SIST ISO 22514-6:2014

INTERNATIONAL ISO
STANDARD 22514-6
First edition
2013-02-15
Statistical methods in process
management — Capability and
performance —
Part 6:
Process capability statistics
for characteristics following a
multivariate normal distribution
Méthodes statistiques dans la gestion des processus — Capabilité et
performance —
Partie 6: Statistiques de capabilité pour un processus caractérisé par
une distribution normale multivariée
Reference number
©
ISO 2013
© ISO 2013
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ii © ISO 2013 – All rights reserved

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Abbreviated terms . 3
5 Process analysis . 4
6 Use of multivariate process capability and performance assessment .4
7 Calculation of process capability and process performance . 4
7.1 Description of Types I and II . 4
7.2 Designation and symbols of the indices . 5
7.3 Types Ιc and ΙΙc process capability index . 8
7.4 Types ΙΙa and Type ΙΙb process capability index .10
8 Examples .11
8.1 Two-dimensional position tolerances .11
8.2 Position and dimension of a slot .16
Annex A (informative) Derivation of formulae .20
Annex B (informative) Shaft imbalance example .25
Annex C (informative) Hole position example .29
Annex D (informative) Construction of the quality function .33
Bibliography .34
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
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ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International
Standards adopted by the technical committees are circulated to the member bodies for voting.
Publication as an International Standard requires approval by at least 75 % of the member bodies
casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 22514-6 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 4, Applications of statistical methods in process management.
ISO 22514 consists of the following parts, under the general title Statistical methods in process
management — Capability and performance:
— Part 1: General principles and concepts
— Part 2: Process capability and performance of time-dependent process models
— Part 3: Machine performance studies for measured data on discrete parts
— Part 4: Process capability estimates and performance measures [Technical Report]
— Part 5: Process capability statistics for attribute characteristics
— Part 6: Process capability statistics for characteristics following a multivariate normal distribution
— Part 7: Capability of measurement processes
— Part 8: Machine performance of a multi-state production process
iv © ISO 2013 – All rights reserved

Introduction
Due to the increased complexity of the production methods and the increasing quality requirements for
products and processes, a process analysis based on univariate quantities is in many cases not sufficient.
Instead, it may be necessary to analyse the process on the basis of multivariate product quantities. This
can, for instance, be in such cases where geometric tolerances, dynamic magnitudes such as imbalance,
correlated quantities of materials or other procedural products are observed.
By analogy with ISO 22514-2, ISO 22514-6 provides calculation formulae for process performance and
process capability indices, which take into consideration process dispersion as well as process location
as an extension to the corresponding indices for univariate quantities. The indices proposed are indeed
based on the classical C and C indices for the one-dimensional case. The motivation for the extension
p pk
to the multivariate case is explained in Annex A.
Examples of possible applications are two-dimensional or three-dimensional positions, imbalance or
several correlated quantities of chemical products.
The dispersion of the measuring results comprises the dispersion of the product realization process and
the precision of the measuring process. It is assumed that the capability of the used measuring system
was demonstrated prior to the determination of the capability of the product realization process.
The calculation method described here should be used to support an unambiguous decision, especially if
— limiting values for process capability indices for multivariate, continuous product quantities are
specified as part of a contract between customers and suppliers, or
— the capabilities of different constructions, production methods or suppliers are to be compared, or
— production processes are to be approved, or
— problems are to be analysed and decisions made in complaint cases or damage events.
NOTE Product realization processes include e.g. manufacturing processes, service processes, product
assembly processes.
INTERNATIONAL STANDARD ISO 22514-6:2013(E)
Statistical methods in process management — Capability
and performance —
Part 6:
Process capability statistics for characteristics following a
multivariate normal distribution
1 Scope
This part of ISO 22514 provides methods for calculating performance and capability statistics for process
or product quantities where it is necessary or beneficial to consider a family of singular quantities in
relation to each other. The methods provided here mostly are designed to describe quantities that follow
a bivariate normal distribution.
NOTE In principle, this part of ISO 22514 can be used for multivariate cases.
This part of ISO 22514 does not offer an evaluation of the different provided methods with respect to
different situations of possible application of each method. For the current state, the selection of one
preferable method might be done following the users preferences.
The purpose is to give definitions for different approaches of index calculation for performance and
capability in the case of a multiple process or product quantity description.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 22514-1, Statistical methods in process management — Capability and performance — Part 1: General
principles and concepts
ISO 22514-2, Statistical methods in process management — Capability and performance — Part 2: Process
capability and performance of time-dependent process models
3 Terms and definitions
For the purpose of this document, the terms and definitions given in ISO 22514-1 and ISO 22514-2 and
the following apply.
3.1
quantity
property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed
as a number and a reference
[ISO/IEC Guide 99:2007, 1.1]
3.2
multivariate quantity
set of distinguishing features
Note 1 to entry: The set can be expressed by a d-tuple, i.e. an ordered set consisting of d elements.
Note 2 to entry: If the single quantities in the set are denoted by x where i = 1, 2…d, the multivariate quantity is
i
T
expressed as the vector x = (x , x , … x ) . Thus, a multivariate quantity can be considered as a feature vector of
1 2 d
a product. The value of the multivariate quantity is represented by a point in the d-dimensional feature space.
Note 3 to entry: The selection of the quantities in a vector is made for specific technical reason.
Note 4 to entry: All single quantities combined in the vector of a multivariate must be measurable in the same
product or object.
Note 5 to entry: If the multivariate quantity is to be described by means of statistics, the vector is to be considered
as a random vector following a d-dimensional multivariate distribution.
EXAMPLE 1 A number of d = 3 quantities like x = colour, x = mass and x = number of defects are combined in
1 2 3
order to use only one statistic for process assessment. The dimension of vector x is d = 3.
EXAMPLE 2 In order to evaluate a boring process, the position of the borehole axis is measured in an
x-coordinate and y-coordinate. The coordinates are combined to the two-dimensional multivariate quantity x
where the component x is the x-coordinate and x is the y-coordinate.
1 2
EXAMPLE 3 Imbalance of a wheel.
3.3
tolerance region
region in the feature space that contains all permitted values of the multivariate quantity (3.2)
Note 1 to entry: The region is limited by lines, surfaces or hyper-surfaces in the d-dimensional space and not
necessarily closed. The form and extension of the region are specified by one or more parameters.
Note 2 to entry: Typical shapes of tolerance regions are rectangles, ellipses (or circles) in the two-dimensional
case, cuboids or hyper-cuboids, ellipsoids or hyper-ellipsoids or composite prismatic shapes. Figure 1 shows
examples of tolerance regions in the two-dimensional space.
Note 3 to entry: The tolerance region is specified based on the required function of the product. Products showing
values outside the region are assumed to not fulfil functional requirements. Those products are considered to be
nonconforming parts.
Note 4 to entry: In order to assess a product with respect to the limits of the tolerance region, the order of the
single quantity in the multivariate quantity and the number d of dimension must be equal to that of the tolerance
region description.
EXAMPLE A tolerance zone as it is defined in ISO 1101 for geometrical product features can be considered as
a tolerance region. In that case, limiting geometrically perfect lines or surfaces correspond to the boundary and
the tolerance correspond to the parameter of the tolerance region.
2 © ISO 2013 – All rights reserved

A) B) C)
a a
d
1 1
x x
x
X
X
Key
A rectangular tolerance region with parameters a , a , x and y
1 2
B circular tolerance region with parameters d, x and y
C triangularly extended rectangular region with parameters a , a , b, x and y
1 2
Figure 1 — Examples of tolerance regions in the two-dimensional space of the bivariate
T
quantity (x , x )
1 2
3.4
process capability
distribution of measured quantity (3.1) values from a process that has been demonstrated to be in
statistical control and which describes the ability of a process to produce quantity values that will fulfil
the requirements for that quantity
Note 1 to entry: The process capability index provides the ability to meet requirements of the measured quantity.
Note 2 to entry: The abbreviation for process capability index is PCI.
3.5
estimated process capability
statistical description of a process capability (3.4)
3.6
process performance
distribution of measured quantity (3.1) values from a process
Note 1 to entry: The process may not have been demonstrated to be in statistical control.
3.7
estimated process performance
statistical description of a process performance (3.6)
4 Abbreviated terms
MMC maximum material condition
PCI process capability index
a
y
y
y
a
b
5 Process analysis
The purpose of process analysis is to obtain sound knowledge of a process. This knowledge is necessary for
controlling the process efficiently, so that the products realized by the process fulfil the quality requirement.
A process analysis is always an analysis of one or more quantities that are considered to be important
to the process.
Product quantities can often be analysed instead of process quantities because product quantities not
only characterize the products, but due to their correlation with process quantities they also characterize
the process creating these products.
The values of the quantities under consideration are typically determined on the basis of samples
taken from the process flow. The sample size and frequency should be chosen depending on the type of
process and the type of product so that all important changes are detected in time. The samples should
be representative for the multivariate quantities under consideration. (Univariate quantity values are
considered in ISO 22514-2.) This part of ISO 22514 describes multivariate capability statistics.
To estimate the PCI, the sample size should preferably be at least 125.
6 Use of multivariate process capability and performance assessment
The purpose of a process capability index is to reflect how well or how badly a process generates qualified
products. The use of PCI for multivariate quantities should reflect this process behaviour better than
PCI for single quantities would. Since a variety of multivariate PCI definitions exists, the selection of a
specific definition to be used will remain in the user’s accountability. However, the following guidance is
given as to when a multivariate PCI should be preferred at all.
A multivariate assessment of process capability and performance is suitable if at least one of the
following circumstances is applicable.
— It is found to be advantageous to describe process capability and performance with only one
comprehensive statistic instead of a high number of single statistics for each product quality quantity.
— The boundary of the tolerance region cannot be expressed independently for all quantities, i.e. at
least one tolerance limit for one quantity is a function of another quantity. This is the case if the
tolerance region is not of rectangular or cuboid shape.
— The single quantities that could be combined to a multivariate one appear to be correlated
among each other.
EXAMPLE In the case of a two-dimensional position tolerance for a borehole axis, the tolerance region is a
circle with defined distances in an x- and y-coordinate direction from the references, see 8.1. The result of the hole
axis measurement will be a value for the x- and y-coordinate. The tolerance limit for the x-coordinate cannot be
expressed independently from the y-coordinate. Thus, a bivariate assessment is to be applied.
7 Calculation of process capability and process performance
7.1 Description of Types I and II
In the multivariate domain, different approaches exist for measuring process capability and process
performance. This part of ISO 22514 describes examples of two different types of indices: Type Ι and
Type ΙΙ. The distinction between the types is based on whether the index is defined based on probability
or defined geometrically by relating the area or volume of a tolerance or process region.
The following description of the types applies:
— Type Ι Based on the probability of conforming or non-conforming products P, the index is calculated
using the relationship between the index and the said probability in a univariate normal case.
4 © ISO 2013 – All rights reserved

— Type ΙΙ The index is calculated as the ratio of the area or volume of the tolerance region to the
area or volume of the region covered by the process variation.
For practical reasons, the multivariate normal distribution mode has been chosen for the calculation
of the statistics which are described in this clause. However, the choice of normal distribution does not
exclude that in special cases other model distributions will describe the reality better. Also, for practical
reasons, in this part of ISO 22514, the process variation region has been chosen to be of ellipsoid shape.
The most important properties of the multivariate normal distribution are explained in Annex A.
Because of that choice, additional transformations should be applied to make the shape of the process
variation intervals comparable to the shape of the tolerance region. Thus, three further principles are to
be distinguished. These are the principles of transforming the shape of the
a) tolerance region into the shape of the process variation interval,
b) process variation interval into the shape of the tolerance region, and
c) tolerance region and/or the process variation into a new function-oriented dimension.
Both the above-mentioned types and the principles can be combined to define a multivariate PCI. Each
combination, however, may not be useful. There is, for instance, no known definition of a type Ιb PCI.
The term “capability” can only be used for processes that have been demonstrated to be in statistical
control using control charts. In the multivariate case, the distinction between special and common
causes is usually more difficult than in the univariate case. If the process has not been demonstrated to
be in statistical control, the term “performance” is used in this part of ISO 22514.
7.2 Designation and symbols of the indices
7.2.1 General
Different symbols are currently used for multivariate index definitions in industry and science. Currently
used symbols try to distinguish between the types of calculation or to specify their use. This part of
ISO 22514 uses the designation C and/or C for basic definitions of calculation. Furthermore, it will
p pk
be distinguished between process capability and process performance in applying the indices by using
capitals “C” for capability and “P” for performance.
7.2.2 Process capability index
Consider a d-dimensional normal distribution N (μ, Σ) with the mean vector μ and covariance matrix Σ.
d
If the tolerance region is not of elliptic shape (circle, ellipse if d = 2 or sphere, ellipsoid if d = 3 or hyper-
sphere, hyper-ellipsoid if d > 3), it is to be transformed into a modified tolerance region that is of elliptic
shape. This is to be done by determining the largest ellipse (or ellipsoid, hyper-ellipsoid) that is centred
at the target and completely fits into the original tolerance region.
In order to calculate the multivariate C index, the normal distribution shall be centred to have the mean
p
at the centre of the elliptic tolerance region. For that normal distribution, determine the largest contour
ellipsoid that is completely contained in the elliptic tolerance region and calculate the probability of the
volume bounded by that contour ellipse under the d-dimensional normal distribution with covariance
matrix Σ and mean at the centre of the elliptic tolerance region. Denote that probability by P. Then, the
multivariate C index is
p
1 P+1
 
−1
C = Φ
p  
3 2
 
The calculation of P, the probability for observations of x within the determined contour ellipse (ellipsoid/
hyper-ellipsoid) for any d can be done by using the relation to the F-distribution. The explanation is
given in Clause A.1.
In order to estimate a C index from d-dimensional data, start by estimating the covariance matrix of the
p
ˆ
multivariate normal distribution from the data. Denote the estimate by Σ and use that covariance matrix
ˆ
to determine the contour ellipsoid and its probability P . Finally, the estimated multivariate C index is
p
ˆ
 
1 P +1
−1
ˆ
C = Φ
 
p
3 2
 
L U
10 U
−5
L
−10
−10 −5 0 5 10
X-coordinate
Key
1 elliptic tolerance zone
2 contour ellipse used for the calculation of the capability index
3 contour ellipse corresponding to the probability zone 99,73 %
Figure 2 — Contour ellipse and tolerance zone used to calculate the capability index for d = 2
In Figure 1, the contour ellipse with probability 99,73 % is completely contained in the contour ellipse
used for the calculation of the index. When this is the case, the index will be larger than 1.
We use the symbol C for this index as for the classical capability index for the univariate normal
p
distribution. The reason is that this calculation method in the one-dimensional case gives the classical
C index. This is explained in Clause A.1.
p
6 © ISO 2013 – All rights reserved
Y-coordinate
7.2.3 Minimum process capability index
–
L X U
10 U




 










 –



 Y 2



 

 


 






0 






−5
−10 L
−10 −5 0 5 10
X-coordinate
Key
1 elliptic tolerance zone
2 contour ellipse used for the calculation of the C index
pk
3 contour ellipse corresponding to the probability 99,73 %
Figure 3 — Tolerance zone and contour ellipse used to calculate the capability index for d = 2
Calculation of the C index involves both the mean and the variance of the distribution, so consider
pk
again a d-dimensional normal distribution with mean μ and covariance matrix Σ. For the N (μ,Σ)
d
distribution, calculate
— the largest contour ellipse (ellipsoid, hyper-ellipsoid) that is completely contained in the elliptic
tolerance region, if μ is contained in the tolerance region, or
— the largest contour ellipse (ellipsoid, hyper-ellipsoid) that is not contained in the tolerance region, if
μ is not contained in the tolerance zone.
Now, the probability, P, of the area (volume) contained in the contour ellipse (ellipsoid, hyper-ellipsoid)
under the N (μ,Σ) distribution is calculated. Finally, the C index is calculated as
d pk
1 P+1
−1
C = Φ
pk  
3 2
 
if μ is in the tolerance region and as
1 1−P
 
−1
C = Φ
pk
 
3 2
 
if μ is not in the tolerance region.
Y-coordinate
We use the same symbol as for the classical C index for the one-dimensional normal distribution. The
pk
reason is that this method of calculation gives the classical index in the one-dimensional case. This is
explained in Clause A.1.
NOTE The described Type Ia finds applications in geometrical dimensioning and tolerancing of position
deviations. Here, the tolerance region usually describes a circular tolerance zone. The symbols often used in that
case are C and C for C and C respectively.
Po Pok p pk
7.3 Types Ιc and ΙΙc process capability index
Type Ιc as well as Type ΙΙc capability indices are characterized by a function-oriented transformation
of the multiple feature characterization into a single feature characterization. By that type, the
multivariate aspect is expressed in the definition of the transforming function q(x), where x describes
the multivariate quantity. This transformation shall represent the functional importance of the single
quantities in x and their interplay. For example, it describes a model for the tolerance region and can be
interpreted as a weighing function, e.g. a loss function or a quantification function that quantifies the
technical functionality.
The calculation of Type Ιc and ΙΙc indices follows four steps; see Figure 4.
Modeling Sampling parts Estimating the Calculating the
tolerance and trans- univariate PCI
region forming data distribution (Type I or II)
Figure 4 — Steps for calculating the type Ιc or ΙΙc process capability indices
The first step concerns the definition of the technical qualification function q(x) over the d-dimensional
tolerance region. This function has a maximum with the value q at the target in the tolerance region.
max
At the boundary of the tolerance region, the q(x) has the value q . In some cases, q and q
bound max bound
can be derived from the technical context of all single quantities in x. In other cases, appropriate values
are q = 1 and q = 0,5. The function q(x) may be expressed in a closed equation or piecewise
max bound
composed. An example for a piecewise composed linear function is given in Figure 5.
8 © ISO 2013 – All rights reserved

Dimenstions in millimetres
20 ±0,2
⊕ 0.1 M A
(d)(d)
Y
20,3 0,33
0,5
20,2
0,67


20,1 
0,83 


2 


















20,0 

0,95
A 0,83
19,9
0,67
0,5
19,8
0,33
0,17
19,7 3
19,6
0,00,1 0,20,3 0,40,5
X
Key
X position
Y width
1 q - contour; limits of the tolerance region
bound
2 q = 1, target
max
3 q(x) contour lines
Figure 5 — Example of a qualification function for a width/symmetry tolerance region under MMC
In Figure 5, the multivariate quantity consists of two geometrical features: width and position. The
tolerance region by utilizing the maximum-material-condition is of composite rectangular and triangular
shape. Further explanations about the example are given in Clause 8. The target where q = 1 is situated
max
at the nominal values. From that point, the qualification function is decreasing towards the tolerance
limits. Thereby different trends may be defined: linear, exponential and others. In Figure 5, the function
q(x) is composed of three linear functions. Completely different constructions of q(x) are also possible.
In the second step, produced parts are to be sampled and measured. The measured values then are to be
complied to the multivariate quantity x and transformed by q(x) into function-related qualification values.
In the third step, based on these values, an appropriate univariate distribution function F(q) is to
be identified. Alternatively, a second transformation to univariate normal may be carried out. If the
qualification function q(x) is monotonically increasing from the boundary to the target and the random
vector X follows a multivariate normal, the distribution density of F(q) will be unimodal.
In the fourth step, based on the identified distribution of the qualification values, the transformed
target and the specification limits, the PCI is calculated. If q = 1 and q = 0,5 are chosen, 0,5 gives
max bound
the lower specification limit and 1 is the upper natural limit. Since q offers only a one-sided tolerance
interval and a one-sided limited distribution, one might only use C for a process assessment.
pk
If there is an Ι-type index, C is calculated analogously to 7.2.3. The probability P is given from the
pk
univariate distribution F(q) by P = 1 – F(q ).
bound
If a Type ΙΙ index is to be calculated, the methods in ISO 22514-2 can be followed. Based on the definition
for C , by choosing X as the median of F(q), choosing L as q and ΔL = X – X , the
pkL mid bound mid 0,135 %
equation for C is given by:
pk
qq−
50% bound
C =
pk
qq−
50%,0135%
where q describe the x % - percentiles of F(q).
x%
Since the C carries information about both variation and location of the process in relation to the
pk
tolerance limit q , one may require an index only for the information about variation. A C can be
bound p
calculated applying the methods in ISO 22514-2. But since the target value of q is the maximum value,
C may be lower than C .
p pk
If, in step 1, a loss function l(x) is defined instead of a qualification function, the function will have a
minimum l at the functional target and a value l at the limits of the tolerance region. Although
min bound
it is not a loss function by the original meaning, the distances D of borehole axes from a target position
can be interpreted as a l(x) - function. It has a minimum of l = 0 at the target and l at the half of
min bound
the position tolerance. The tolerance region is a circle.
7.4 Types ΙΙa and Type ΙΙb process capability index
7.4.1 General
Multivariate process capability indices of Type ΙΙ follow the principle of putting in relation an extent
of the tolerance region to an extent of the process variation. These extents are expressed in areas or
volumes. The area or volume of the tolerance region is denoted by V and V denotes the area or
tol proc
volume of the process variation region. Thus, for the index, the following can be defined:
a
 
V
tol
C =
 
p
V
 proc 
 
The exponent a is introduced to give the possibility of reducing the area or volume back to one dimension.
Thus, a is normally a = 1/d. Otherwise, a is set to a = 1.
In order to make the areas or volumes comparable, a transformation of the shape of the regions may
be necessary. Type IIa indices transform the original tolerance region into a modified tolerance region
that is of the shape of the process variation region (e.g. elliptic/ ellipsoid/ hyper-ellipsoid in the case
of a multivariate normal distribution). For Type IIb indices, a transformation is made for the process
variation region. In this case, the shape of the process variation region is adapted to the shape of the
[9]
tolerance region. Comparisons between IIa and IIb are given in Reference.
Since this index only gives information about the process variance in relation to tolerance, the index
may be supported by one or more indices giving information about the location of the mean vector μ in
relation to the target.
7.4.2 Type ΙΙa
The modified tolerance region is again defined as the largest ellipse (or ellipsoid, hyper-ellipsoid) that
is centred at the target and completely contained within the original tolerance region; see Figure 2 and
10 © ISO 2013 – All rights reserved

Figure 3. In the case of a tolerance region where the parameters x , x … x denote the half distance
t1 t2 td
from the tolerance limit to the target, the volume V is given by:
tol
d
d/2
π
V = ⋅ x
tolt∏ i
 d 
Γ 1+ i=1
 
 
For estimating the C , the volume of the 99,73 % ellipsoid is to be estimated by:
p
22d/
()π⋅χ
99,73
V = ⋅ S
proc
d
 
Γ 1+
 
 
S is the determinant of S (Annex A). This index is supported by the value of 1/D to give an information
about the process location in relation to the target µ . D is estimated as:
T
n
−1
ˆ
D =+1 xS− μμμx − μ
() ()
0 0
n−1
Both are combined to a C value that is estimated as
pm
V
tol
ˆ
C =⋅
pm
ˆ
V D
proc
[7]
Examples for this PCI are described in Reference.
7.4.3 Type ΙΙb
[8]
A Type ΙΙb PCI is defined by Reference. The shape of the process variation interval, which is elliptic is
transformed to the shape of the tolerance region. In the case of a rectangular (cuboid or hyper-cuboid)
tolerance region, this is the smallest rectangle (cuboid, hyper-cuboid) that enfolds given ellipse (ellipsoid,
hyper-ellipsoid) Based on the projection intervals of the ellipse (ellipsoid, hyper-ellipsoid) of each one
dimension L and U, the PCI is defined.
[8]
Examples for this PCI are described in Reference.
8 Examples
8.1 Two-dimensional position tolerances
8.1.1 Type Ιa Index
On a produced part, the midpoint of a drilled hole is measured. The nominal value is 80 mm in the X
direction and –116,5 mm in the Y direction as shown in Figure 6. The diameter of hole is given as Ø 50
with a tolerance of ± 0,05 mm. Information on geometrical tolerances is given in ISO 1101.
A
Ø50 ±0,05
⊕
Ø0,5 AB
Figure 6 — Measurement task is position of a hole
One hundred sets of values from produced parts were measured (see Table 1).
The X- and Y- values were plotted in Figure 8 and the variation interval calculated. The method used to
calculate the interval can be found in Annex A.
Number of measured parts n = 100
Specification limits: L = 79,750 U = 80,250
X X
L = −116,750 U = −116,250
Y Y
12 © ISO 2013 – All rights reserved
116,5
B
Table 1 — Measured values and a calculated deviation
Nr. dev. D X-coord. Y-coord. Nr. dev. D X-coord. Y-coord. Nr. Dev D X-coord. Y-coord.
1 0,038 79,976 −116,470 36 0,090 79,995 −116,410 71 0,107 79,986 −116,394
2 0,094 79,993 −116,406 37 0,097 80,002 −116,403 72 0,073 80,016 −116,429
3 0,086 80,031 −116,420 38 0,113 80,027 −116,390 73 0,069 79,995 −116,431
4 0,041 79,968 −116,475 39 0,021 79,995 −116,520 74 0,108 79,975 −116,395
5 0,105 79,973 −116,399 40 0,085 80,010 −116,416 75 0,118 79,965 −116,387
6 0,092 79,983 −116,410 41 0,110 80,005 −116,390 76 0,122 79,971 −116,382
7 0,099 80,008 −116,401 42 0,081 80,004 −116,419 77 0,119 79,978 −116,383
8 0,086 80,014 −116,415 43 0,055 79,966 −116,457 78 0,118 79,999 −116,382
9 0,075 80,020 −116,428 44 0,097 80,013 −116,404 79 0,024 80,008 −116,477
10 0,076 79,979 −116,427 45 0,078 80,021 −116,425 80 0,094 80,005 −116,406
11 0,064 79,978 −116,440 46 0,118 79,989 −116,383 81 0,056 80,007 −116,444
12 0,086 80,016 −116,416 47 0,111 79,988 −116,390 82 0,093 80,032 −116,413
13 0,067 79,990 −116,434 48 0,057 79,987 −116,445 83 0,139 79,958 −116,368
14 0,120 79,992 −116,380 49 0,101 80,012 −116,400 84 0,122 79,990 −116,378
15 0,103 79,999 −116,397 50 0,067 80,017 −116,435 85 0,126 79,994 −116,374
16 0,119 80,016 −116,382 51 0,099 80,000 −116,401 86 0,089 80,029 −116,416
17 0,086 80,038 −116,423 52 0,101 79,995 −116,399 87 0,110 80,000 −116,390
18 0,118 80,018 −116,383 53 0,139 79,999 −116,361 88 0,084 80,010 −116,417
19 0,116 80,005 −116,384 54 0,086 80,002 −116,414 89 0,121 80,000 −116,379
20 0,118 80,071 −116,406 55 0,095 80,068 −116,433 90 0,131 79,992 −116,369
21 0,072 79,941 −116,458 56 0,103 79,990 −116,397 91 0,122 79,992 −116,378
22 0,097 79,984 −116,404 57 0,178 80,035 −116,325 92 0,062 79,990 −116,439
23 0,029 79,986 −116,475 58 0,107 79,980 −116,395 93 0,098 79,999 −116,402
24 0,093 80,043 −116,418 59 0,182 79,978 −116,319 94 0,086 79,986 −116,415
25 0,047 80,027 −116,538 60 0,099 80,000 −116,401 95 0,097 79,986 −116,404
26 0,090 80,031 −116,415 61 0,080 79,995 −116,420 96 0,092 80,020 −116,410
27 0,097 80,005 −116,403 62 0,133 79,996 −116,367 97 0,095 79,984 −116,406
28 0,122 80,024 −116,380 63 0,088 80,000 −116,412 98 0,133 79,980 −116,369
29 0,081 80,040 −116,430 64 0,107 79,948 −116,406 99 0,132 79,981 −116,369
30 0,094 80,006 −116,406 65 0,101 80,015 −116,400 100 0,058 80,033 −116,452
31 0,099 79,986 −116,402 66 0,081 79,990 −116,420
32 0,094 79,982 −116,408 67 0,087 80,009 −116,413
33 0,111 79,942 −116,405 68 0,067 80,004 −116,433
34 0,135 79,975 −116,367 69 0,130 79,960 −116,376
35 0,103 80,014 −116,398 70 0,121 80,007 −116,379
Based on the values in Table 1, two sets of control charts were constructed.
The control chart for x- and y- coordinates show a process out-of-control. Therefore, only the process
performance can be calculated.
−116,36
80,03
UCL UCL
−116,38
80,02
80,01
−116,40
=
tar x
=
80,00
tar x
−116,42
79,99
−116,44
79,98 LCL
−116,46
79,97
LCL
05 10 15 20 05 10 15 20
UCL
UCL
0,10
0,12
0,08
0,08
0,06 –
–
tar R
tar R
0,04
0,04
0,02
LCL
0,00
0,00
Figure 7 — X and R control charts for X-coordinate and Y-coordinate
–
L x U
U
−116,3













 


 –
−116,4 

  

 y

















−116,5


−116,6
−116,7
L
79,8 79,9 80,0 80,1 80,2
X-coordinate [mm]
Figure 8 — Graphical presentation of the position tolerances with reference region (3) and
specified tolerance
Results:
ˆ
Process performance index:    P =24, 3
p
ˆ
Minimum performance index: P =14, 8
pk
ˆ ˆ
The limits for the 95 % confidence interval for P are calculated to P = 1,99 and P = 2,88.
p
p,low p,up
For P :
pk
ˆ
ˆ
= 1,19 and P = 1,48
P
pk,up
pk,low
14 © ISO 2013 – All rights reserved
X-coordinates [mm]
Y-coordinate [mm]
Y-coordinates [mm]
8.1.2 Type ΙΙc index by calculating the capability index using the distance from target
The target location (,xy ) = (80, −116,5) is specified for the centre of the hole in Figure 6. The location
of each (,xy) is measured. The coordinates (,xy) denote the centre of the hole drilled. The deviation
from the target location is
2 2
Dx=−()xy+−()y
0 0
The actual calculated values for the distances D can be found in Table 1.
All deviations are plotted in a histogram shown in Figure 9. The maximum permissible deviation is
0,25 mm, because the tolerance zone is a circle centred on the target and with a diameter of 0,5 mm. The
maximum permissible deviation is the radius of that circle.
– –
L x U L x U
30 30 99,999   0,001
99,99 0,01
25 25
99,9 0,1
0,5
20 20





15 15 5








 15



80 



10 10 

 30



60 



 50






30  80

5 5 






0 
0,00 0,05 0,10 0,15 0,20 0,25
Target [mm] FND [<>0] →
0,00 0,05 0,10 0,15 0,20 0,25
Target [mm] FND [<>0] →
Figure 9 — Histogram and probability plot
The distribution model for the actual data set will be a Rayleigh distribution if the production is centred
around the target. However, in this special case where all the values are above the target, the normal
distribution fits well.
Absolute frequency →
Relative frequency [%] →
P [%] →
← 1 − P [%]
0,14
UCL
0,12
0,10 =
tar x
0,08
LCL
0,06
05 10 15 20
UCL
0,12
0,08
–
tar R
0,04
0,00
Figure 10 — Control chart
The data do not show stability in the control chart (see Figure 10). In such cases, only a probability index
P can be calculated.
pk
Calculation of the capability indices:
— The capability index cannot be calculated because of no lower specification limit exist.
Minimum capability index:

UD−
02,,50− 096
50%
C = = =18, 1
PoKu
DD− 0,,181−0096
99,865% 50%
8.2 Position and dimension of a slot
A slot is to be placed in a part as shown in Figure 5. The technical function is to carry a second part in
a specific position. The width of the groove is 20 ±0,2 mm. To ensure correct fitting of the parts, the
position tolerance for the slot in relation to “A” is given as 0,1 mm when the dimension of the slot is at
Ⓜ
its maximum material size (19,8 mm). This maximum material condition is indicated by the symbol .
As a result, a part can still be considered acceptable if the deviation of the slot position does not exceed
the value of 0,1 mm plus the difference between the actual width of the slot and the maximum material
size. The tolerance region in the situation where the maximum material condition shall not be applied is
given by the rectangle in the chart of Figure 5. The extended tolerance because of MMC is shown by the
right triangular extension of that rectangle.
In the first step, q(x) is defined for the multivariate quantity x where x = width and x = position. The
1 2
function chosen is composed of three linear functions q , i = 1, 2, 3 of the type q = a x +a x +a . Their
i i 1i 1 2i 2 0i
coefficients are set to appropriate values in order to give values of q = 1 in the target and q = 0,5
max bound
at the tolerance limits.
16 © ISO 2013 – All rights reserved
Target [mm]
Table 2 — Measured values and the calculated qualification values
Width Position Width Position Width Position
No. q No. q No. q
mm mm mm mm mm mm
1 0,744 20,102 0,06 18 0,828 20,069 0,09 35 0,862 20,033 0,116
2 0,845 20,062 0,11 19 0,899 20,04 0,091 36 0,879 20,027 0,1
3 0,858 20,016 0,102 20 0,807 20,007 0,123 37 0,671 20,131 0,074
4 0,846 20,035 0,127 21 0,838 20,065 0,083 38 0,865 20,026 0,107
5 0,829 20,068 0,075 22 0,781 20,087 0,071 39 0,897 20,035 0,097
6 0,906 20,038 0,091 23 0,866 20,053 0,09 40 0,777 20,001 0,135
7 0,763 20,002 0,144 24 0,888 20,045 0,096 41 0,771 20,009 0,146
8 0,789 20,084 0,074 25 0,897 20,041 0,102 42 0,823 20,026 0,132
9 0,798 20,001 0,122 26 0,768 20,093 0,084 43 0,84 20,012 0,108
10 0,797 20,081 0,069 27 0,824 20,07 0,084 44 0,832 20,01 0,111
11 0,841 20,063 0,09 28 0,85 20,06 0,116 45 0,875 20,05
...