limit values c m ;Grenz and c mk ;Grenz :
characteristics with two-sided tolerance zones:
cˆm ≥ cm;limit and cˆmk ≥ cmk ;limit
(3.1)
characteristics with one-sided tolerance zone only6)
cˆmk ≥ cmk ;limit
(3.2)
with an effective random sample size of n e ≥ 50
Unless otherwise agreed, the following capability limit values apply:
cm;limit = 2,0
cmk ;limit = 1,67
In cases where, with reasonable effort, an examination can only be carried out with an effective
random sample size less than 50, the resulting uncertainty of the capability indexes determined
must be taken into consideration by applying correspondingly higher limit values as follows.
The determination of limit values for effective random sample sizes smaller than 50 is related here
to limit values that comply with the requirement from (3.1) or (3.2) for the population to be
investigated with 95% probability (lower confidence range limits). With the assumption of a
normally distributed population, these result from the upper confidence range limit of the standard
deviation
σ o = σˆ ⋅
49
χ 52%; 49
(3.3)
and the statistical percentage range for the production dispersion
x 99 ,865 % 
1 
49

⋅ σˆ
⋅
 = µˆ ± u 99 ,865% ⋅ 1 +
x 0 ,135 % 
2 ⋅ 50  χ 52%; 49

(3.4)
where
u 99 ,865 % = 3 ,0 : quantile of the standardized normal distribution
χ 52%; 49 = 33 ,9 : quantile of the chi-square distribution with a degree of freedom of f = 49 (see
also [1])
6) Although no limit value for Cm is specified for characteristics with upper limited tolerance zone only, a Cm value is to be determined
and specified, since from the comparison of this value to the Cmk value, it is possible to recognize whether an undesirable situation
shift exists. For example, if the Cmk value is higher than the Cm value, the distribution lies closer to the natural limit value zero than
to the upper limiting value.
Page 19
VW 101 30: 2005-02
By transformation and substitution in the evaluation formulas (2.1) and (2.2), this results in the
capability limit values for the population
cm ≥ cm;limit ⋅
cmk ≥ cmk ;limit ⋅
χ 52%; 49
49
= 0,832 ⋅ cm;limit
χ 52%; 49
1
⋅
= 0,824 ⋅ cmk ;limit
49
1+ 1
100
(3.5)
(3.6)
For effective random sample sizes ne < 50 , this results in the following fitted capability limit
values7)
cˆm ≥ cm;limit ⋅ 0,832 ⋅
f
χ
2
5%; f

f
1 
 ⋅
cˆmk ≥ cmk ;limit ⋅ 0,824 ⋅ 1 +
2
 2 ⋅ ne  χ 5%; f
(3.7)
(3.8)
with the degree of freedom
f = ne − 1
(3.9)
Example:
For specified capability limit values of cm;limit = 2,0 , cmk ;limit = 1,67 and an effective random sample
size of n e = 20 , according to the formulas (3.7) to (3.9) the following fitted limit values result for
characteristics with two-sided tolerance zones:
ĉ m ≥ 2 ,0 ⋅ 0 ,832 ⋅
20 − 1
= 2 ,28
10 ,1
1  20 − 1

ĉ mk ≥ 1,67 ⋅ 0 ,824 ⋅ 1 +
= 1,93
⋅
2 ⋅ 20 
10 ,1

7) The determination of the adapted capability limit values using formulas (4.7) to (4.9) is also used for populations without normal
distribution, since there are no other methods for these at this time and thus at least a usable consideration of a random sample size
that is less than 50 is carried out.
Page 20
VW 101 30: 2005-02
3.4
Statistical tests
Generally, the measurements of a machine capability investigation shall not exhibit any
─
unexpectedly great deviation of single measured values (outliers) in comparison to the
dispersion of other measured values,
─
significant change of the production location during the sampling and
─
significant deviation from the expected distribution model.
Otherwise additional systematic influences on production have to be taken into consideration. The
causes should then be known for this behavior and their effect should be acceptable in order to
fulfill the requirement of a secure manufacturing process.
Therefore, to test the above mentioned criteria the appropriate statistical tests shall be used during
a machine capability investigation. Since these tests are described in detail in standards and
standard statistics references, they will only be indicatedby the following references:
The following tests are to be carried out in the scope of a machine capability investigation:
─
Test for outliers using the distribution-free test according to Hampel in modified form (see
Volkswagen standard VW 10133)
─
Test for change of the production location using the distribution-free Run-Test according to
Swed-Eisenhard (see [1])
─
Test for deviation from the normal distribution according to Epps-Pulley (see ISO 5479)
─
Test for deviation from any specified distribution model using the chi-square test (see [1])
The statistical tests all proceed according to the following system:
─
Setting up the null hypothesis H0 and the alternative hypothesis H1, e.g.
H0: The population of the measured values of the characteristic considered has normal
distribution
H1: The population of the measured values of the characteristic considered does not have
normal distribution
─
Specification of the confidence level γ = 1- α or probability of error α
─
Setting up the formula for the test variable
─
Calculating the test value from the random sample values according to the test variable
formula
─
Determining the threshold value of the test distribution
─
Comparison of the test value to the threshold value for decision of whether a contradiction to
the null hypothesis exists and thus the alternative hypothesis is valid
It should be noted that in a statistical test with the specified confidence level γ possibly only a
contradiction to the null hypothesis can be proven, e.g. that a significant deviation of the measured
values from a normally distributed population exists. If no contradiction to the null hypothesis
results from the test result, this is not a confirmation of the validity of the null hypothesis. In this
case, it cannot be proven with the defined confidence level that a normally distributed population
exists, for example. Then, analogously to the legal principle “if in doubt, find for the defendant,” a
decision is simply made for assumption of the null hypothesis.
The probability of error α indicates the risk of rejecting the null hypothesis on the basis of the test
result, although it applies (α risk). However, not just any small value can be specified for the
probability of error, since that would increase, e.g., the risk of not discovering an actual deviation
from a normal distribution (β risk).
Page 21
VW 101 30: 2005-02
4
Carrying out a machine capability investigation
A machine capability investigation (MFI) is carried out according to the sequence shown in Figures
9 to 11.
Start
Start
4.1Prüfmittelanwendung
Test equipment use
4.1
4.2 Sampling
4.2 Stichprobenentnahme
Conditions for
Bedingungen
zur
MFI fulfilled
MFU
erfüllt
no
No
nein
yes
ja
4.3
regulationfür
for
4.3Special
Sonderregelung
restricted MFI
eingeschränkte
MFU
Data analysis
4.44.4
Datenauswertung
4.5 Documentation
Dokumentation
4.6Ergebnisbeurteilung
Results evaluation
4.6
yes
ja
Auswertungswie
Evaluation
repeat
derholung
4.7Machinenoptimierung
Machine optimization
4.7
yes
ja
no
Machinecapable
fähig
Machine
ja
yes
no
nein
Feasible
machbare
machine
Machinenoptimi
optimization
erung
no
nein
4.8
Handling ofnicht
incapable
4.8
Behandlung
fähiger
machines
Machinen
End
Ende
Figure 9 - Sequence of a machine capability investigation
Page 22
VW 101 30: 2005-02
4.4
Datenauswertung
4.4
Data analysis
Auswahl
zu
4.4.1 4.4.1
Selection
of thedes
expected
erwartenden
distribution
model
Verteilungsmodelles
4.4.2
4.4.2Test
Testauf
for Ausreißer
outliers
Outliers
Ausreißer
present
vorhanden
yes
ja
4.4.3
Ausreißer
derthe
4.4.3
Take
outliers aus
out of
Berechnung
der
statischen
calculation of statistics
Kennwerte nehmen
no
nein
4.4.4 Test for
auf change
Änderung
der
of the
Fertigungslage
production location
4.4.5Test
Test for
aufdeviation
Abweichung
4.4.5
fromvom
the
festgelegten
Verteilungsmodell
specified
distribution
model
Abweichung
vom
Deviation from
Verteilungsmodell
distribution model
yes
ja
4.4.8
Verteilungsfreie
Distribution-free
Auswertung
evaluation
no
nein
normal
distribution
Normalverteilung
yes
nein
no
4.4.7
Auswertung
nach
4.4.7
Evaluation
according
festgelegtem
Modell
to specified model
4.4.6
Auswertung
nach
4.4.6
Evaluation
according
Normalverteilung
to normal
distribution
Continued at
Fortsetzung
in 4
4
Figure 10 - Sequence of data analysis
Page 23
VW 101 30: 2005-02
4.6 Results evaluation
Outliers present
yes
yes
no
Outliers due to
incorrect
measurements
yes
Change of
production
location
no
yes
Deviation from
distribution
model
Other
distribution
model possible
no
no
Evaluation repeat
Continued at 4
yes
Capability indexes
less than limit values
yes
Cause known and
effect acceptable
no
no
Machine capable
Continued at 4
Machine incapable
Continued at 4
Figure 11 - Sequence of the results evaluation
no
Page 24
VW 101 30: 2005-02
4.1
Test equipment use
Test equipment shall only be used for the MFI that has been released by the responsible
department for the planned test process.
4.2
Sampling
An MFI relates to only one production characteristic or one machine parameter. Generally the
single measurements of the random sample are recorded for evaluation. In the case of manually
recorded measurements in a class subdivision of the value range (tally), the frequency distribution
of the classified measurements can also be recorded instead8).
So that essentially only the machine influence is recorded in an MFI, the following conditions shall
be complied with in the production of random sample parts:
─
A uniform blank batch and a uniform preparation (supplier, material) shall be ensured during
the investigation. During the MFI, the machine or system shall always be operated by the
same operator.
─
The premachining quality of the characteristics to be evaluated must correspond to the
required production specifications.
─
The number of parts to be produced (random sample size) should generally be 50. If this
random sample size is difficult to obtain for economic or technical reasons, a smaller one is
also permissible. Then the corresponding higher limit values according to Table 3 or the
formulas (3.7) and (3.8) have to be complied with. However, the random sample size (i.e.
without outliers) must be at least 20.
─
The parts shall be produced immediately after each other and numbered according to the
manufacturing sequence. All specified characteristics shall be tested on each part.
─
The MFI shall only be carried out with the machine at operating temperature. “Operating
temperature“ is to be defined for each use case.
─
The test parts are to be produced under the standard production conditions required for the
machine (i.e. with the cycle time and the machine adjustment parameters as in standard
production).
─
Depending on the project, special specifications shall be set so that at the beginning of the
MFI, it is ensured that, e.g. the tooling is broken in and that the end of the tooling lifetime
does not lie within the MFI.
─
Tooling change, manual tooling adjustments or other changes of machine parameters shall
not be carried out during the MFI. Automatic tooling corrections due to integrated measuring
controls are excepted from this.
─
If there are machine malfunctions during the MFI that influence the characteristic to be
investigated, the MFI must be started over again.
─
The measuring method must be specified before the investigation and agreed upon between
supplier and customer.
─
During production of different parts (different part numbers, e.g. steel shaft / cast iron shaft) on
one machine that can additionally have different characteristics, MFIs are to be carried out for
all these parts.
8) Since this form of manual data recording still occurs frequently in practice, it is taken into consideration in this standard, although the
possibilities of MFI are somewhat restricted by it.
Page 25
VW 101 30: 2005-02
4.3
Special regulation for restricted MFI
If the conditions named in 4.2 cannot be met completely, in justified cases a restricted MFI can be
carried out, for which special regulations are to be agreed upon between supplier and customer
and documented under the note „Restricted MFI.“
4.4
Data analysis
4.4.1
Selection of the expected distribution model
The expected distribution model depends on the characteristic type. For the most important types
of characteristics (see also VW 01056), the assigned distribution models are to be taken from
Table 2.
Table 2 - Assignment of characteristic types and distribution models
Characteristic type
Length dimensions
Diameter, radius
Straightness
Flatness
Roundness
Cylindricity
Profile of any line
Profile of any surface
Parallelism
Squareness
Slope (angularity)
Position
Coaxiality/concentricity
Symmetry
Radial runout
Axial runout
Roughness
Unbalance
Torque
Legend:
Distribution model
N
N
B1
B1
B1
B1
B1
B1
B1
B1
B1
B2
B2
B1
B2
B2
B1
B2
N
N: Normal distribution
B1: Type I absolute value distribution
B2: Type II absolute value distribution (Rayleigh distribution)
For characteristic types not listed, in most cases an assignment of a distribution can be carried out
according to the following rules:
─
a normal distribution for characteristics with two-sided tolerance zones or one-sided tolerance
zone only
─
and a type I or II absolute value distribution for characteristics with one-sided tolerance zone
only
Page 26
VW 101 30: 2005-02
4.4.2
Test for outliers
First a determination of whether the measured values recorded contain outliers is made using the
distribution-free outlier test according to VW 101 33. Outliers are measured values that lie so far
from the other measured values that it is highly probable that they do not come from the same
population as the remaining values, e.g. erroneous measurements. The outlier test shall be carried
out with a confidence level of 99%.
4.4.3
Take outliers out of the calculation of statistics
If outliers are identified, these are not considered in the calculation of the statistics. However, the
outliers shall not be deleted. Rather they shall be marked accordingly in the graphic representation
of the single value curve and their number shall be indicated in the documentation.
4.4.4
Test for change of the production location
Using the distribution-free Run Test according to Swed-Eisenhard (see [1]), a determination shall
be made of whether the production location has changed systematically during the sampling. A
systematic change of the production location can occur, e.g., due to the influence of temperature or
due to tooling wear (trend curve). This test shall be carried out with a confidence level of 95%.
If only the frequency distribution of classified measured values was recorded, this test cannot be
used.
4.4.5
Test for deviation from the specified distribution model
The recorded measured values are to be tested to see whether they exhibit a significant deviation
from the distribution model that was defined for the characteristic involved. To do this, in the case
of a specified normal distribution, the Epps-Pulley test (see ISO 5479) shall be used and in the
case of a different specified model, e.g. with a type I or II absolute value distribution, the chi-square
test (see [1]) shall be used with a confidence level of 95%. A deviation from the specified
distribution model can occur, e.g. due to different material batches during the sampling (mixed
distribution, see example 3 in Section 5).
A deviation from the specified distribution model can occur, e.g. due to sampling from different
tools (mixed distribution, see also Section 3, in Figure 5).
4.4.6
Evaluation according to normal distribution
In the case of a specified normal distribution or one that is approximated according to criteria
(1.13), (1.22), in which the measured values do not exhibit any significant deviation from the
distribution model, the calculation of the capability indexes is carried out according to formulas (2.1)
to (2.5), depending on the tolerancing, whereby the dispersion range limits are determined
according to (2.6).
4.4.7
Evaluation according to specified model
In the case of a different specified distribution model, e.g. type I or II absolute value distribution, in
which the measured values do not exhibit any significant deviation from the distribution model, the
calculation of the capability indexes are carried out according to formulas (2.1) to (2.5), whereby
the statistics of the distribution to be fitted are determined according to formulas (2.15) and (2.16)
or (2.24) and (2.25) respectively with the use of the approximated function (2.18) or (2.27)
respectively and the dispersion range limits can be calculated according to the approximated
functions (2.19) or (2.28) respectively.
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4.4.8
Distribution-free evaluation
If the statistical test results in a contradiction between the recorded measured values and the
specified distribution model, or if not fitting distribution model can be found for the considered
production characteristic, a distribution-free calculation of the capability indexes is carried out
according to formulas (2.29) to (2.40).
4.5
Documentation
The documentation of an MFI regarding a characteristic must contain the following information and
representations:
Header data:
─
Department, coordinator and preparation date
─
Information regarding the part
─
Designation, nominal dimension and tolerance of the characteristic
─
Machine data
─
Test equipment data
─
Production time period
Results:
─
Graphical representation of the single value curve with the random sample mean values with
limit lines of the tolerance zone (as long as single values were recorded)
─
Histogram with a fitted distribution model, limit lines of the tolerance zone and dispersion
range, as well as mean value and/or median value line
─
Representation in the probability grid with the fitted distribution model, limit lines of the
tolerance zone and dispersion range, as well as mean value and/or median value line (see [2])
─
Number of measured values
─
Number of measured values evaluated or outliers found
─
Estimated value of the production location
─
Estimated values of the dispersion range limits or the estimated value of the dispersion range
─
The distribution model used
─
The result of the test for change of the production location
─
The result of the test for deviation from the specified distribution model
─
Calculated capability indexes Cm and Cmk (to two digits after the decimal)
─
Required limit values for Cm and Cmk
References and notes:
─
If necessary, reference to restricted MFI
─
If necessary, special agreements between supplier and customer
─
If necessary, special events during the sampling
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4.6
Results evaluation
Whether a machine can be evaluated capable with respect to production of a considered
characteristic depends on the following result evaluation:
If outliers occur during an evaluation, their cause has to be clarified. Outliers shall only be caused
by incorrect measurements or be wrongly identified as such using the outlier test based on the
specified probability of error of 1%. Otherwise the machine shall be evaluated as incapable. If more
than 5% of the recorded measured values or more than 2 values are identified as outliers, there
shall be a test of whether the test process is faulty. Then the MFI is to be repeated if necessary.
If the production location has changed significantly during the sampling, generally its cause
has to be known and the effect has to be acceptable in order to meet the requirement for machine
capability (see last paragraph of this section for exception).
If a distribution-free evaluation exists due to a significant deviation from the specified distribution
model and if no other distribution model can be assigned to the considered characteristic without
contradiction, the cause must be known and the effect must be acceptable9), in order to meet the
requirement for machine capability (see last paragraph of this section for exception).
Unless otherwise agreed, the capability indexes determined with an effective random sample size
of n e ≥ 50 (i.e. without outliers) must meet the requirement
ĉ m ≥ 2 ,0 and ĉ mk ≥ 1,67 for a characteristic with two-sided tolerance zones
ĉ mk ≥ 1,67
for a characteristic with one-sided tolerance zone only
in order for the machine to be evaluated as capable. In this case, for comparison to the limit values,
the capability indexes determined are to be rounded to two digits after the decimal so that, e.g., a
determined value of ĉmk = 1.66545 , will still meet the requirement after the resulting rounding to
1,67.
With an effective random sample size of 20 ≤ n e < 50 , correspondingly higher limit values shall be
complied with. The fitted limit values are given in Table 3 for a few random sample sizes. If there
is agreement on other limit values on the basis of n e ≥ 50 , the corresponding fitted limit values are
to be determined according to formulas (3.7) to (3.9).
Table 3 - Limit values for machine capability for 20 ≤ ne ≤ 50
ne
ĉ m ≥
ĉ mk ≥
20
25
30
35
40
45
50
2,28
2,19
2,13
2,08
2,05
2,02
2,00
1,93
1,85
1,79
1,75
1,72
1,69
1,67
If a capability index results that is lower than the corresponding limit value, the machine is to be
evaluated as incapable.
9) In the case of a very small tolerance (T < 5 µm) there is frequently a significant deviation from the specified distribution model
because of a restricted measured value resolution
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With a determined capability index of c mk ≥ 2 ,33 (corresponds to 14 σ) and additionally
c m ≥ 2 ,67 (corresponds to 16 σ) for characteristics with two-sided tolerance zones, the machine
can be evaluated as capable with respect to the considered characteristic, even independent of a
significant location change or deviation from the specified distribution model.
4.7
Machine optimization
For the case where the machine capability cannot be documented with respect to the tested
characteristic, measures for machine optimization are necessary. To do this, the corresponding
influences are to be identified (e.g. using DOE statistical test methodology) and eliminated.
4.8
Handling incapable machines
If it is not possible to achieve machine capability with machine optimizations that are economically
reasonable, there shall first be an investigation using statistical tolerance calculation according
to VW 01057 of whether a tolerance extension is possible to achieve the machine capability. If
machine capability cannot be achieved using this measures, a decision is to be made whether the
machine will be accepted according to special regulations agreed upon in writing or not. These
special regulations shall contain the following points:
─
Reasons for the acceptance
─
Risk and cost considerations
─
If necessary, restricting production and additional test conditions
─
Specification of the responsibility
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5
Examples
Example 1:
Shaft diameter with a nominal dimension of 20 mm, a lower limiting value of Gu = 19,7 mm and an
upper limiting value of Go = 20,3 mm
Using the statistical tests, no outliers resulted, no significant change of the production location
occurred and there was no significant deviation from an expected normal distribution from n = 50
measured values of the random sample. The following random sample statistics are determined:
µˆ = x = 20 ,05 and σˆ = s = 0 ,05
Therefore, according to formula (2.11), the following estimated values of the dispersion range limits
result for the normally distributed populations:
x̂ 0 ,135 % = µˆ − 3 ⋅ σˆ = (20 ,05 − 3 ⋅ 0 ,05 )mm = 19 ,9 mm and
x̂ 99 ,865% = µˆ + 3 ⋅ σˆ = (20 ,05 + 3 ⋅ 0 ,05 )mm = 20 ,2 mm
and from that, ultimately, the following capability indexes result:
ĉ m =
G o − Gu
20 ,3 − 19 ,7
=
= 2 ,0 and
x̂ 99 ,865 % − x̂ 0 ,135% 20 ,2 − 19 ,9
 Go − µˆ
µˆ − Gu  20 ,3 − 20 ,05
ĉ mk = min 
;
= 1,67
=
 x̂ 99 ,865 % − µˆ µˆ − x̂ 0 ,135%  20 ,2 − 20 ,05
Because of the capability indexes, it is thus verified that the machine just meets the capability
requirements with respect to the considered shaft diameters.
Figure 12 shows the evaluation result
Gu
µ̂
x̂ 99 ,865%
Go
Frequency
Häufigkeit
x̂ 0 ,135%
19,70
19,75 19,80
19,85
19,90
19,95
20,00
20,05
20,10
20,15
20,20
20,25
20,30
Messwert value
Measured
Figure 12 - Example of a production with the model of a normal distribution and the
capability indexes cm = 2,0 and cmk = 1,67
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Example 2:
Hole with a maximum permissible position deviation of Go = 0,2 mm.
From the n = 50 measurements of the random sample, the statistical tests did not result in any
outliers, no significant location change and no significant deviation from an expected type II
absolute value distribution. The following random sample statistics were determined:
µˆ = x = 0 ,038 mm and σˆ = s = 0 ,02 mm
The random sample statistics result in the ratio
µˆ 0 ,038
=
= 1,9
σˆ
0 ,02
Since, because of the random dispersion of the random sample statistics, this value is lower than
the limit value 1,9131 according to condition (2.23), the ratio is set to this limit value, which in turn
results in an eccentricity of z = 0.
In this way, the second parameter value of the type II absolute value distribution to be fitted is
calculated according to special case (2.26) as follows
σ N = 1,526 ⋅ σˆ = 1,526 ⋅ 0 ,02 mm = 0 ,0305 mm
The estimated values of the dispersion range limits result from formula (2.27):
x̂ 99 ,865% = 5 ,5485 ⋅ σˆ = 5 ,5485 ⋅ 0 ,02 mm = 0 ,111mm
x̂ 0 ,135 % = 0 ,0773 ⋅ σˆ = 0 ,0773 ⋅ 0 ,02 mm = 0 ,0016 mm
Finally, according to formulas (2.3) and (2.4), the following capability indexes result:
ĉ m =
ĉ mk =
Go
0 ,2
=
= 1,83
x̂ 99 ,865 % − x̂ 0 ,135% 0 ,111 − 0 ,0016
Go − µˆ
0 ,2 − 0 ,038
=
= 2 ,22
x̂ 99 ,865% − µˆ 0 ,111 − 0 ,038
Figure 13 illustrates the evaluation result
Go
µ̂
x̂ 99 ,865 %
Häufigkeit
Frequency
x̂ 0 ,135%
0,00
0,02
0,04
0,06
0,08
0,10
0,12
0,14
0,16
0,18
0,20
Messwert value
Measured
Figure 13 - Example of a production with the model of a type II absolute value distribution
and the capability indexes ĉ m = 1,83 and ĉ mk = 2 ,22
Because of the determined statistic Cmk, it is verified that the machine meets the requirement well
with respect to the position deviation of a hole. Although no limit value is specified for statistic Cm in
the case with an upper limited tolerance zone, comparison to the Cmk value provides information on
the production location and the smaller Cm value indicates that the location lies closer to the natural
limit zero than to the upper limiting value.
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Example 3:
Shaft diameter with a nominal dimension of 20 mm, a lower limiting value of Gu = 19,7 mm and an
upper limiting value of Go = 20,3 mm From n = 50 measured values of the random sample, due to
the statistical tests, no outliers resulted and no significant location change resulted, but a significant
deviation from an expected normal distribution did. Therefore, a distribution-free evaluation is
carried out according to Section 3.2.2. To do so, the following random sample statistics were
determined:
x̂ 50% = ~
x = 20 ,02 mm , x max = 20 ,19 mm and x min = 19 ,85 mm
Correction factor according to formula (2.38) and Table 1: k =
6
6
=
= 1,33
d n 4 ,5
Range according to formula (2.37): R = x max − x min = (20 ,19 − 19 ,85 )mm = 0 ,34 mm
According to formula (2.36): x c =
x max + x min
20 ,19 + 19 ,85
=
mm = 20 ,02 mm
2
2
Estimated values for dispersion range limits according to formula (2.35):
x̂ o 
20 ,246
R 
0 ,34 
mm
mm = 
 = x c ± k =  20 ,02 ± 1,33 ⋅
x̂ u 
2 
2 
19 ,794
Thus the formulas (2.29) and (2.30) resulted in the capability indexes:
ĉ m =
Go − Gu
20 ,3 − 19,7
=
= 1,33
x̂ o − x̂ u
20 ,246 − 19 ,794
G − x̂ 50% x̂ 50% − G u 
20 ,3 − 20 ,02
ĉ mk = min  o
;
= 1,24
=
 x̂ o − x̂ 50% x̂ 50% − x̂ u  20 ,246 − 20 ,02
Figure 14 illustrates the evaluation result.
Gu
x̂ 50%
x̂ o
Go
Häufigkeit
Frequency
x̂ u
19,7
19,75
19,8
19,85
19,90
19,95
20,00
20,05
20,10
20,15
20,20
20,25
20,30
Messwert value
Measured
Example 14 - Example of a production without defined distribution model with the capability
indexes cm = 1,33 and cmk = 1,24
From the capability indexes determined, it can be seen that the machine does not meet the
capability requirement with respect to the considered characteristic. The significant deviation from
an expected normal distribution supplies interesting information in this context. Because of this, an
optimization potential can be recognized, in this case through mixed distribution.
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VW 101 30: 2005-02
6
Referenced standards10)
VW 010 56
Drawings; Form and Position Tolerances
VW 010 57
Statistical Tolerance Calculation of Dimension Chains
VW 101 33
Test auf Ausreißer (Test for Outliers – currently only available in German)
DIN 55319
Quality Capability Statistics (QCS)
ISO 5479
Statistical Interpretation of Data - Tests for Departure from the Normal Distribution
7
Referenced literature
[1] Graf, Henning, Stange, Willrich, Formeln und Tabellen der angewandten mathematischen
Statistik, Springer publishing group, third edition 1987
[2] Kühlmeyer M., Statistische Auswertungsmethoden für Ingenieure, Springer publishing group,
2001
10) In this Section, terminological inconsistencies may occur as the original titles are used.
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VW 101 30: 2005-02
8
Keyword index
Key word
A
Absolute frequency a
Absolute cumulative frequency A
α-Risk
Alternative hypothesis
Fitted capability limit values
Outlier
Confidence level γ
B
Conditions for the machine capability
investigation
β-Risk
Absolute value distribution, type I
Absolute value distribution, type II
Machine at operating temperature
C
Capability
Chi-square distribution
D
Data analysis
Density function f(x)
Documentation
E
Restricted machine capability investigation
Effective random sample size ne
Epps-Pulley test
Results evaluation
Expected value for the w-distribution dn
Eccentricity z
F
Determination of capability
Capability indexes cm and cmk
Capability limit values
Production location
Production variation
Production sequence
Degree of freedom
G
Limit values of machine capability
H
Hampel test
Frequency distribution
Upper limiting value Go
I
Probability of error α
Page
10, 17
17
20
20
19, 28
20, 26
20, 26
24
20
5, 11
7, 13
24
3
18
25
4
27
25
10, 24
20, 26
28
17
7
8
3, 9
18, 28
3, 28
3, 8
24
18
18, 28
20
10, 17
3, 9
20
Page 35
VW 101 30: 2005-02
Key word
K
Class width ∆x
Classified measured values
Correction factor k
L
Location
M
Machine optimization
Machine malfunctions
Median value
Characteristic type
Characteristic value
Measuring method
Lower limiting value Gu
Mixed distribution
Mean value µ
N
normal distribution
Null hypothesis
Zero offset
P
Parameter of a distribution
Test variable
Test value
Test equipment use
Q
Quantile
- of the standardized normal distribution
- of the chi-square distribution
R
Rayleigh distribution
Radial deviation r
Blank batch
Rounding of capability values
Run test
S
Estimation / estimated value
Threshold value
Standard production conditions
Significant change / deviations
Range R
Standard deviation σ
Standardized normal distribution
- U Transformation
- Distribution function Φ(ug)
- Probability density function φ(u)
Statistical tests
Statistical tolerance calculation
Page
17
10, 17
17
3
29
24
16
4, 25
3
24
3, 9
26, 32
4, 10
4, 10
20
6
3
20
20
24
9
18
18
7
7
24
28
20
8, 9
20
24
20
17
4, 10
5
5
5
5
20
29
Page 36
VW 101 30: 2005-02
Key word
Statistical percentage range
Sampling
Random sample range
Dispersion range limits
Swed-Eisenhard test
T
Test
- for outliers
- for specified distribution model
- for change of production location
Tolerance expansion
Tolerance zone
Toleranced characteristic
- on one side, upward
- on one side, downward
- on two sides
Trend curve
V
Variance σ2
Distribution
Distribution-free estimate
Distribution function F(x)
Distribution model
Confidence range limit
Premachining quality
W
Probability p
Probability density function f(x)
Probability grid
Weibull distribution
Tooling change / adjustment
w-Distribution
Z
Random influences
Page
18
24
24
3, 10
20
20, 26
20, 26
20, 26
29
3, 8
4
9, 16
9, 16
9, 16
26
4
4
16, 27
5
4
18
24
5
4
27
8
24
17
3