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5. Statistical TolerancingQuality Management in the Bosch Group | Technical Statistics
Edition 08.1993
1993 Robert Bosch GmbH
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Statistical Tolerancing
Table of Contents:
1. Introduction ................................................................................. 4
2. Terms........................................................................................... 5
3. Arithmetic Tolerancing ................................................................ 7
4. Theoretical Basis for Statistical Tolerancing...............................104.1 Central Limit Theorem of Statistics ............................................104.2 Law of Error Propagation............................................................154.3 Distribution of Characteristics and Tolerance .............................174.3.1 Variance of a Rectangular Distribution .......................................184.3.2 Variance of a Triangular Distribution .........................................19
5. Statistical Tolerancing ................................................................205.1 The Reduction Factor ..................................................................245.2 Calculation of the Statistical Assembly Tolerance ......................275.3 Calculation of Individual Tolerances
for a Given Assembly Tolerance ..............................................275.4 The General Case ........................................................................285.5 Process Capability Indices ..........................................................29
6. Prerequisites for Statistical Tolerancing......................................31
7. Statistical Control of the Individual Dimensions .........................33
8. Generalisation for arbitrary Dimension Chains ...........................35
9. Bibliography ...............................................................................36
10. Appendix ....................................................................................38
Index ...........................................................................................48
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1. Introduction
Statistical tolerancing is a method to determine tolerances on the basis ofstatistical principles.The purpose is to determine a tolerance for the assembly dimension, which is ageometric combination of several individual dimensions.
Statistical tolerancing is based on the idea that random positive and negativedeviations, present in actual dimensions of individual parts, from theirrespective nominal dimensions will neutralize each other as a rule, when theseindividual parts are stacked in an assembly. Thereby the most unfavorablecase that can be thought of, for example exclusive combination of maximumpossible positive dimensional deviations (arithmetic tolerancing) is veryimprobable.
When the tolerance for an assembly dimension is given, then, under theaforesaid consideration, greater tolerances result for the individual dimensionsthan those resulting from mere arithmetical calculation.
The resultant advantages with respect to the production technique andproduction costs have been explained in detail in many papers andpublications on this subject and have frequently been highlighted already inthe titles, as the following examples show (faithful translations of the Germantitles):
Realising advantages in production by applying statistical principles totolerance calculation [1]Coupling of tolerances in theory and practice: A possibility to reduceproduction costs [4]Economic tolerancing [16]Determining optimum tolerances on the basis of dimension chain theory andthe theoretical properties of the production process [17]
The dates of publication of these and other publications on StatisticalTolerancing (cp. Bibliography) show that the fundamentals of this methodwere indeed known long back. Its development can be traced back to thetwenties of this century (cp. [9]).
Considering this fact as well as the advantages of Statistical Tolerancing,pointed out in the literature, it is surprising that this method, which is alsodenoted as probability-theoretical method for calculation of dimensionchains, is not consistently taken into consideration.
New activities in the sphere of Statistical Tolerancing attribute this to theexpensive calculation procedures to be applied for general cases, which havebecome economically possible only of late with the developments in thesphere of computer hardware and software.
Within the scope of this booklet a short summary will be provided on thetheoretical background of Statistical Tolerancing, the constraints listed out andthe problems of practical application presented.
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2. Terms
The assembly of individual parts within a technical system (an assembly) leadsoften to the situation, that independent dimensions of individual parts form aninterrelated chain of linear dimensions, viewed purely from geometric aspects,which in its totality becomes the so-called assembly dimension.It is called, suggestively, a dimension chain, consisting of many individualdimensions.
The dimension chain is called linear, if all the individual dimensions can bepresented through arrows, parallel to each other, which form a closed chain oflines. The direction of the arrows denotes the additive or subtractive (positiveor negative) contributions of the individual dimensions.
A dimension is called positive (negative), if, with the change of this dimensionand all other dimensions of the chain remaining constant, the assemblydimension changes in the same (opposite) direction.
Fig. 2.1: Tolerance chain graphDimension chain consisting of individual nominal dimensions NiTop: only positive contributory directionBottom: positive and negative contributory direction
Apart from this there are plane (two-dimensional) and spatial (three-dimensional) dimension chains. Plane (two-dimensional) dimension chainsoften result from interaction of rotating parts or movement of rocker arms orcams, where angles and radii play a part (cp. for example [25]).
It is clear, that in such nonlinear dimension chains complicated correlations ofindividual dimensions can occur, which can however be presented in most ofthe cases through analytical functions (but occasionally can only bedetermined by recursive calculation).
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A simple example of a (two-element) nonlinear dimension chain is that of twoholes, whose position on two axes perpendicular to each other is given by therespective distance from a reference point (intersection point of the axes).
Fig. 2.2: Example of a two-element nonlinear dimension chain
The relative distance of the two holes is then defined by
z x y= +2 2 . (2.1)
Tolerances are determined, according to DIN 7182 [11] (cp. also [12]-[15]) byindicating a maximum dimension GU and a minimum dimension GL :T G GU L= − .
The midpoint of the tolerance interval is
C G GU L= +2
. (2.2)
If the nominal dimension of a characteristic does not coincide with thismidpoint, then the tolerance is specified by way of a minimum and maximumallowance with reference to the nominal dimension.
Hint (not contained in the German edition): The notations used in this subjectfield are not standardised and not uniform. Therefore it is sometimes not quiteclear what the assembly criterion of interest is. As can be seen from Fig. 2.1Ns can denote the nominal assembly dimension or an assembly tolerance gap(clearance). It should be clear from the context, however, what is meant.
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3. Arithmetic Tolerancing
Analogous to the linking of individual dimensions to form an assemblydimension, individual tolerances are linked to form an overall assemblytolerance.Assembly tolerances are often determined on the basis of theoreticalconsiderations or on the basis of experience, taking into account theirsignificance for:
- function- reliability- lifetime- optical quality (appearance)- compliance with legal and official regulations.
As result of a design calculation, tolerances for individual dimensions are thendetermined. To the contrary, if individual tolerances are given, then theassembly dimension can be calculated by inversion of this concept.
The actual dimensions of parts must, as per DIN 7182, be within the tolerancezone bounded by the minimum dimension and the maximum dimension. If themost unfavorable case is assumed, that all actual dimensions of the individualparts of an assembly have the maximum permissible deviation from thenominal dimension, i.e., are lying at the limit of the tolerance zone, allindividual tolerances Ti must be added arithmetically to obtain the assemblytolerance, independent of the respective contributory direction of theindividual dimensions (cp. Fig. 2.1).
T T T T Ta k ii
k
= + + + ==1 2
1
... . (3.1)
This instance is called as Arithmetic Tolerancing or Worst CaseTolerancing. Other names for this procedure are: Maximum-MinimumMethod or Method of Absolute Interchangeability [15].
Remark 1:
Equation 3.1 is valid only when all the tolerance zones of the individualdimensions are symmetric to the respective individual nominal dimensions,i.e., nominal dimension and midpoint of the tolerance interval are the same.Otherwise, the lower and upper allowances are to be given consideration inaccordance with the contributory direction of the dimensions (cp. Fig. 2.1).The tolerance zone of the assembly dimension is then, in general, alsoasymmetric to the nominal assembly dimension.
Remark 2:
The term absolute interchangeability of a part denotes its property that,without any limitation to the aforementioned design criteria (function,reliability, ...) of the assembly, the part must be able to be replaced by anyother part from the same production batch.
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Example for arithmetic tolerancing:
Given are five individual parts with dimensions and tolerances. Required arethe assembly dimension and its tolerance after assembly (cp. [12]).
Fig. 3.1: Tolerance stack example with nominal dimensions and tolerances
Fig. 3.2: Tolerance chain graphSchematic representation of the nominal dimensionsconsidering the contributory direction
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Calculation of the arithmetical assembly tolerance with the help of thecalculation sheet.It is to be noted that in columns 5 and 6 prefixes and indices of the allowancechange when the nominal dimension is negative (according to the contributorydirection).
For the nominal dimension -23.8 the AL = -0.02 becomes AU' = +0.02.
1 2 3 4 5 6 7 8
Arithmetic addition of the tolerances
Dimension NAllowancesaccording to
drawing
Allowancesaccord. to
contrib. direct.Ti C
AL AU AL' AU'
A +44.8 -0.02 +0.02 -0.02 +0.02 0.04 +44.8
B -23.8 -0.02 +0.02 0.02 -23.79
C -3.5 -0.01 +0.01 -0.01 +0.01 0.02 -3.5
D -8.7 +0.02 -0.02 0.02 -8.71
E -8.7 +0.02 -0.02 0.02 -8.71
+ +44.8
- -44.7
+0.1 -0.07 +0.05 0.12 +0.09
s AsL AsU Ta Cs
As assembly dimension we find 01 0 070 05. .
.−+ or 0 09 0 06. .± .
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4. Theoretical Basis for Statistical Tolerancing
The calculation procedures of Statistical Tolerancing are based essentiallyupon three basic principles which will be presented in the following sections:
• the central limit theorem of statistics• the law of error propagation• the distribution of characteristic values within a tolerance zone
(distribution of characteristics and tolerance)
4.1 Central Limit Theorem of Statistics
When a quality control chart from a process, which is controlled by SPC, isobserved it becomes immediately evident, that parts of a mass production arenever totally identical and the individual values of variables always deviate,more or less, from the desired target value (nominal value).
This becomes easily understandable, when it is explained with an example, saya lathe, as to what lot of errors and causes for errors can have an impact on theprocess result (cp. [5] and [9]):
- combination of measuring and drive system- deviation in scale- bearing of scale- deformation due to insufficient rigidity of machine- distortion of workpiece- elastic deformation of workpiece and cutting tool- wear of bearing- inherent vibrations (resonances) of machine- variations in speed and feed rate.
These are only a few examples of influence factors which can be assigned tothe machine. Apart from this, other external influences are also responsible forthe process result (characteristic values of manufactured parts), which can besummarised, together with the aforementioned, into 5 groups: Man, Machine,Material, Method, Environment.
Allocation of an influencing factor to one of these groups is, however, notprecise in every case.
If we once set aside processes with systematic changes of the process average,such as trend and between-batch variation, a process analysis often shows adistribution of characteristic values, which can be described at leastapproximately by the Gaussian normal distribution.
This is a consequence of the central limit theorem of statistics. Expressedsomewhat in a free language, it means: due to random interaction of manyinfluence quantities (addition of random variables), the target quantityobtained is approximately normally distributed.
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With this abstraction, which includes consideration of a characteristic of a partas a random variable, the phenomenon of deviation experienced in practice inproduction, viz., actual dimension differing from the desired nominaldimension, can be described mathematically. The fact defined by the centrallimit theorem of statistics may be explained by a simple example.
The result of a throw with a regular dice is a random number x , which canassume the values 1, 2, 3, 4, 5 and 6. The probability that a 6 is thrown is1/6. The same applies to all other values. The respective probability functionf x( ) can thus be represented as follows:
Fig. 4.1: Probability function of a regular dice. Pure mathematical calculationresults in the value of 3.5 as the average of many throws.
If the total score from a throw of two dice is the considered random variable,the representation of the probability function of this random variable shows atriangular form (Fig. 4.2).
Fig. 4.2: Probability function of the random variable:Total score from a throw of two dice
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The least possible result is 2, the maximum possible result is 12. Theprobability of getting a total score of 7 is obtained by dividing the number ofall combinations yielding a total score of 7 by the number of all possiblecombinations (36).
There are 6 possibilities to get the total score 7 and they are: (1,6), (2,5), (3,4),(4,3), (5,2), (6,1), and in all 36 possible results from the random processconsidered here:
(1,1), (1,2), ..., (1,6), (2,1), (2,2), ..., (2,6), ..., (6,6).
Thus it is: f ( )7 636
16
= = .
The random variable total score as results from a throw with 4 dice has aprobability function, whose representation is already very similar to the curvedepicting the density function of the normal distribution (continuous line).
Fig. 4.3: Probability function of the random variable:total score as results from a throw with 4 dice
The mean of this random variable is 4 35 14⋅ =. . Thus we obtain, as randomresult, values which are often very close to this mean and only occasionallybig deviations from the mean, for example, the value 24, which corresponds tothe throw result (6, 6, 6, 6).
Said otherwise, to arrive at this result each of the four random variables (dice)must assume a value (6) which differs very much from the mean 3.5.
The probability for this result is very small: 16
14
< % .
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This principle is of basic importance for the theory of Statistical Tolerancing.
Statistical Tolerancing makes use of the fact, that in a random combination ofa sufficiently big number of individual parts (elements) positive and negativedimensional deviations neutralize each other. The result from the exampleshown above can be applied suitably to the properties of a dimension chain.
In order that the assembly dimension of k parts assumes a value whichdeviates very much from the nominal assembly dimension, actual dimensionsof all k individual parts must simultaneously be at their respective tolerancelimit. The probability for this situation is however very small for a sufficientlybig number of k . Assuming a random selection of parts in assembly, thedistribution of the random variable assembly dimension is approximately anormal distribution.
Figure 4.4 should make this point clear.
Fig. 4.4: Schematic representation of an assembly consisting of 4 parts whoseindividual dimensions are subject to a rectangular distribution. In all66 assemblies are arranged in sequence. The assembly dimension orthe clearance which is referred to the top edge (broken line) isapproximately normally distributed.
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The figure shows schematically the interaction of four individual dimensions,having identical tolerance and following rectangular distribution, for anexample of 66 assemblies in all, which are arranged here in sequence. It isseen, that the clearance N s (broken line) drawn at the top edge shows bigdeviations from the average value only seldom.
The range µ σ± ⋅3 of the normal distribution, approximately resulting for theassembly dimension, is obviously somewhat smaller than the arithmeticallycalculated tolerance, which is shown by the width of big rectangle.
The randomness of combination of parts is only one of the many conditions(see Section 6), which must be satisfied, so that this feature of statistics ismade use of with advantage for determining tolerances.
Remark:
Based on the dice example, it was shown as to how the probability function ofa sum of random variables is determined, which can only assume values ofwhole numbers (1,...,6). This mathematical operation is also called asconvolution.
The convolution thus denotes the process of linking the probabilityfunctions of random variables. This can be applied also to continuousvariables, i.e., such variables which can assume any real value.
Linear dimensions of parts can be considered as continuous variables, becausethey can assume any interim value on a scale.
In the case of measuring instruments with analogue display, such as slidecaliper rule, micrometer screw or indicating precision gauge, this situation isobvious, in contrast to those with digital display. Strictly speaking, a lineardimension can always assume discrete values only, whose minimum distanceis limited by the resolution of the measuring instrument or the measurementprocedure.
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4.2 Law of Error Propagation
The Gaussian law of error propagation describes how the measurement errorsof several independent measurands xi affect a target quantity z , which iscalculated in accordance with a functional correlation
( )z f x x xk= 1 2, , ... , : (4.1)
szx
szx
szx
sk
k2
1
2
12
2
2
22
2
2≈
⋅ +
⋅ + +
⋅
∂∂
∂∂
∂∂
... . (4.2)
z is, then, generally only an indirectly measurable quantity. For example, thearea of a rectangle is determined by measuring the length of sides andmultiplying the results of measurement with each other.The accuracy with which z can be indicated depends on the accuracy of themeasurands xi . In general, the average xi and the standard deviation si of asequence of measurements are given for each respective xi .
In the above expression ∂∂zxi
denotes the partial derivatives of the function z
with respect to the variables xi . They are to be calculated at xi .
The derivation of the formula for sz , includes the expansion of z in a Taylorsseries and overlooks the terms of higher order.
Because of this reason, instead of the equals sign the wave symbol is used inthe formula, indicating that the two terms are approximately equal.
The application of the law of error propagation is understood easily with anexample.
We consider two Ohmic resistors, on which a series of measurements werecarried out. As a result of this measurement series, the mean and the standarddeviation are given in each case, the latter being a measure for the averageerror:
R 1 47 0 8= ± . Ω R 2 68 11= ± . Ω .
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In the event of parallel connection, the total resistance R is calculated asfollows:
RR RR R
=⋅+
1 2
1 2
R =⋅+
=47 6847 68
27 8Ω
Ω. .
To calculate the corresponding average error, first it is necessary to determinethe partial derivatives of R with respect to R1 and R2 :
( ) ( )∂∂RR
R
R R1
22
1 2
2
2
268
47 680 35=
+=
+= .
( ) ( )∂
∂RR
R
R R2
12
1 2
2
2
247
47 680167=
+=
+= . .
The average error of the total resistance is obtained by substituting theseexpressions in the law of error propagation:
sR2 2 2 20 35 0 8 0167 11 011= ⋅ + ⋅ =. . . . .
=sR 0 33. .
Accordingly the result is:
R = ±27 8 0 3. . Ω .
The law of error propagation becomes simple to present, when the function f ,which defines the link of the independent variables xi is a sum.
Then the partial derivatives are all equal to one and we find:
s s s sz k2
12
22 2= + + +... . (4.3)
Accordingly, for the variances of the respective underlying (parent)populations the following is valid:
σ σ σ σz k2
12
22 2= + + +... . (4.4)
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This means, the variance of an indirect measurand, which is determined byaddition of independent individual measurands, is equal to the sum of thevariances of these individual measurands.
Hint:The aforementioned equation is identical with a general relationship existingbetween the variances of several independent random variables. It can bederived, also totally independent of the law of error propagation, on the basisof statistical computation rules (see, for example [10] p. 155).
Example:If the two resistors are connected in series, we find:
( ) ( )s s sR R R2 2 2 2 2
1 20 8 11= + = +. .Ω Ω ⇔ =sR 136. Ω
thus R = ±115 14. Ω .
4.3 Distribution of Characteristics and Tolerance
As per DIN 7182, actual dimensions of parts must be within the respectivetolerance ranges. However, there is no stipulation for the distribution of thecharacteristic values in a tolerance zone.
Based on the assumption of the most unfavourable case, that all values of thecharacteristic lie at the limit of the tolerance range, the need for arithmetictolerancing was derived in Section 3.
Instead of this, in the scope of the calculation procedures of StatisticalTolerancing it is assumed that the values of a characteristic are subject to acertain distribution within the tolerance range, bounded by the limiting values.On the other hand, a distribution is characterised, in general, by the parametersmean and standard deviation.
Therefore, it is necessary to consider in detail the relationship between thetolerance of a characteristic and the standard deviation of specific distri-butions.
We confine ourselves, in the following sections, to discussions on the normal,triangular and rectangular distribution. Apart from them, the literature alsodeals with a trapezoidal distribution having different slopes of flanks. (cp. [6],p. 153).
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4.3.1 Variance of a Rectangular Distribution
As indicated already in Section 4, we consider in the following the actualdimension of a parameter of a part merely as random variable X . Theprobability, that the variable X assumes a value x in the range between x andx d x+ , is equal to the product f x d x( ) ⋅ . f x( ) is the probability densityfunction.
The probability density function for an individual dimension, whose values aredistributed uniformly over the entire tolerance range bounded by GL and GU
( T G GU L= − ), has the value f xT
( ) = 1 within this range and is equal to zero
for all x outside. For the mean µ is valid: µ = +G GL U
2(4.5)
Fig. 4.5: Probability density function f x( ) of a rectangular distribution
The graphical representation of this function explains the term rectangulardistribution. The area of the rectangle corresponds to the probability, that Xassumes any value x between GL and GU . It has the value one. The variance
σ 2 of a continuous distribution is defined by
( ) ( )σ µ2 2= − ⋅ ⋅
−∞
+∞
x f x dx . (4.6)
As in the case considered here, f x( ) is equal to zero outside the interval [GL ,GU ], it is enough to carry out the integration over this interval, for calculatingthe variance of the rectangular distribution:
σ 22 2
21
12= − +
⋅ ⋅ = x G G
Tdx TL U
G
G
L
U
. (4.7)
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4.3.2 Variance of a Triangular Distribution
The density function of the triangular distribution in the interval [GL , µ ] isgiven by:
( ) ( )f xT
x GL= ⋅ −42 . (4.8)
The area of the triangle corresponds to the probability, that X assumes anyvalue x between GL and GU . Its value is equal to one. As the base of thistriangle corresponds to the width of the rectangle (cp. rectangular distri-bution), f x( ) must have the value 2/T at the position
µ = +G GL U
2. (4.9)
Fig. 4.6: Probability density function of a triangular distribution
Because of the symmetry of the density function, it is enough to determine theintegral in the limits GL and µ and to multiply by two, for calculating thevariance of the triangular distribution.
( ) ( )σ µµ
2 222 4= ⋅ − ⋅ ⋅ − ⋅ x
Tx G dxL
G L
(4.10)
The result is σ 22
24=T
. (4.11)
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5. Statistical Tolerancing
In the subsections of Section 4, a few theoretical relationships are explained,which form the basis of Statistical Tolerancing. Consequently, fulfillment ofthe conditions associated with them is an important aspect, which is decisivefor the practical applicability of Statistical Tolerancing. However, for the timebeing, let us set aside this problem.
First of all, we consider in the following a dimension chain consisting ofnormally distributed individual dimensions. If the individual dimension isinterpreted formally as a random variable, then the variance of the assemblydimension σ s
2 which is once again a random variable as sum of these random
variables, can be calculated as the sum of the variances σ i2
of all individualdimensions, according to the law of error propagation, as per section 4.2:
σ σs ii
k2 2
1=
= . (5.1)
In order that this equation can be converted into a relation between thetolerances of the individual dimensions Ti and the tolerance of the assembly
dimension Ts we need to know the ratios Tssσ
and Tiiσ
.
For the triangular and rectangular distribution of individual dimensions, suchrelations exist already in the form of equations 4.7 and 4.11. They werederived under the assumption that the distribution occupies the tolerance rangefully, just as it is shown by the figures 4.5 and 4.6.
In case of the normal distribution we use the approach Ti i= ⋅6 σ , i.e., theindividual tolerance corresponds to the ± 3 σ -range of the distribution of theindividual dimension.
In practice, this information could be obtained, e.g., from the processcapability index Cp of a process controlled with SPC:
CT
p =⋅6 σ
. (5.2)
Our approach Ti i= ⋅6 σ corresponds, as such, to the assumption Cp = 1.
According to the central limit theorem of statistics random combination ofsufficiently many individual dimensions produces an assembly dimension,which is approximately normally distributed, independent of the distributionsof the individual dimensions.
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Now a value must be specified also for the ratio Tssσ
. This is determined based
on the acceptable fraction of assemblies, whose dimensions may lie outside the
specified tolerance zone NT
ss±
2.
For reasons of simplicity we use Ts s= ⋅6 σ . (5.3)
The permissible proportion nonconforming is p ≈ 0 27. % in this case. Thus,
if Ti i= ⋅6 σ and Ts s= ⋅6 σ are used in the equation σ σs ii
k2 2
1=
=
(Equation 5.1), we find:
σ NN i
i
k
N ii
k
k
T TT T T T T2
2 2
1
2
112
22 2
36 36= = = = + + +
= = ... .
(5.4)
Remark:
To identify the distributions that have been taken as a basis for the individualdimensions, the tolerance of the assembly dimension T is provided with anindex: N for normal distribution, D for triangular distribution and R forrectangular distribution.
The tolerance of the assembly dimension of a dimension chain, consisting ofnormally distributed individual dimensions is, thus, calculated from the sum ofsquares of the individual tolerances. Therefore, this is also called as quadratictolerancing.
If the individual dimensions are distributed according to a triangular or arectangular distribution, then, by using the variances of these distributions inequation 5.1, calculated as per 4.3.1 and 4.3.2, we find once again with theassumption Ts s= ⋅6 σ :
σ DD i
i
k
D k
T TT T T T2
2 2
112
22 2
36 24624
= = ⇔ = ⋅ + + += ...
(Triangular distribution) (5.5)
and σ RR i
i
k
R k
T TT T T T2
2 2
112
22 2
36 12612
= = ⇔ = ⋅ + + += ... .
(Rectangular distribution) (5.6)
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If the tolerances of the assembly dimensions TN , TD and TR are compared witheach other, then the ratios 1 : 1.2 : 1.7 become apparent. (5.7)
Example 5.1: For the individual dimensions given in the following table, thestatistical tolerance for the assembly dimension should be calculated andcompared with that one calculated arithmetically.
DimensionNo.
Individual nominaldimension Ni
AllowanceAi
ToleranceTi
1 3 ± 0.01 0.02
2 8 ± 0.01 0.02
3 7 ± 0.01 0.02
4 14 ± 0.01 0.02
Normal: T TN ii
= = ⋅ = ⋅ == 2
1
424 0 02 2 0 02 0 04. . .
Triangular: T TD ii
= ⋅ = ⋅ ⋅ ==
624
624
2 0 02 0 052
1
4
. .
Rectangular: T TR ii
= ⋅ = ⋅ ⋅ ==
612
612
2 0 02 0 072
1
4
. .
Arithmetical: Ta = ⋅ =4 0 02 0 08. .
As a result of Statistical Tolerancing, in this example a smaller value isobtained for the assembly tolerance compared to the arithmetically calculatedtolerance Ta . The difference is, obviously, maximum for normally distributedindividual dimensions (cp. ratios 5.7).
Remark: With the assumption Ts s= ⋅8 σ , corresponding to p ≈ 64 ppm, hadthe value TR = 0.9 been the result for rectangular distribution of the individualdimensions, which is greater than Ta . Accordingly, it is not sensible to use a
greater ratio Tssσ
, just to obtain, by mere calculation, a smaller fraction of
nonconforming assemblies.
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If the actual individual dimensions are within their tolerance ranges, then thefraction nonconforming with respect to the arithmetically determinedtolerance, is equal to zero.
This result is plausible, when the three distributions under consideration arerepresented in graphical form.
Fig. 5.1: Graphs of the density functions of Normal, Triangularand Rectangular distribution
Obviously, in case of normal distribution, the individual values concentratemostly around the midpoint of the tolerance zone. The gain from statisticaltolerancing, compared to arithmetic tolerancing, is the most with normallydistributed individual dimensions.
If the assembly dimension is assumed to be given, then by inverting the abovecalculation steps, it can be shown, that under specific conditions and throughapplication of Statistical Tolerancing the individual tolerances can be selectedsignificantly greater (looser) than those resulting from arithmetic tolerancing.
The aforementioned considerations make the special advantages of StatisticalTolerancing clear: the smaller assembly tolerance, resulting from the giventolerances for individual dimensions, means for example higher functionalreliability or longer life; the greater tolerances for the individual dimensionsresulting from a given assembly tolerance lead in general to highermanufacturing reliability or more economic production. Naturally, thecomparison is always with reference to arithmetic tolerancing.
Statistical Tolerancing bases itself, however, on a number of restrictiveconditions, which have not been listed explicitly in the foregoing sections, buthave been assumed as complied with without exception. These prerequisitesare listed in detail and discussed in Section 6.
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5.1 The Reduction Factor
The ratios as per equation 5.7 as well as the results of example 5.1 showclearly the advantages of Statistical Tolerancing, in principle, for the specialcases of individual dimensions which are subject to the normal, triangular orrectangular distribution.
However, it shall not be overlooked, that with the selection of the approachTs s= ⋅6 σ (cp. equation 5.3) a permissible proportion of assemblies isaccepted ( p = 0 27. % ), whose dimensions lie outside the tolerance zone(fraction nonconforming). Then, there is no unrestricted interchangeabilitywithin the assemblies any more.
Further, four components have been assumed for the calculations in the scopeof the example. The explicit results are, therefore, valid only under theassumptions k = 4 and p = 0 27. % .
The resultant reduction in assembly tolerance Ts , which can be achieved fromStatistical Tolerancing, compared to the arithmetically calculated assemblytolerance Ta , can be described in form of the reduction factor r and can bedetermined, in general, for the given special distributions, as long as all kindividual dimensions follow, in each case, the same distribution and allindividual tolerances Ti are equal:
T r Ts a= ⋅ . (5.8)
Instead of the special option Ts s= ⋅6 σ corresponding to p = 0 27. % (two-sided), we substitute in equations 5.4, 5.5 and 5.6 T us p s= ⋅ ⋅−2 1 2/ σ and usethe number k of combined individual dimensions.
2 1 2⋅ −u p / is the random variation range of the approximately normallydistributed assembly dimension, in which ( )%1 − p all values lie.
From this follows for:
a) normally distributed individual dimensions (assumption Ti i= ⋅6 σ ):
T u uT
N p N pi
i
k
= ⋅ ⋅ = ⋅ ⋅− −=2 2
361 2 1 2
2
1/ /σ
Tu
k TNp
i= ⋅ ⋅−1 2
3/
rTT
u k Tk T
uk
N
a
p i
i
p= = ⋅⋅
⋅=
⋅− −1 2 1 2
3 3/ / (5.9)
- 25 -
b) individual dimensions being subject to a triangular distribution:
Tu
Tu
k TDp
ii
kp
i=⋅
⋅ = ⋅ ⋅−
=
−
224 61 2 2
1
1 2/ /
rTT
u k Tk T
u
kD
a
p i
i
p= = ⋅
⋅⋅
=⋅
− −1 2 1 2
6 6/ /
(5.10)
c) individual dimensions being subject to a rectangular distribution:
Tu
Tu
k TRp
ii
kp
i=⋅
⋅ = ⋅ ⋅−
=
−
212 31 2 2
1
1 2/ /
rTT
u k Tk T
uk
R
a
p i
i
p= = ⋅⋅
⋅=
⋅− −1 2 1 2
3 3/ / . (5.11)
Accordingly, the reduction factor r will be the smaller, the higher theproportion p (acceptable fraction nonconforming) and the greater the numberk of individual dimensions.
The following table shows a few values for r , which have been calculatedunder the assumption p = 0 27. % , u p1 2 3− =/ .
Number of individual dimensions k
Distribution of theindividual dimensions
4 6 8
Normal distribution 0.50 0.41 0.35
Triangular distribution 0.61 0.50 0.43
Rectangular distribution 0.87 0.71 0.61
Table 5.1: Reduction factor r for different distributionsand numbers of individual dimensionsof a dimension chain
- 26 -
It is obvious, that Statistical Tolerancing can yield, depending upon thedistribution of the individual dimensions, assembly tolerances which areconsiderably smaller (tighter) than that resulting from arithmetic tolerancing.
Difficulties, concerning applicability of Statistical Tolerancing and makinguse of the theoretically possible advantages in practice, grow because of theneed to know exactly the distributions which are underlying the individualdimensions and the stability of the parameters affecting the location and thevariation of these distributions (cp. section 6).
As a way out, it is often suggested in the literature, to assume basically arectangular distribution as model distribution for the individual dimen-sions. With this there is no need for an expensive monitoring of process (cp.section 7), but instead, one must be satisfied with a comparatively smalladvantage (higher reduction factor).
The following figure shows the dependence of reduction factor r upon k (thenumber of elements in dimension chain) and nonconforming fraction p forindividual dimensions following a rectangular distribution.
Figure 5.2: Reduction factor r depending upon k (number of elements in thedimension chain) and p (fraction nonconforming; proportion ofassemblies exceeding the specification limits for the assemblydimension) for individual dimensions which are subject to arectangular distribution.
- 27 -
5.2 Calculation of the Statistical Assembly Tolerance
With the use of the reduction factor r , calculation of the statistical assemblytolerance becomes very simple. Under the assumption, that individualdimensions follow a rectangular distribution,
ru
kp=⋅
−1 2
3/ . (5.11)
With p = 0 27. % , (u p1 2 3− =/ ) and k = 4 for the example 5.1 (Section 5)results:
r = 0 866. and T r T mm mmR a= ⋅ = ⋅ =0 866 0 08 0 07. . . .
5.3 Calculation of Individual Tolerancesfor a Given Assembly Tolerance
Also here, let us assume that the individual dimensions are subject to arectangular distribution.
Because of T r TR a= ⋅ and T k Ta i= ⋅
is T r k TR i= ⋅ ⋅ and TTr kiR=⋅
.
If we apply this relationship to Example 5.1 and use for TR the assemblytolerance 0.08 mm, then under the same conditions as in 5.2 ( p = 0 27. % ,k = 4, individual tolerances are of equal size):
TTr k
mmmmi
R=⋅
=⋅
=0 080 866 4
0 023..
. .
The resultant individual tolerance Ti is thus somewhat greater than that isobtained with mere arithmetical calculation:
TTk
mmmmi
a= = =0 08
40 02
.. .
- 28 -
5.4 The General Case
The special cases, which were considered so far, wherein all individualdimensions have the same distribution and the same tolerance, will comeacross in practice really very seldom. In most of the cases here, it is anassembly consisting of same parts (same drawing dimensions, for example,transformer sheets).
In general, the designer has to take into account individual tolerances whichare different from each other and the distributions followed by the individualdimensions differ. This general case is covered already in the discussions sofar.
According to 5.1, 2 1 2⋅ −u p/ is that random variation range of theapproximately normally distributed assembly dimension, wherein ( )%1 − p ofall values are contained.After having selected p , the assembly tolerance can be calculated withT us p s= ⋅ ⋅−2 1 2/ σ , where σ s is defined by the law of error propagation
(equation 4.3 or 4.4): T us p k= ⋅ ⋅ + + +−2 1 2 12
22 2
/ . . .σ σ σ .
The σ i are substituted on the basis of actually available tolerances anddistributions of individual dimensions (information from production).
If for example
N 1 is normally distributed with σ 11
4=T
,
N 2 is subject to a triangular distribution with σ 22
24=T
,
N 3 follows a rectangular distribution with σ 33
12=T
,
N 4 is normally distributed with σ 44
6=T
,
N 5 is normally distributed with σ 55
5=T
,
then the tolerance for the assembly dimension results in
T uT T T T T
s p= ⋅ ⋅
+
+
+
+
−2
4 24 12 6 51 21
22
23
24
25
2
/ .
- 29 -
5.5 Process Capability Indices
The process capability index Cpk is a measure for the variation of acharacteristic within the tolerance zone, considering the average position ofthe process.
CD
pk =⋅3 σ
D is the minimum distance of the mean of the distribution to one of thespecification limits GU or GL respectively (cp. Fig 5.3).
Fig. 5.3: Representation to illustrate Cpk
Assuming that the process is centred, i.e., the process average coincides
with the midpoint of the tolerance interval (thus D T=2
), then Cpk is equal to
Cp :
CT
p =⋅6 σ
.
We will consider in the following the impact of statistics on the Cp value ofthe assembly dimension when assembling several parts whose characteristicsfollow the same distribution.
- 30 -
The Cp value of an assembly dimension is, of course, not a process capabilityindex in the original sense. However, it can be used as a formal variable, forexample, to assess the functional reliability of the assemblies.
We designate the Cp value of the i-th individual characteristic of a dimension
chain with ( )Cp i, and the Cp value of the assembly dimension with ( )C p s
:
( )CT
p i
i
i=
⋅6 σ ( )C
Tp s
s
s=
⋅6 σ.
Now, if for Ts the arithmetically calculated assembly tolerance T k Ta i= ⋅ is
used and σ σs ik= ⋅ , which is based on equation 5.1, is substituted in the
expression for ( )C p s, then results:
( ) ( )CT k T
kk Cp s
s
s
i
ip i
=⋅
=⋅
⋅ ⋅= ⋅
6 6σ σ.
Thus, when arithmetically tolerancing a dimension chain having k elements,the Cp value of the assembly dimension is greater by a factor of k , than theCp value of the individual dimension.
Therefore, it is possible, that the Cp value of the assembly dimension exceedsa minimum value for Cp , although this is not reached by the Cp values of theindividual dimensions.
Contrarily, the value ( )C p s in case of statistical tolerancing, depends upon
the selection of the ratio Tssσ
.
As in section 5, if Ts s= ⋅6 σ (Equation 5.3), then
( )CT
p s
s
s
s
s=
⋅=
⋅⋅
=6
66
1σ
σσ
.
The considerations above are independent of the specific distribution.
- 31 -
6. Prerequisites for Statistical Tolerancing
As pointed out already a number of times in the text so far, application ofStatistical Tolerancing assumes a number of prerequisites as fulfilled, whichare once again referred to in this section and subjected to a critical scrutiny asfor their possibility of fulfillment and checking in practice.
The prerequisites in detail (cp. Supplement to QSP 0407 [13]):
1. The individual dimensions are uncorrelated random variables.
It is conceivable, that two or more parts of an assembly are the same (samedrawing dimensions), especially if the assembly is symmetric.
If these parts are produced directly in succession then it is very probable, thata current deviation of the process average (process-inherent trend, between-batch variation) leads to a deviation of the actual dimensions of these parts inthe same direction. If such parts go to the assembly station in an orderedsequence (e.g. transfer line or sequential interim storage in boxes), then thereis also matched or sequential grouping of assembly with great probability. Inthis case, a statistical compensation (neutralization) of dimensional deviationsin the sense of the Central Limit Theorem is not at all possible or possibleonly with constraints.
2. The dimension chain is linear.
This prerequisite presumes, of course, the knowledge of the interaction of theindividual parts with respect to the assembly dimension. In case of nonlineardimension chains, the functional correlation of the individual dimensions mustbe known and be able to be described through analytical functions.
The statistical tolerancing of the assembly dimension of a linear dimensionchain uses the law of error propagation in its simple form - equation 4.3 or 4.4.In case of nonlinear correlations, their partial derivatives are to be givenconsideration as per equation 4.2. If the nonlinear correlation is known ordescribable only approximately, it is to be borne in mind that approximationerrors can have strong influence because of the law of error propagation. Weunderline the fact, that nonlinear dimension chains can lead to complicatedapplications of the law of error propagation (cp. Section 8).
3. The dimension chain consists at least of four links.
As the figures 4.2 and 4.3 show, on the basis of the discrete random variabletotal score from a throw of k dice, the sum of k random variables leads,from k = 4 and above, to a distribution, which can sufficiently good beapproximated by the normal distribution. The combination, exclusively ofextreme actual dimensions, is sufficiently improbable only from k = 4onwards.
- 32 -
The paraphrases sufficiently good and sufficiently improbable are indeedas vague as the formulation permissible fraction nonconforming. Suchphrases are always closely linked with the assessment of assemblies fromconsideration of their design such as functional reliability, reliability (lifetime)or economics of production.
4. The individual tolerances are of the same magnitude
Expressed in simple terms, the statistical compensation of negative andpositive deviations of actual dimensions from nominal dimensions canfunction only when the random variation ranges of the actual dimensions andtherewith also their tolerances are approximately equal.
It is shown in [l] by simulation that with growing difference in size oftolerances in a dimension chain the effect of narrow distributions on thetotal distribution (distribution of the assembly dimension) gets diminished.With a ratio between sizes of 5:1, the effect of the smaller tolerances is highlysuppressed (see [1], page 63).
The prerequisite includes the requirement that the distribution of theindividual dimension makes the maximum use of tolerance concerned. If thetolerance is only used up, because a small distribution due to process trenddrifts systematically within the tolerance range, then Statistical Tolerancingmay not be applied. In this case arithmetical tolerancing is to be resorted to.
5. The distributions of the individual dimensions of the dimension chainare known
The condition of knowing the stability of distribution of individual dimensionsover time is ultimately a condition relating to the properties of the productionprocesses concerned. Accordingly these must be controlled as for averageposition and variation and must always be centred on the midpoint.The distribution of a characteristic of a part never follows exactly an idealdistribution as observed in Section 5.Apart from this, in a concrete case an estimation of the parameters of adistribution on the basis of process data (random samples) is always affectedwith an error dependent upon the sample size.
In the following we quote, without comment, from a few literature (seebibliography) published on the theme Statistical Tolerancing, which deal withthe problems referred to above. (Hint: Faithful translations of the originalquotations. The quoted literature is only available in German.)
[l] P. 29: To establish as called for in the beginning a close associationbetween tolerance and type of distribution for Statistical Tolerancing as aprecondition information is necessary on the distribution of dimensions withinthe tolerance range. It is necessary, that these conditions are checked carefully,because deviations in production from the assumptions made for calculationget themselves magnified in the convolution process, and because thesedeviations influence the total distribution, especially the important peripheralzones.
- 33 -
[1] P. 34/35: It is valid in general, that special distributions are always to begiven consideration in statistical error propagation, because they possiblyinfluence very much the properties of the total distribution.
[l] P. 69: The distribution of the counterparts influences, during convolution,very much the propagation of a distribution deviating from the standard.
[l] P. 37/38: To assess the stability over time, the change of statistical data ofa production process is studied by one or many repetitions of a measurementseries, ... While evaluating, it is noticed that only 2 of 17 production processesshow total concurrence in x and s.
[l] P. 38: The same result occurs if a process is governed by a slow trend (i.e.,takes long time to become apparent), which does not show itself up in fullwithin one measurement series. Then, it is very difficult to assess a probabledistribution of measurement values.
[l] P. 39: So, also the constant drifting of a dimension as a result of shift incutting edge, the Trend, is inconsistent in parts in terms of steadiness andspeed. These influences, which are not purely random, but cannot be excluded,lead to the observed result that the distributions and other statistical data showa wide variability and their reproducibility is low.
[3] P. 113: As has been said already, by giving consideration to the type ofdistribution, certainly we gain by way of maximum possible tolerance rangefor individual dimensions and thereby ease production; but with this advantagesomething is also lost, because there must be constant vigilance to see thatprerequisites remain valid; one must monitor the distribution and control theprocess.
[4] P. 138: It can be seen from this example, that under the assumption, theproduction is known, significant ease in production can be realized.
7. Statistical Monitoring of the Individual Dimensions
The prerequisites for Statistical Tolerancing call for consistency over time ofthe characterisitcs of the distributions and concurrence of average ofdistribution with the respective midpoints of tolerance zones. The assumption,that is occasionally found in the literature (see for example [5] p. 67), viz.,these conditions can be guaranteed by Statistical Process Control (SPC) ofconcerned production process is correct only with its own limitations. Wewish to make this point clear in this section.
The control charts, maintained within the scope of SPC, enable to recognize ashift in location and/or a change in variation of the distribution underobservation at the earliest possible time.
- 34 -
It would all the more be extremely uncomfortable, if a control chart wouldalready respond at small deviations, for example, of sample average fromcenterpoint C; as such small deviations can result very much from randomnature of sampling; and consequently a process which is absolutely stablecentrally would be interfered with unnecessarily.
According to this, a quality control chart which is functioning in an ideal wayshould respond with 100% certainty only when the prevailing change inprocess actually exceeds a given limit and it should not respond with the samecertainty otherwise. A real control chart does not possess this property by wayof an on/off switch. Real shifts in average position of the characteristicdistribution are indicated only with a certain probability. This probability isthe greater the more the actual average position µ t differs from the targetvalue C (or µ ) (see Fig. 7.1). This action probability as a function of shift inaverage is described by an action characteristic.
The course of this curve depends upon the type of control chart and the samplesize n opted for (see [7] and [8]). Figure 7.1 shows the action characteristicsof an average chart with sample size n as parameter.
Fig. 7.1: Action characteristics of the average chart;sample size n as parameter
For the case, where a shift in average of a process under control around ± 1σshould be shown with a probability of 90%, then a sample size of about n = 15would be necessary. In practice, however, a sample size of n = 5 is often used.The inspection expenses thus would have to be tripled.
- 35 -
8. Generalisation for Arbitrary Dimension Chains
It is obvious, that all the cases and examples considered in sections 5.1 to 5.3are always special cases which can be represented and calculated in the simpleform shown only under highly restrictive assumptions.
In the general cases, where
- any distribution of the individual dimensions,- any combination of these distributions,- any ratio for the magnitude of the individual tolerances, and- a nonlinear dimension chain
should be permitted, Statistical Tolerancing would be possible only with thehelp of computer.
An appropriate computer program would have to be able to determine theprobability density function of the assembly dimension, which arises from aknown functional relationship of individual dimensions following in each caseknown distributions, by calculation of the so-called convolution integral and todetermine again the random variation limits of the assembly dimension, usingthe density function.
If it would be desired to avoid the expensive determination of convolutionintegrals by computation, which can be represented as closed expressions onlyfor simple analytical functions such as sums and products (of randomvariables), then it would be necessary to apply the law of error propagation inits general form (Equation 4.2). In this case, however, the analytical ornumerical determination of partial derivatives would be necessary.
Finally there would be still the possibility, to simulate the combination ofdistributions of individual dimensions (Monte-Carlo-Methods).
Although the aforementioned altematives could be realised in form ofcomputer programs, the practicability of a computer-aided StatisticalTolerancing is, in the general case, not very probable.
Despite all the new developments in hardware and software spheres, theproduction processes, their controllability and sufficiently reliable statisticaldescription of them remain the decisive factors for economic application ofStatistical Tolerancing.
Further the above concepts can apply always only to calculation of tolerancesfor assembly dimensions from individual dimensions following given (known)distributions. The inversion of calculation, i.e., calculation of tolerances forindividual dimensions from a given tolerance for the asembly dimension asshown for example in 5.3 is not possible in the general case.
- 36 -
9. Bibliography
[1] F. Böttger: Erzielung von Fertigungsvorteilen durch Anwendungstatistischer Gesetze auf die Toleranzberechnung,Dissertation, TH Aachen, 1961
[2] H. Röper: Statistische Maßtolerierung von Großserienprodukten, Konstruktion 41 (1989) S.313-318
[3] E. Schlötel: Toleranzfestlegung unter Berücksichtigung statistischerGesichtspunkte,Qualitätskontrolle, 13. Jahrgang (1968) Heft 9, S.111
[4] F. Goubeau: Toleranzkopplungen in Theorie und Praxis, Feinwerktechnik, Jahrgang 63, Heft 4, 1959, S.133-142
[5] K. Blaier: Erstellung eines Rechnerprogramms zur statistischen Tolerierung von Toleranzketten, Diplomarbeit, Uni Stuttgart, 1990
[6] G. Kirschling: Qualitätssicherung und Toleranzen, Springer-Verlag Berlin, 1988
[7] Bosch Schriftenreihe, Heft Nr. 7: SPC
[8] DGQ-Schrift Nr. 16-30 Qualitätsregelkarten
[9] Balakschin: Technologie des Werkzeugmaschinenbaus, VEB Verlag Technik, Berlin, 1953
[10] E. Kreyszig: Statistische Methoden und ihre Anwendungen, 7. Auflage, Göttingen, 1979
[11] DIN 7182, Maße, Abmaße, Toleranzen und Passungen,Grundbegriffe, 1986
[12] Bosch Norm N13 B11, Toleranzen und Passungen, Grundbegriffe, 1975 Bosch Norm N13 B94, Toleranzrechnung, 1980
[13] Qualitätssicherungsplan 0407, Toleranzehrlichkeit
[14] DIN 7186 Statistische Tolerierung, 1974
[15] TGL 19115/01 - TGL 19115/04 (DDR-Standard), Berechnung von Masz- und Toleranzketten, 1983
- 37 -
[16] E. Rusch: Wirtschaftliches Tolerieren,QZ 16 (1971), Heft 1, S.19-23
[17] H. Trumpold, Ch. Beck: Optimale Toleranzfestlegung unter Berücksichtigung
der Maßkettentheorie und der statistischen Eigenschaften des Fertigungsprozesses, Fertigungstechnik und Betrieb,
21 (1971) H.4, S.242-246
[18] E. Görler: Berücksichtigung der Lage und Form statistischer Verteilungen bei der Tolerierung von Maßketten, Wiss. Z. d. Techn. Hochsch. Karl-Marx-Stadt, 21 (1979) H. 2, S.203-210
[19] E. Görler: Lage und Form statistischer Verteilungen bei der Tolerierung von Maßketten, Feingerätetechnik, 28. Jahrg., Heft 7, 1979, S.314-316
[20] E. Görler: Rationelle Verfahren bei der statistischen Qualitäts-analyse in der mechanischen Fertigung,
Fertigungstechnik und Betrieb 26 (1976) 3, S.142-145
[21] W. Hanka: Die Häufigkeitsverteilung der Summentoleranzen von zusammengebauten Teilen, Werkstattstechnik 55. Jahrg., 1965, Heft 12, S.590-593
[22] A. Kettmann: Festlegung von Toleranzen nach statistischen Gesichtspunkten, Konstruktion,
13. (1961) Heft 5, S.202-204
[23] K. Stange: Die Bestimmung von Toleranzgrenzen mit Hilfe statistischer Überlegungen, Qualitätskontrolle 14. Jahrg. (1969) Heft 5, S.57-63
[24] Stuhlmann, Schmidt: Statistische Tolerierung als Problem der Fertigung
und der Prüfung,Werkstattstechnik, 55. (1965) H.12 S.585-589
[25] R. Müller: Optimierung der Funktionssicherheit mechanischer Maßketten durch Statistische Tolerierung, QZ 16 (1971) H. 12, S.269-271
- 38 -
10. Appendix
Revision - Normal Distribution
Solutions to Excercises
The Probability Plot
Standard Normal Distribution
Table of the Standard Normal Distribution
- 39 -
Revision - Normal Distribution (Excercises)
Relays were tested for series production start-up. A functionally decisivecharacteristic is the so-called response voltage. With 50 relays, thefollowing values of the response voltage Uresp were measured in volts.
6.2 6.5 6.1 6.3 5.9 6.0 6.0 6.3 6.2 6.46.5 5.5 5.7 6.2 5.9 6.5 6.1 6.6 6.1 6.86.2 6.4 5.8 5.6 6.2 6.1 5.8 5.9 6.0 6.16.0 5.7 6.5 6.2 5.6 6.4 6.1 6.3 6.1 6.66.4 6.3 6.7 5.9 6.6 6.3 6.0 6.0 5.8 6.2
a) Calculate x and s with pocket calculator!
b) The customer asks for U Vresp = 68. . Estimate the proportion of relayswhich does not satisfy the customers requirement (fraction noncon-forming)!
c) What is the value of response voltage that will be exceeded by maximum0.1% of all relays manufactured?
d) Determine with the help of x , s and the upper specification limitUSL V= 6 8. the Cpk value of the production!
- 40 -
Solutions to Excercises
To a) Mean x = 615. Standard deviation s = ≈0 2998 0 3. .
To b) Substitution in the transformation equation results in
u =−
=6 8 615
0 32 17
. ..
. .
Table value ( )Φ 2 17 0 985. .=
Fraction nonconforming p = − =1 0 985 15. . %
To c) p u= − =1 01Φ ( ) . % or: Φ ( ) .u = 0 999
Read from the table that value of u , at which Φ ( )u is as close aspossible to the value 0.999. We find u = 3 09. .
If the transformation equation u x xs
= − is resolved for x and the
values are substituted, this produces
x x u s= + ⋅ = + ⋅ ≈615 3 09 0 3 7 1. . . . .
Not more than 0.1% of all relays will have a response voltage whichis in excess of 7.1 V.
To d) CUSL x
spk =−
⋅=
−⋅
≈3
68 6153 0 3
0 7. .
..
- 41 -
The Probability Plot
If a normal distribution is being spoken of, one usually associates this termwith the Gaussian bell-shaped curve. This curve is a representation of theprobability density function f x( ) of the normal distribution:
( )f x ex
=⋅
⋅−
−
1
2
12
2
σ π
µσ .
The normal distribution gives the probability for each value x that the randomvariable X will assume a value between − ∞ and x . One obtains the distri-bution function Φ ( )x of the normal distribution by integrating the densityfunction given above:
( )Φ x e dvvx
=⋅
⋅− ⋅
−
−∞
12
12
2
σ π
µσ
Φ ( )x corresponds to the area below the Gaussian curve up to the value x .
The graphical representation of this function follows an s-shaped curve.Strictly speaking, we should always think of this curve, when we talk of anormal distribution.
If the y-axis of this representation is distorted in such a manner, that the s-shaped curve becomes a straight line, a new coordinate system arises, theprobility paper. Here the x-axis remains unchanged. Because of this corre-lation, the representation of a normal distribution in this new coordinatesystem produces a straight line.
This fact is made use of to test a given data set for normality graphically. Aslong as the number of measurement values is sufficiently big, one candetermine the relative frequencies of values within the classes of a groupingand draw a histogram. If the corresponding cumulative relative frequencies areplotted versus the upper class limits on probability paper and a sequence ofdots is obtained, which approximately lie on a staight line, then it can beconcluded that the values of the data set are approximately normallydistributed.
- 42 -
The Standard Normal Distribution
A normally distributed random quantity X with the mean µ and standarddeviation σ is converted to an equally normally distributed random quantity U
through the transformation UX
=− µσ
. The average of U is zero, and its
standard deviation is one.
This fact is illustrated by the following figure.Here µ x = 238. and σ x = 05. .
In this example we find: u =−
=25 238
0 52 4
..
. .
The special normal distribution N(0,1) is called standard normal distribution.
The distribution function Φ ( )u indicates the probability with which therandom quantity U assumes a value between − ∞ and u .
Φ ( )u corresponds to the area below the Gaussian curve up to the value u .
The total area under the bell-shaped curve has the value one.
The values for Φ ( )u can be taken from the table.For example, we find Φ (2.4) = 0.9918.
The proportion of all X , which are greater than 25, corresponds to theproportion of U, which exceed the value u = 2.4. Accordingly it is 1 - 0.9918 ≈ 0.8%.
The following representations show a few examples from which the meaningof the terms and the table values become apparent.
- 43 -
Please note: Φ (-u)=1-Φ (u) D(u)=Φ (u)-Φ (-u)
- 44 -
u Φ (-u) Φ (u) D(u)0.01 0.496011 0.503989 0.0079790.02 0.492022 0.507978 0.0159570.03 0.488033 0.511967 0.0239330.04 0.484047 0.515953 0.0319070.05 0.480061 0.519939 0.0398780.06 0.476078 0.523922 0.0478450.07 0.472097 0.527903 0.0558060.08 0.468119 0.531881 0.0637630.09 0.464144 0.535856 0.0717130.10 0.460172 0.539828 0.0796560.11 0.456205 0.543795 0.0875910.12 0.452242 0.547758 0.0955170.13 0.448283 0.551717 0.1034340.14 0.444330 0.555670 0.1113400.15 0.440382 0.559618 0.1192350.16 0.436441 0.563559 0.1271190.17 0.432505 0.567495 0.1349900.18 0.428576 0.571424 0.1428470.19 0.424655 0.575345 0.1506910.20 0.420740 0.579260 0.1585190.21 0.416834 0.583166 0.1663320.22 0.412936 0.587064 0.1741290.23 0.409046 0.590954 0.1819080.24 0.405165 0.594835 0.1896700.25 0.401294 0.598706 0.1974130.26 0.397432 0.602568 0.2051360.27 0.393580 0.606420 0.2128400.28 0.389739 0.610261 0.2205220.29 0.385908 0.614092 0.2281840.30 0.382089 0.617911 0.2358230.31 0.378281 0.621719 0.2434390.32 0.374484 0.625516 0.2510320.33 0.370700 0.629300 0.2586000.34 0.366928 0.633072 0.2661430.35 0.363169 0.636831 0.2736610.36 0.359424 0.640576 0.2811530.37 0.355691 0.644309 0.2886170.38 0.351973 0.648027 0.2960540.39 0.348268 0.651732 0.3034630.40 0.344578 0.655422 0.3108430.41 0.340903 0.659097 0.3181940.42 0.337243 0.662757 0.3255140.43 0.333598 0.666402 0.3328040.44 0.329969 0.670031 0.3400630.45 0.326355 0.673645 0.3472900.46 0.322758 0.677242 0.3544840.47 0.319178 0.680822 0.3616450.48 0.315614 0.684386 0.3687730.49 0.312067 0.687933 0.3758660.50 0.308538 0.691462 0.382925
u Φ (-u) Φ (u) D(u)0.51 0.305026 0.694974 0.3899490.52 0.301532 0.698468 0.3969360.53 0.298056 0.701944 0.4038880.54 0.294598 0.705402 0.4108030.55 0.291160 0.708840 0.4176810.56 0.287740 0.712260 0.4245210.57 0.284339 0.715661 0.4313220.58 0.280957 0.719043 0.4380850.59 0.277595 0.722405 0.4448090.60 0.274253 0.725747 0.4514940.61 0.270931 0.729069 0.4581380.62 0.267629 0.732371 0.4647420.63 0.264347 0.735653 0.4713060.64 0.261086 0.738914 0.4778280.65 0.257846 0.742154 0.4843080.66 0.254627 0.745373 0.4907460.67 0.251429 0.748571 0.4971420.68 0.248252 0.751748 0.5034960.69 0.245097 0.754903 0.5098060.70 0.241964 0.758036 0.5160730.71 0.238852 0.761148 0.5222960.72 0.235762 0.764238 0.5284750.73 0.232695 0.767305 0.5346100.74 0.229650 0.770350 0.5407000.75 0.226627 0.773373 0.5467450.76 0.223627 0.776373 0.5527460.77 0.220650 0.779350 0.5587000.78 0.217695 0.782305 0.5646090.79 0.214764 0.785236 0.5704720.80 0.211855 0.788145 0.5762890.81 0.208970 0.791030 0.5820600.82 0.206108 0.793892 0.5877840.83 0.203269 0.796731 0.5934610.84 0.200454 0.799546 0.5990920.85 0.197662 0.802338 0.6046750.86 0.194894 0.805106 0.6102110.87 0.192150 0.807850 0.6157000.88 0.189430 0.810570 0.6211410.89 0.186733 0.813267 0.6265340.90 0.184060 0.815940 0.6318800.91 0.181411 0.818589 0.6371780.92 0.178786 0.821214 0.6424270.93 0.176186 0.823814 0.6476290.94 0.173609 0.826391 0.6527820.95 0.171056 0.828944 0.6578880.96 0.168528 0.831472 0.6629450.97 0.166023 0.833977 0.6679540.98 0.163543 0.836457 0.6729140.99 0.161087 0.838913 0.6778261.00 0.158655 0.841345 0.682689
- 45 -
u Φ (-u) Φ (u) D(u)1.01 0.156248 0.843752 0.6875051.02 0.153864 0.846136 0.6922721.03 0.151505 0.848495 0.6969901.04 0.149170 0.850830 0.7016601.05 0.146859 0.853141 0.7062821.06 0.144572 0.855428 0.7108551.07 0.142310 0.857690 0.7153811.08 0.140071 0.859929 0.7198581.09 0.137857 0.862143 0.7242871.10 0.135666 0.864334 0.7286681.11 0.133500 0.866500 0.7330011.12 0.131357 0.868643 0.7372861.13 0.129238 0.870762 0.7415241.14 0.127143 0.872857 0.7457141.15 0.125072 0.874928 0.7498561.16 0.123024 0.876976 0.7539511.17 0.121001 0.878999 0.7579991.18 0.119000 0.881000 0.7620001.19 0.117023 0.882977 0.7659531.20 0.115070 0.884930 0.7698611.21 0.113140 0.886860 0.7737211.22 0.111233 0.888767 0.7775351.23 0.109349 0.890651 0.7813031.24 0.107488 0.892512 0.7850241.25 0.105650 0.894350 0.7887001.26 0.103835 0.896165 0.7923311.27 0.102042 0.897958 0.7959151.28 0.100273 0.899727 0.7994551.29 0.098525 0.901475 0.8029491.30 0.096801 0.903199 0.8063991.31 0.095098 0.904902 0.8098041.32 0.093418 0.906582 0.8131651.33 0.091759 0.908241 0.8164821.34 0.090123 0.909877 0.8197551.35 0.088508 0.911492 0.8229841.36 0.086915 0.913085 0.8261701.37 0.085344 0.914656 0.8293131.38 0.083793 0.916207 0.8324131.39 0.082264 0.917736 0.8354711.40 0.080757 0.919243 0.8384871.41 0.079270 0.920730 0.8414601.42 0.077804 0.922196 0.8443921.43 0.076359 0.923641 0.8472831.44 0.074934 0.925066 0.8501331.45 0.073529 0.926471 0.8529411.46 0.072145 0.927855 0.8557101.47 0.070781 0.929219 0.8584381.48 0.069437 0.930563 0.8611271.49 0.068112 0.931888 0.8637761.50 0.066807 0.933193 0.866386
u Φ (-u) Φ (u) D(u)1.51 0.065522 0.934478 0.8689571.52 0.064256 0.935744 0.8714891.53 0.063008 0.936992 0.8739831.54 0.061780 0.938220 0.8764401.55 0.060571 0.939429 0.8788581.56 0.059380 0.940620 0.8812401.57 0.058208 0.941792 0.8835851.58 0.057053 0.942947 0.8858931.59 0.055917 0.944083 0.8881651.60 0.054799 0.945201 0.8904011.61 0.053699 0.946301 0.8926021.62 0.052616 0.947384 0.8947681.63 0.051551 0.948449 0.8968991.64 0.050503 0.949497 0.8989951.65 0.049471 0.950529 0.9010571.66 0.048457 0.951543 0.9030861.67 0.047460 0.952540 0.9050811.68 0.046479 0.953521 0.9070431.69 0.045514 0.954486 0.9089721.70 0.044565 0.955435 0.9108691.71 0.043633 0.956367 0.9127341.72 0.042716 0.957284 0.9145681.73 0.041815 0.958185 0.9163701.74 0.040929 0.959071 0.9181411.75 0.040059 0.959941 0.9198821.76 0.039204 0.960796 0.9215921.77 0.038364 0.961636 0.9232731.78 0.037538 0.962462 0.9249241.79 0.036727 0.963273 0.9265461.80 0.035930 0.964070 0.9281391.81 0.035148 0.964852 0.9297041.82 0.034379 0.965621 0.9312411.83 0.033625 0.966375 0.9327501.84 0.032884 0.967116 0.9342321.85 0.032157 0.967843 0.9356871.86 0.031443 0.968557 0.9371151.87 0.030742 0.969258 0.9385161.88 0.030054 0.969946 0.9398921.89 0.029379 0.970621 0.9412421.90 0.028716 0.971284 0.9425671.91 0.028067 0.971933 0.9438671.92 0.027429 0.972571 0.9451421.93 0.026803 0.973197 0.9463931.94 0.026190 0.973810 0.9476201.95 0.025588 0.974412 0.9488241.96 0.024998 0.975002 0.9500041.97 0.024419 0.975581 0.9511621.98 0.023852 0.976148 0.9522971.99 0.023295 0.976705 0.9534092.00 0.022750 0.977250 0.954500
- 46 -
u Φ (-u) Φ (u) D(u)2.01 0.022216 0.977784 0.9555692.02 0.021692 0.978308 0.9566172.03 0.021178 0.978822 0.9576442.04 0.020675 0.979325 0.9586502.05 0.020182 0.979818 0.9596362.06 0.019699 0.980301 0.9606022.07 0.019226 0.980774 0.9615482.08 0.018763 0.981237 0.9624752.09 0.018309 0.981691 0.9633822.10 0.017864 0.982136 0.9642712.11 0.017429 0.982571 0.9651422.12 0.017003 0.982997 0.9659942.13 0.016586 0.983414 0.9668292.14 0.016177 0.983823 0.9676452.15 0.015778 0.984222 0.9684452.16 0.015386 0.984614 0.9692272.17 0.015003 0.984997 0.9699932.18 0.014629 0.985371 0.9707432.19 0.014262 0.985738 0.9714762.20 0.013903 0.986097 0.9721932.21 0.013553 0.986447 0.9728952.22 0.013209 0.986791 0.9735812.23 0.012874 0.987126 0.9742532.24 0.012545 0.987455 0.9749092.25 0.012224 0.987776 0.9755512.26 0.011911 0.988089 0.9761792.27 0.011604 0.988396 0.9767922.28 0.011304 0.988696 0.9773922.29 0.011011 0.988989 0.9779792.30 0.010724 0.989276 0.9785522.31 0.010444 0.989556 0.9791122.32 0.010170 0.989830 0.9796592.33 0.009903 0.990097 0.9801942.34 0.009642 0.990358 0.9807162.35 0.009387 0.990613 0.9812272.36 0.009137 0.990863 0.9817252.37 0.008894 0.991106 0.9822122.38 0.008656 0.991344 0.9826872.39 0.008424 0.991576 0.9831522.40 0.008198 0.991802 0.9836052.41 0.007976 0.992024 0.9840472.42 0.007760 0.992240 0.9844792.43 0.007549 0.992451 0.9849012.44 0.007344 0.992656 0.9853132.45 0.007143 0.992857 0.9857142.46 0.006947 0.993053 0.9861062.47 0.006756 0.993244 0.9864892.48 0.006569 0.993431 0.9868622.49 0.006387 0.993613 0.9872262.50 0.006210 0.993790 0.987581
u Φ (-u) Φ (-u) D(u)2.51 0.006037 0.993963 0.9879272.52 0.005868 0.994132 0.9882642.53 0.005703 0.994297 0.9885942.54 0.005543 0.994457 0.9889152.55 0.005386 0.994614 0.9892282.56 0.005234 0.994766 0.9895332.57 0.005085 0.994915 0.9898302.58 0.004940 0.995060 0.9901202.59 0.004799 0.995201 0.9904022.60 0.004661 0.995339 0.9906782.61 0.004527 0.995473 0.9909462.62 0.004397 0.995603 0.9912072.63 0.004269 0.995731 0.9914612.64 0.004145 0.995855 0.9917092.65 0.004025 0.995975 0.9919512.66 0.003907 0.996093 0.9921862.67 0.003793 0.996207 0.9924152.68 0.003681 0.996319 0.9926382.69 0.003573 0.996427 0.9928552.70 0.003467 0.996533 0.9930662.71 0.003364 0.996636 0.9932722.72 0.003264 0.996736 0.9934722.73 0.003167 0.996833 0.9936662.74 0.003072 0.996928 0.9938562.75 0.002980 0.997020 0.9940402.76 0.002890 0.997110 0.9942202.77 0.002803 0.997197 0.9943942.78 0.002718 0.997282 0.9945642.79 0.002635 0.997365 0.9947292.80 0.002555 0.997445 0.9948902.81 0.002477 0.997523 0.9950462.82 0.002401 0.997599 0.9951982.83 0.002327 0.997673 0.9953452.84 0.002256 0.997744 0.9954892.85 0.002186 0.997814 0.9956282.86 0.002118 0.997882 0.9957632.87 0.002052 0.997948 0.9958952.88 0.001988 0.998012 0.9960232.89 0.001926 0.998074 0.9961472.90 0.001866 0.998134 0.9962682.91 0.001807 0.998193 0.9963862.92 0.001750 0.998250 0.9965002.93 0.001695 0.998305 0.9966102.94 0.001641 0.998359 0.9967182.95 0.001589 0.998411 0.9968222.96 0.001538 0.998462 0.9969232.97 0.001489 0.998511 0.9970222.98 0.001441 0.998559 0.9971172.99 0.001395 0.998605 0.9972103.00 0.001350 0.998650 0.997300
- 47 -
u Φ (-u) Φ (-u) D(u)3.01 0.001306 0.998694 0.9973873.02 0.001264 0.998736 0.9974723.03 0.001223 0.998777 0.9975543.04 0.001183 0.998817 0.9976343.05 0.001144 0.998856 0.9977113.06 0.001107 0.998893 0.9977863.07 0.001070 0.998930 0.9978593.08 0.001035 0.998965 0.9979303.09 0.001001 0.998999 0.9979983.10 0.000968 0.999032 0.9980653.11 0.000936 0.999064 0.9981293.12 0.000904 0.999096 0.9981913.13 0.000874 0.999126 0.9982523.14 0.000845 0.999155 0.9983103.15 0.000816 0.999184 0.9983673.16 0.000789 0.999211 0.9984223.17 0.000762 0.999238 0.9984753.18 0.000736 0.999264 0.9985273.19 0.000711 0.999289 0.9985773.20 0.000687 0.999313 0.9986263.21 0.000664 0.999336 0.9986733.22 0.000641 0.999359 0.9987183.23 0.000619 0.999381 0.9987623.24 0.000598 0.999402 0.9988053.25 0.000577 0.999423 0.9988463.26 0.000557 0.999443 0.9988863.27 0.000538 0.999462 0.9989243.28 0.000519 0.999481 0.9989623.29 0.000501 0.999499 0.9989983.30 0.000483 0.999517 0.9990333.31 0.000467 0.999533 0.9990673.32 0.000450 0.999550 0.9991003.33 0.000434 0.999566 0.9991313.34 0.000419 0.999581 0.9991623.35 0.000404 0.999596 0.9991923.36 0.000390 0.999610 0.9992203.37 0.000376 0.999624 0.9992483.38 0.000362 0.999638 0.9992753.39 0.000350 0.999650 0.9993013.40 0.000337 0.999663 0.9993263.41 0.000325 0.999675 0.9993503.42 0.000313 0.999687 0.9993743.43 0.000302 0.999698 0.9993963.44 0.000291 0.999709 0.9994183.45 0.000280 0.999720 0.9994393.46 0.000270 0.999730 0.9994603.47 0.000260 0.999740 0.9994793.48 0.000251 0.999749 0.9994983.49 0.000242 0.999758 0.9995173.50 0.000233 0.999767 0.999535
u Φ (-u) Φ (-u) D(u)3.51 0.000224 0.999776 0.9995523.52 0.000216 0.999784 0.9995683.53 0.000208 0.999792 0.9995843.54 0.000200 0.999800 0.9996003.55 0.000193 0.999807 0.9996153.56 0.000185 0.999815 0.9996293.57 0.000179 0.999821 0.9996433.58 0.000172 0.999828 0.9996563.59 0.000165 0.999835 0.9996693.60 0.000159 0.999841 0.9996823.61 0.000153 0.999847 0.9996943.62 0.000147 0.999853 0.9997053.63 0.000142 0.999858 0.9997173.64 0.000136 0.999864 0.9997273.65 0.000131 0.999869 0.9997383.66 0.000126 0.999874 0.9997483.67 0.000121 0.999879 0.9997573.68 0.000117 0.999883 0.9997673.69 0.000112 0.999888 0.9997763.70 0.000108 0.999892 0.9997843.71 0.000104 0.999896 0.9997933.72 0.000100 0.999900 0.9998013.73 0.000096 0.999904 0.9998083.74 0.000092 0.999908 0.9998163.75 0.000088 0.999912 0.9998233.76 0.000085 0.999915 0.9998303.77 0.000082 0.999918 0.9998373.78 0.000078 0.999922 0.9998433.79 0.000075 0.999925 0.9998493.80 0.000072 0.999928 0.9998553.81 0.000070 0.999930 0.9998613.82 0.000067 0.999933 0.9998673.83 0.000064 0.999936 0.9998723.84 0.000062 0.999938 0.9998773.85 0.000059 0.999941 0.9998823.86 0.000057 0.999943 0.9998873.87 0.000054 0.999946 0.9998913.88 0.000052 0.999948 0.9998963.89 0.000050 0.999950 0.9999003.90 0.000048 0.999952 0.9999043.91 0.000046 0.999954 0.9999083.92 0.000044 0.999956 0.9999113.93 0.000042 0.999958 0.9999153.94 0.000041 0.999959 0.9999183.95 0.000039 0.999961 0.9999223.96 0.000037 0.999963 0.9999253.97 0.000036 0.999964 0.9999283.98 0.000034 0.999966 0.9999313.99 0.000033 0.999967 0.9999344.00 0.000032 0.999968 0.999937
- 48 -
Index
Action characteristic 34Allowance 9, 22Assembly
dimension 5, 7, 13, 20, 22tolerance 7, 9, 22, 27
Average chart 34
Bell-shaped curve 41, 42
Central limit theorem 10, 11, 20Clearance 6, 13Contributory direction 5, 7, 8, 9Control chart 10, 33, 34Convolution 14, 35
Density function 12, 18, 19, 23Dimension
chain 5, 6, 13, 20, 35maximum 6, 7minimum 6, 7nominal 6
Distributionfunction 41, 42normal 10, 13, 21, 41rectangular 18, 21standard normal 42trapezoidal 17triangular 19, 21
Fraction nonconforming 24, 25
Gaussian bell-shaped curve 41
Interchangeability 7, 24
Law of error propagation 15, 16
Monte-Carlo-Methods 35
Probability 11density function 18, 41function 11, 12paper 41plot 41
Process capability index 20, 29Proportion nonconforming 21
Random variable 11, 12, 13Random variation range 24, 28Reduction factor 24, 25, 26, 27
Specification limit 29Statistical Process Control 33
Tolerance 22chain graph 5, 8stack 8zone 7, 17
Tolerancingarithmetic 7, 8statistical 10, 20, 35worst case 7
Variance 18, 19
Robert Bosch GmbH
C/QMM
Postfach 30 02 20
D-70442 Stuttgart
Germany
Phone +49 711 811-4 47 88
Fax +49 711 811-2 31 26
www.bosch.com
