qs-STAT Evaluation Strategy - Q-DAS

qs-STAT Evaluation Strategy

Q-DAS Process Capability (06/2013)

qs-STAT Evaluation Strategy 1/73

Version: A © 2015 Q-DAS GmbH & Co. KG, 69469 Weinheim qs_STAT_ES_PC_ea.docx

Content 1 General ............................................................................................................................................ 4

1.1 Basic Information about the Evaluation Strategy ................................................................... 4

1.2 Functional Design of the Evaluation Strategy ......................................................................... 5

1.3 Overview: Options for Describing Mathematical Functions ................................................... 7

2 Description of the Single Options .................................................................................................... 8

2.1 Preparation .............................................................................................................................. 8

2.1.1 Takeover Tab ................................................................................................................... 8

2.1.2 Classification Tab ............................................................................................................. 8

2.1.3 Outliers Tab ..................................................................................................................... 9

2.1.3.1 Plausibility Limits ....................................................................................................... 10

2.1.3.2 Natural Boundaries .................................................................................................... 10

2.1.3.3 Procedure with Incomplete Subgroups ..................................................................... 11

2.1.4 Positional Tolerances Tab .............................................................................................. 11

2.1.4.1 What Is a Position? .................................................................................................... 12

2.1.4.2 Calculation of the True Position Value ...................................................................... 12

2.1.4.3 Creation of True Position Tolerance .......................................................................... 13

2.1.5 General Tab ................................................................................................................... 14

2.1.5.1 Study Type ................................................................................................................. 14

2.1.5.2 Trend Compensation ................................................................................................. 15

2.1.5.3 Test Procedures ......................................................................................................... 16

2.2 “Variation Constant?“ – Test for Fluctuations of Variation .................................................. 18

2.3 “Location Constant?“ – Test for Fluctuation of Averages ..................................................... 19

2.4 “Defined Distribution by Measuring Unit?“ – Specified Distribution .................................... 20

2.5 Select Distribution ................................................................................................................. 21

2.6 Default ND/WD ...................................................................................................................... 21

2.6.1 Selection of Distribution – General Options .................................................................. 22

2.6.1.1 Distributions with Offset ........................................................................................... 22

2.6.1.2 Best Possible Distribution .......................................................................................... 23

2.7 Test for Distribution / Test for Normality .............................................................................. 24

2.7.1 Normal Distribution Tests.............................................................................................. 25

2.7.2 Arbitrary Distribution Tests ........................................................................................... 26

2.8 Find Distribution Model ........................................................................................................ 26

2.8.1 Distributions Tab ........................................................................................................... 27

2.8.2 General Options Tab ...................................................................................................... 27

2.8.2.1 Distributions with Offset ........................................................................................... 28

2.8.2.2 Best Possible Distribution .......................................................................................... 28



2.9 “Momentary Normal Distribution?“ ..................................................................................... 29

3 Distribution Time Models in Accordance with ISO 21 747 ............................................................ 30

3.1 Distribution Time Models A1 and A1* .................................................................................. 35

3.1.1 Analysis Chart for Location: Shewhart Average Chart (x-chart) .................................... 35

3.1.1.1 Calculation Method for Control Limits in 𝒙-chart ..................................................... 36

3.1.1.2 Stability Criteria for Shewhart 𝒙-chart (Analysis Chart) ............................................ 36

3.1.2 Analysis Chart for Variation: Shewhart s-chart ............................................................. 36

3.1.2.1 Calculation Method for Control Limits in Shewhart s-chart ...................................... 37

3.1.2.2 Stability Criterion for the Shewhart s-chart (Analysis Chart) .................................... 37

3.1.3 Distribution Time Models A1 and A1* – Stable/Not Stable .......................................... 38

3.1.3.1 Distribution Time Models A1 and A1* – Q-QIS ......................................................... 39

3.2 Distribution Time Models A2 and A2* .................................................................................. 40

3.2.1 Analysis Chart for Location: Johnson Average Chart ..................................................... 41

3.2.1.1 Calculation Method for Control Limits in Johnson Chart .......................................... 41

3.2.1.2 Stability Criteria of the Johnson Chart (or Pearson Chart) ........................................ 43


3.2.2.1 Calculation Method for Control Limits in the s-chart ................................................ 43

3.2.2.2 Stability Criteria for the Shewhart s-chart (Analysis Chart)....................................... 44

3.3 Distribution Time Models C2,4, B and D ................................................................................. 44

3.3.1 Analysis Chart for Location: Extended Shewhart Chart ................................................ 44

3.3.1.1 Calculating the External Standard Deviation 𝝈𝑨 ....................................................... 45

3.3.1.2 Calculating Control Limits .......................................................................................... 45

3.3.1.3 Stability Criteria of the Extended Shewhart Average Chart ...................................... 46


3.3.2.1 Calculation Method for Control Limits in the s-chart ................................................ 47

3.3.2.2 Stability Criteria of the Shewhart s-chart .................................................................. 47

4 Requirements ................................................................................................................................ 48

4.1 Requirements Variable Characteristics ................................................................................. 48

4.1.1 Target Values / QCC Stable Tab ..................................................................................... 48

4.1.1.1 Capability Indices ....................................................................................................... 49

4.1.1.2 Preliminary Capability Indices ................................................................................... 49

4.1.1.3 Warning Limits for Insufficient Number of Values (Limits for Conditionally Capable)

49

4.1.1.4 Confidence Intervals for the Potential Capability Index Cp ....................................... 49

4.1.1.5 Confidence Interval for the Critical Capability Index Cpk ........................................... 50

4.1.2 Target Values / QCC Unstable Tab ................................................................................ 51

4.1.3 AIAG Pp/Cp Requirements Tab ..................................................................................... 51



4.1.4 Requirements Tab (for Evaluating the Capability of the Characteristic) ....................... 52

4.1.5 Total Part Evaluation Tab .............................................................................................. 53

4.1.5.1 Example for a Total Part Evaluation (Test_all.dfq File) ............................................. 54

4.1.5.2 Execute Part Grading for Individuals Option ............................................................. 55

4.1.6 Additional Conditions Stable Tab .................................................................................. 57

4.1.7 Additional Conditions Unstable Tab .............................................................................. 58

4.1.8 AFNOR Settings Tab ....................................................................................................... 58

4.2 Requirements Attribute Characteristics ................................................................................ 59

4.2.1 Quality Control Chart ..................................................................................................... 59

4.2.1.1 Calculation of Control Limits – x-chart Binomial Distribution ................................... 60

4.2.1.2 Calculation of Control Limits – x-chart Poisson Distribution ..................................... 63

4.2.2 Requirements for Attribute Characteristics .................................................................. 65

4.2.2.1 Cpk for an Attribute Characteristic Value (Binomial) ................................................. 66

4.2.2.2 Cpk Value for an Attribute Characteristic (Poisson) ................................................... 68

4.3 Positional Tolerances Po/Pok:MPo2 ........................................................................................ 69

4.3.1 Calculation Method Tab ................................................................................................ 69

4.3.1.1 Meaning of a Dispersion Ellipse ................................................................................ 70

4.3.1.2 Calculation of the Potential Capability Index Po ....................................................... 70

4.3.1.3 Calculation of the Critical Capability Index Pok .......................................................... 71

4.3.2 Target Values Tab .......................................................................................................... 71

4.3.2.1 Minimum Values for Capability Indices ..................................................................... 71

4.3.2.2 Automatic Adaptation of Target Values .................................................................... 71

4.3.2.3 Warning Limit for Insufficient Values (Conditionally Capable Limit) ........................ 71

4.3.3 Requirements Tab ......................................................................................................... 72

4.3.3.1 Selection of Acceptance Criteria ............................................................................... 72

4.3.4 Additional Conditions Tab ............................................................................................. 73



1 General The Q-DAS Process Capability (06/2013) evaluation strategy is a design pattern providing the basis

for the configuration of an own evaluation strategy. Even though we designed this evaluation

strategy with due care, please do not consider it to be the “one and only“ evaluation strategy

recommended by Q-DAS.

1.1 Basic Information about the Evaluation Strategy

When a user opens a data set in the program qs-STAT ME 10 (Process Capability Analysis module),

the program takes the single evaluation steps according to the evaluation strategy automatically.

Then the software compares the calculation results automatically to the requirements adjusted in

the evaluation strategy and evaluates them.

The advantage is obvious now. The test planning engineer and the QM specialist define the

requirements for capability indices and the corresponding evaluation steps in an evaluation strategy.

This strategy is write-protected and adjusted in the program. Write protection ensures that all

employees achieve comparable and reliable evaluation results company-wide by using this

evaluation strategy.

All the produced parts frequently include a high number of characteristics. The program assesses the

capability of all characteristics automatically and correctly according to the adjusted evaluation

strategy. Users do not have to apply formulas and evaluation methods manually but may focus on

the interpretation of their evaluation results in practice.

The evaluation strategy leads to comparable and correct evaluation results company-wide. It makes it easier to conduct capability analyses for machines and manufacturing processes, especially in case you have to evaluate many characteristics simultaneously.



1.2 Functional Design of the Evaluation Strategy

Figure 1-1 shows five levels (0 to 4) with different functions.

Figure 1-1: Functional design of the evaluation strategy

Level 1: Evaluating stability in order to find a distribution time model

The program conducts tests for fluctuation of averages and for changes in the variation at the first

level. Depending on the test results, the software is able to assign a characteristic to a distribution

time model in accordance with ISO 21 747.

A reasonable selection of approved distribution models is allocated to distribution time models A1 to

D. The program chooses the model that best fits the characteristic.

At level 1, the characteristic is assigned to one of the distribution time models A1 to D and the program selects the distribution model that best fits the characteristic.



Level 2: Evaluating stability (µ,σ) based on analysis charts

At the second level, the program allocates a suitable analysis quality control chart to the

characteristic. Based on this chart, the software conducts tests for fluctuation of averages and for

changes in the variation again. These results are important to classify a characteristic distribution as

stable or not stable.

The stability evaluation of a characteristic distribution influences the requirements applying to the

corresponding capability indices. Different requirements apply to characteristics classified as not

stable compared to characteristics classified as stable.

When completing level 2, the characteristic distribution is classified as stable or not stable.

Level 3: Calculation of capability indices

The third level defines the methods for calculating capability indices and calculates these indices. The

next step is to compare the indices to the requirements defined at level 0 and to evaluate them.

At level 3 the program calculates and evaluates the capability indices for the respective characteristic. The evaluation is based on the requirements adjusted at level 0.

Level 4: Calculation of an online quality control chart for the program O-QIS

This level is about calculating a SPC quality control chart by using the program qs-STAT. The

calculated SPC control chart is saved to the data set (QCC menu – Modify calculated SPC QCC –

activate Quality control chart is designated to be saved option).

By opening the data set in the program O-QIS – a SPC program for process control – the software

imports the SPC quality control chart as an online quality control chart. The chart is available for

process control at the work station now.

At the end of level 4 the SPC quality control chart is calculated. In case you save the control chart to the data set, the program O-QIS takes it over as an online quality control chart upon loading the data set.

Please find more details about O-QIS in the online help of the program (qs-STAT).



1.3 Overview: Options for Describing Mathematical Functions

The following graphic shows different numbers displayed in green ellipses. These numbers refer to

different options whose functions are explained in the corresponding subsections of chapters 2 and

3.

Figure 1-2: Overview of the Q-DAS Process Capability evaluation strategy (06/2013)

In order to show the overview of the evaluation strategy as displayed above, select Configuration -

Evaluation.

You may click on any of these icons. After clicking it, the corresponding dialog box opens. Please note

that you cannot open several dialog boxes at the same time. In order to open another dialog box, you

have to close the open dialog box first.



2 Description of the Single Options

2.1 Preparation

Click on Preparation in order to open the corresponding dialog box. This dialog box provides different

options for loading a file or a data set from the database.

2.1.1 Takeover Tab

Figure 2-1: Options for takeover from data file

When loading a file, the program does not take over any settings by default. By not taking over any

settings, the software ensures that the results are calculated in accordance with this evaluation

strategy.

2.1.2 Classification Tab

Figure 2-2: Options available under Preparation - Classification

Classes are necessary, inter alia, to draw a histogram or to conduct the Chi-squared test. This

evaluation strategy applies the Classification with regard to resolution classification model by

default.



The program takes several steps of calculation in order to gain a classification. The first step is to

calculate the number of classes nclasses.

2-1 Number of classes

50;

3

)ln(

)10ln(

101min

Nnclasses

N refers to the number of values available for the characteristic. The calculated number of classes is

often a decimal fraction amounting to between 1 and 50.

Calculating the minimum resolution REmin is the second step. The program distinguishes the

minimum value for the resolution entered in the characteristics mask from the minimum difference

between two (numerically different) values of a characteristic. So the program uses the smaller one

of the two values as the minimum resolution REmin.

2-2 Minimum resolution REmin =min{minimum difference between values; entered resolution}

Now the program takes step 3 and calculates the multiple of the resolution VRE. The program divides

the range R by the number of classes nclasses and by the minimum resolution REmin. Since it is important

to round up to the nearest whole number, the formula always adds 0,5 to the fraction.

2-3 Multiple of the resolution

5.0;1maxint

min classes

REnRE

RegerV

The fourth and last step is to calculate the class width wclass.

2-4 Class width minREVw REclass

The first class limit is gained by subtracting the distance = REmin/ 2 from the median. The program

thus forms classes starting from the inside. However, some classifications do not cover the upper or

lower extreme value in exceptional cases and the number of classes subsequently needs to be raised

by 1. Only in these special cases, the applied number of classes exceeds the number calculated in

step 1.

2.1.3 Outliers Tab

By default, the evaluation strategy deletes values beyond plausibility or natural limits automatically

when loading data.

„Deleting values automatically“ does not mean that these values are removed physically from the file or the data set in the database. Instead, these values are marked by the program and will not be considered in the evaluation.



Figure 2-3: Options available under Preparation - outliers

2.1.3.1 Plausibility Limits

Plausibility limits are the limits that the values of this characteristic usually cannot exceed. As an

example, the plausibility limits for a turned part with a nominal diameter of 30 mm might be 40 mm

(upper limit) and 20 mm (lower limit). You can find plausibility limits – if specified by the test

planning engineer / user – in the characteristics mask (Edit menu – Characteristics mask).

The program deletes values beyond the plausibility limits automatically upon loading the file. In case of subsequently entered limits you have to recalculate the data by tapping <F9>.

2.1.3.2 Natural Boundaries

Natural boundaries are limits occurring naturally e.g. due to laws of chemistry or laws of physics. As

an example, water cannot get hotter than 100°C in case of “normal” atmospheric air pressure.

If the test planning engineer / user entered natural limits, they will be displayed in the characteristics

mask (Edit menu – Characteristics mask).

The program deletes values beyond natural limits automatically upon loading the file. In case of subsequently entered limits you have to recalculate the data by tapping <F9>.



2.1.3.3 Procedure with Incomplete Subgroups

The characteristics mask (Edit menu – Characteristics mask) shows the applied subgroup size in the

Subgroup size input field. In order to gain the number of subgroups, a total of N values is divided by

the subgroup size and leads to m = N/n subgroups.

Example of an Incomplete Subgroup

In case a subgroup size of n = 5 is specified in the characteristics mask under Subgroup size, the first

subgroup consists of values 1 to 5, the second subgroup of values 6 to 10 and the third subgroup

includes values 11 to 15, etc.

In case the seventh value field does not contain any value and is empty (gap), the second subgroup is

incomplete and will not be considered in the evaluation.

However, the last subgroup of a characteristic is an exception. Even if it is incomplete, the program

will consider it in the evaluation.

2.1.4 Positional Tolerances Tab

Figure 2-4: Options available under Preparation – Positional tolerances



2.1.4.1 What Is a Position?

Here is an example for the specification of the nominal position of a hole: 30 mm to the left (x-

coordinate) and 20 mm to the top (y-coordinate). With this in mind, specifying two (or more)

coordinates defines the nominal position. The superordinate characteristic true position includes the

absolute deviation of the actual position from the nominal position.

Figure 2-5: Example of a true position consisting of an x-coordinate, a y-coordinate and the superordinate characteristic true position.

2.1.4.2 Calculation of the True Position Value

The characteristics mask shows the nominal value for the x-coordinate Xnom and the nominal value for

the y-coordinate Ynom in the Nominal value input field (Edit menu – Characteristics mask).

The values mask includes the actual values of the x-position Xact and of the y-position Yact (Edit menu

– Values mask).

First, the program calculates the deviation X = Xact – Xnom or Y= Yact – Ynom and then it calculates the

superordinate characteristic true position (2√∆𝑥2 + ∆𝑦2).

Figure 2-6: Example – true position in the values mask (nominal positions in the characteristics mask)

Figure 2-6 shows the example data of the actual position X1act = 30,566 mm and Y1act = 19,969 mm in

the values mask. The Nominal value input field in the characteristics mask contains the nominal

positions of the x-coordinate and y-coordinate (Xnom = 30 mm and Ynom = 20 mm). Now the program

calculates the true position as follows.



1,134 = 2 × √(30,566 − 30,000)2 + (19,969 − 20,000)2

Figure 2-7: Jus a single absolute value l, but true position = 2 l

2.1.4.3 Creation of True Position Tolerance

The program calculates the true position tolerance according to the adjusted calculation method

(“diameter“ instead of “radius“).

Figure 2-8: Example of true positions plotted on x/y plot including the diameter of the tolerance cycle = 2 mm

Y=YIst-YSoll

X=XIst-XSoll



Figure 2-9: Tolerance of the true position USL = 2 mm, calculated automatically

The following specifications were entered in order to gain the example shown in Figure 2-8.

X-coordinate in characteristics mask

Nominal value = 30, Up. Spec. Lim. = 31 and Lo. Spec. Lim. = 29.

Y-coordinate characteristics mask

Nominal value = 20, Up. Spec. Lim. = 21 and Lo. Spec. Lim. = 19

The program now calculates the tolerance for the superordinate characteristic true position on condition that the two-dimensional true position leads to a tolerance cycle. The true position tolerance is not calculated for tolerance ellipses.

In other words, the tolerances for the single coordinates have to be equal in order to gain a clear true

position tolerance that can be calculated.

2.1.5 General Tab

This tab shows the settings referring to the study type, trend compensation and test procedures –

however, the latter only with respect to summary (all tests).

2.1.5.1 Study Type

In case you specify the study type in the evaluation strategy you want to output in reports, you may

enter an individual description of the study type here or – by clicking the Text from database button

– select a text from the database. By default, the program does not apply any text to describe the

study type.



Figure 2-10: Options available under Preparation - General

2.1.5.2 Trend Compensation

Figure 2-11: Settings for trend compensation

The trend compensation is applied in conjunction with the value chart of individuals.



Figure 2-12: Test_09.dfq test example – trend compensation with the value chart of individuals

This function helps to “straighten“ e.g. saw-like value charts through calculations and to eliminate

trends in the capability index. An interval of values including trends must include a minimum of 10

individual values in order to start a trend compensation.

2.1.5.3 Test Procedures

The carry out all tests option only affects the results in the Summary (all tests) window (Numerics

menu – Test procedures – Summary (all tests)). After activating this option, only this very window

also shows the results of the tests that are not activated in the evaluation strategy.



Figure 2-13: Summary (all tests) window

By opening the tests individually, the program still shows the note “The test could not be carried out“ for all procedures that are not activated in the evaluation strategy.

The last column lists the values of the test statistics for the tests. In case the background is displayed

in green and in case there is no star next to the test statistic, the null hypothesis was not rejected (H0

applies). A test statistic displayed against an orange or red background means the null hypothesis

was rejected (H1 applies). In addition, the stars next to the test statistics indicate the level of

significance .

= 5% = 1% = 0,1% One star (*) Two stars (**) Three stars (***)



2.2 “Variation Constant?“ – Test for Fluctuations of Variation

Figure 2-14: Icon representing the test for constant variation

By clicking the Variation constant? icon, the dialog providing settings regarding tests for equal

variation opens.

A total of N values corresponding to the characteristic are displayed in the values mask. The

characteristics mask shows the applied subgroup size in the Subgroup size input field. A total of N

values is divided by the subgroup size and leads to m = N/n subgroups. The test of Levene checks

whether the variation within the m subgroups can be considered as constant (equal).

Figure 2-15: Test for equality of variances – Levene test

The test of Levene analyzes the equality of variances

Null hypothesis H0: 𝜎12 = 𝜎2

2 = ⋯ = 𝜎𝑛2

(All subgroups are taken from populations with equal variance.)

Alternative hypothesis H1: : 𝜎𝑖2 ≠ 𝜎𝑗

2

(At least two subgroups are taken from populations with different variances.) In case the test discovers significant fluctuations of the variation, the evaluation follows the H1 branch of the tree diagram in the evaluation strategy, otherwise it follows the H0 branch.



The test of Levene is a test that is not based on any distribution. It thus applies to a non-normal population, too.

2.3 “Location Constant?“ – Test for Fluctuation of Averages

By clicking the Location constant? icon, the dialog providing settings regarding tests for fluctuation of

averages opens.

Figure 2-16: Icon representing the test for fluctuation of averages

A total of N values corresponding to the characteristic (values) is divided by the subgroup size and

leads to m = N/n subgroups. The characteristics mask shows the applied subgroup size in the

Subgroup size input field.

The test of Kruskal and Wallis checks the equality of locations based on the median values.

Null hypothesis H0: ��1 = ��2 = ⋯ = ��𝑛 (All subgroups are taken from populations with the same median.)

Alternative hypothesis H1: ��𝑖 ≠ ��𝑗

(At least two subgroups are taken from populations with different median values.)

In case the test discovers a significant deviation between median values, the evaluation follows the

H1 branch of the tree diagram in the evaluation strategy, otherwise it follows the H0 branch.



Figure 2-17: Test for equality of locations – test of Kruskal and Wallis

Even the test of Kruskal and Wallis does not require that the single subgroups are taken from a

normally distributed population.

2.4 “Defined Distribution by Measuring Unit?“ – Specified Distribution

The program includes some default distribution models for certain measurement quantities.

Figure 2-18: Icon for Defined distribution by measuring unit?

In case of measurement quantities with a defined distribution, the evaluation follows the tree

diagram to the Select distribution icon. In case of undefined measurement quantities or

measurement quantities without any specified distribution, the evaluation follows the tree diagram

to the Default ND/WD icon.



2.5 Select Distribution

Click on the Select distribution icon to open the dialog box providing settings for the definition of

distribution models for single measurement quantities. The following measurement quantities are

assigned to a distribution model by default.

Measurement quantity Model distribution

Straightness Folded normal distribution

Flatness Folded normal distribution

Roundness Folded normal distribution

Cylindricity Folded normal distribution

Profile (line) Folded normal distribution

Surface (form) Folded normal distribution

Angularity Folded normal distribution

Perpendicularity Folded normal distribution

Parallelism Folded normal distribution

Concentricity Folded normal distribution

Symmetry Folded normal distribution

Runout Folded normal distribution

Total runout Folded normal distribution

X-coordinate Normal distribution

Y-coordinate Normal distribution

Z-coordinate Normal distribution

Maximum profile height Rz Folded normal distribution

Distance Normal distribution

Unbalance Folded normal distribution

Torque Normal distribution

Table 2-1: List of defined distribution models for specific measurement quantities

Note: The test planning engineer / user may define the measurement quantity for the characteristic

in the characteristics mask and thus define a specified distribution model for this quantity.

Figure 2-19: Extract from the characteristics mask – characteristic with a defined measurement quantity

2.6 Default ND/WD

By clicking the Default ND/WD icon, a dialog box opens providing options for undefined

measurement quantities or measurement quantities without any specified distribution model. You

may distinguish between two different cases.



One-sided characteristic Only an upper or lower specification limit exists. The specified model is the Weibull distribution.

Two-sided characteristic The normal distribution is used as the specified distribution model.

2.6.1 Selection of Distribution – General Options

Figure 2-20: Distributions – General options tab

2.6.1.1 Distributions with Offset

Some distribution models are limited to the left by the point of origin. In other words, such

distribution models have their starting point in the point of origin. Some examples are the Weibull

distribution, the logarithmic normal distribution and the folded normal distribution. However, you

may add a move or offset parameter to these distribution models. It helps you to shift the left-hand

limit to any nonzero value. This is what we refer to as a distribution with offset.

You calculate the best suitable value for this offset based on the subgroup values. In some cases, this

offset might reach a value that is virtually undesirable, e.g. in case of a negative value for the offset

of the maximum profile height Rz characteristic. This is the reason why the Calculate the best offset

considering the tolerance limits option is enabled by default. With this option enabled, the program

cannot apply a value exceeding the natural limit. In case of the maximum profile height Rz example



mentioned above, the offset would be set to zero (= without offset) instead of using the best suitable

negative offset beyond the natural limit.

2.6.1.2 Best Possible Distribution

The program calculates the regression coefficient for all distribution models available for selection

(see Table 2-2). This coefficient indicates the rate of change between the theoretical distribution

function and the empirical distribution function of the subgroup data. The regression coefficient may

reach any value between 0% and 100%. The higher the value, the lower the rate of change. The

program normally chooses the distribution model with the highest regression coefficient.

Normal distribution

Logarithmic normal distribution

Folded normal distribution Folded at 0

Rayleigh distribution Folded at 0

Folded normal distribution Not folded at 0

Rayleigh distribution Not folded at 0

Weibull distribution Table 2-2: List of permissible unimodal distributions (distribution time models A1 and A2)

Exceptions

Predefined distribution models A distribution model is already defined for some measurement quantities. If the predefined

distribution model is not rejected by a goodness-of-fit test (see test for distribution), it will be

retained even though the program calculated a higher regression coefficient for other distribution

models.

Several distributions have the same regression coefficient In case the goodness-of-fit test calculates an equal regression coefficient for several distribution models, the program selects the model listed at the end of the list of distributions. This might apply e.g. to folded normal distributions that are not folded at zero (distance between the folding point and the average amounts to more than the sum of three standard deviations).



2.7 Test for Distribution / Test for Normality

By clicking on the Test for distribution or Test for normality icon, you open a dialog box providing

settings for goodness-of-fit tests.

Figure 2-21: Test for distribution / Test for normality icons

These tests analyze the specified distribution.

Null hypothesis H0: The distribution model assumed for the population from which the values are taken is correct.

Alternative hypothesis H1: The distribution model assumed for the population from which the values are taken is not correct.

In case the null hypothesis is rejected (H1 path), the evaluation follows the tree diagram to the Find

distribution model icon. If the null hypothesis is not rejected (H0 path), the evaluation follows the tree

diagram to the ND icon.



Figure 2-22: Goodness-of-fit test

2.7.1 Normal Distribution Tests

The applied normal distribution test depends on the number of individual values. This helps to obtain

the best possible accuracy of the test results.

The number of values exceeds 200 (n > 200).

In case there are more than 200 values available, the program conducts the test for asymmetry and

the test for kurtosis.

The number of values exceeds 50 but is equal or less than 200 (50 < n 200).

If the number of values lies within this interval, the program starts the Epps-Pulley test for normality.

The number of values is equal or less than 50 (n 50).



In case there are at most 50 values available, the program conducts the Shapiro-Wilk test for

normality.

Depending on the number of values, the program only conducts a single goodness-of-fit test, i.e. the results of all the other tests that have not been conducted are not available.

Figure 2-23: Note displayed when the test is not available

Independent of the number of values, the Anderson-Darling test for normality is always conducted.

However, the result is only available in the test summary (Numerics menu – Test procedures –

Summary).

2.7.2 Arbitrary Distribution Tests

Any non-normal, continuous distribution is analyzed by using the Chi-squared test.

Null hypothesis H0: The distribution model assumed for the population from which the values are taken is correct.

Alternative hypothesis H1: The distribution model assumed for the population from which the values are taken is not correct.

The result depends on the selected classification model. In other words, changing the number of

classes or the class limits affects the result of the Chi-squared test of goodness of fit.

2.8 Find Distribution Model

In case the predefined distribution model was rejected by a normal distribution test or the arbitrary

distribution test, the program finds another distribution matching the values.

Figure 2-24: Find distribution model icon

By clicking on the Find distribution model icon, a list of possible distributions opens. In searching for a

distribution, the list contains all approved distribution models.



2.8.1 Distributions Tab

In searching for a distribution model, this tab shows all approved distribution models.

Figure 2-25: Approved distributions in searching for a distribution model

2.8.2 General Options Tab

Figure 2-26: Find distribution model – General options



2.8.2.1 Distributions with Offset

These settings concern the following distribution models.

Logarithmic normal distribution

Weibull distribution

Folded normal distribution By calculating the distributions listed above without offset, they are limited to the left at zero by the point of origin. These distributions are not defined for any value being less than zero. By adding an offset, you can move the left-hand limit to any other nonzero value. With the Calculate the best offset considering the tolerance limits option enabled, the offset cannot

exceed the natural limit.

2.8.2.2 Best Possible Distribution

In order to find the distribution matching the data the best possible way, the program uses the

highest value for the regression coefficient as a criterion. This coefficient indicates the rate of

change between the theoretical distribution function and the empirical distribution function of the

characteristic values. These coefficients always lie within an interval of 0% r 100%. The higher the

value, the lower the rate of change between the model and the data. The program normally chooses

the distribution model with the highest regression coefficient.

Exceptions

Predefined distribution models A distribution model is already defined for some measurement quantities. If the predefined

distribution model is not rejected by a goodness-of-fit test (see test for distribution), it will be

retained even though the program calculated a higher regression coefficient for other distribution

models.

Several distributions have the same regression coefficient In case the goodness-of-fit test calculates the same regression coefficient for several distribution models, the program selects the model listed at the end of the list of distributions. This might apply e.g. to folded normal distributions that are not folded at zero (distance between the folding point and the average amounts to more than the sum of three standard deviations).



2.9 “Momentary Normal Distribution?“

Figure 2-27: Momentary normal distribution? icon

By clicking the Momentary ND? icon, the dialog box for the respective test procedure opens (Others

tab).

Table 2-3: Configuration of the test for subgroup normality

The extended Shapiro-Wilk test is selected as the test for subgroup normality by default (level of

significance = 5%).

Null hypothesis H0: The subgroups are taken from a normally distributed population.

Alternative hypothesis H1: The subgroups are not taken from a normally distributed population.

Depending on the test result, the program follows the H0 path or the H1 path.

In case H0 is not rejected, the resulting distribution might be a normal distribution; otherwise, the

resulting distribution is considered to be a mixed distribution.



3 Distribution Time Models in Accordance with ISO 21 747 The ISO 21 747 standard divides the distribution of a characteristic into the four distribution time

model categories A to D. This classification of characteristic distributions is based on the behavior of

the location parameter µ (or µ) and the dispersion σ over time. The following table gives a brief

overview of the different categories of distribution time models.

Constant location parameter Variable location parameter

Constant dispersion A1 and A2 C1 to C4

Variable dispersion B D

The allocation of a distribution to a distribution time model simplifies, inter alia, the selection of an

analysis quality control chart suitable for the characteristic distribution. The analysis chart provides

the second and decisive stability evaluation. It classifies the distribution of characteristic values as

stable or not stable.

Figure 3-1: Two stages of stability evaluation at level 1 and level 2

The software uses a two-stage stability evaluation for the location parameter and the dispersion of the characteristic distribution.

The first stage is based on different test procedures (test of Levene, test of Kruskal and Wallis) and allocates the distribution to distribution time models A1 to D.

The second stage of stability evaluation uses analysis charts and leads to the final stability evaluation.



The result of the second stage is the decisive factor in order to classify the characteristic distribution

as stable or not stable.

Note: The software uses different analysis charts for various distribution time models.

The stability evaluation based on analysis quality control charts decides on the requirements applying

to a characteristic.

There are different Cp/Cpk or Pp/Ppk requirements for characteristic distributions classified as stable and distributions of characteristic values classified as not stable.

ISO 21 747 – Calculation methods for capability indices M1

There are four methods available for the calculation of capability indices. The ISO/FDIS 21 747

standard refers to them as methods M1 to M4.

Method Mode of Calculation

M1l,d General geometric method

M2l,d,a Explicit inclusion of additional variation

M3l,d,a Alternative method of explicit inclusion of additional variation

M4 Calculation of fractions nonconforming Table 3-1: Calculation methods for capability indices (ISO/FDIS 21 747)

The program applies the calculation method M1l,d. M1 refers to the “general geometric method“. The

letters in the index indicate that the method is used for the estimation of the location and dispersion.

Estimators for location l

The index l refers to the estimator for location. ISO 21 747 includes five estimators for location.

l Note

1 Arithmetic mean of all individual values

2 Median of all individual values

3 Median of the distribution model adapted to the individual values

4 Average of the subgroups’ averages

5 Average of the subgroups’ medians Table 3-2: Index indicating the estimators for location according to ISO 21 747

The software uses the estimator for location l = 1 by default in case of normal distribution. This

estimator refers to the arithmetic mean of all values µ =1

𝑛∑ 𝑥𝑖

𝑛𝑖=1 . In case of any other distribution,

the software applies the estimator for location l=3 by default, i.e. the median of the distribution

model adapted to the characteristic values.



Estimators for dispersion d

The second index d refers to the estimator for dispersion. ISO 21 747 contains six estimators for

dispersion.

d Dispersion Estimator for sigma

1 1ˆ6ˆ

m

sˆ

2

i

1

2 2ˆ6ˆ

4

i

2cm

sˆ

3 3ˆ6ˆ

2

i

3dm

Rˆ

4 4ˆ6ˆ

2

i4 xx1n

1ˆ

5 Rˆ not applicable

6 %135,0%865,99 XXˆ not applicable

Table 3-3: Index indicating the estimator for dispersion according to ISO 21747

The software uses d = 6 by default. This dispersion represents the distance between the 0,135%

quantile and the 99,865% quantile of the distribution model adapted to the values of the

characteristic.



Calculating the potential process capability or process performance index Cp or Pp based on

method M13,6 according to ISO 21 747

Figure 3-2: Potential capability index calculated through method M13,6 in accordance with ISO 21 747

3-1 Performance or capability index Pp / Cp as per ISO 21 747

%135,0%865,99 XX

LSLUSLPp

Note: The software uses the same formula for indices Pp and Cp.



Calculating Ppk or Cpk through method M13,6 according to ISO 21 747

Figure 3-3: Critical capability or performance index Ppk or Cpk calculated in accordance with ISO 21 747

3-2 Critical index for lower specification limit

%135,0%50

%50

XX

LSLXPpkL

3-3 Critical index for upper specification limit

%50%865,99

%50

XX

XUSLPpkU

3-4 Critical capability or performance index Ppk / Cpk pkUplkLpk P;PminP

First, the program calculates the critical index for the lower and upper specification limit. The statistic

with the lower value is now taken over as the critical capability index.

Note: The software uses the same formula for indices Ppk and Cpk. In case of method M11,6, the

program uses the average as an estimator for location instead of the median of the distribution.



3.1 Distribution Time Models A1 and A1*

The distribution time model A1 or A1* represents a normal distribution including location and

dispersion parameters (µ and σ) being constant over time.

Figure 3-4: Distribution time model A1 - analysis

After allocating the characteristic distribution to a distribution time model, an analysis quality control

chart finally evaluates the stability of the location parameter and dispersion.

The program uses the Shewhart average chart by default in order to evaluate the stability of the

location parameter corresponding to the respective characteristic distribution. It applies the

Shewhart chart of standard deviations for evaluating the stability of the dispersion.

3.1.1 Analysis Chart for Location: Shewhart Average Chart (x-chart)

Figure 3-5: Configuration of the location chart

The default chart type for the stability evaluation for the location parameter is the Shewhart average

chart (x-chart).



3.1.1.1 Calculation Method for Control Limits in ��-chart

Control limits are calculated based on a probability of acceptance (non-interference probability) of

Pa = 99,73%. They are determined by means of the calculation method ”normal calculation“ as

follows.

3-5 Upper control limit as per calculation method “normal calculation“ n

uUCL

ˆ

ˆ%865,99

3-6 Lower control limit as per calculation method ”normal calculation“ n

uLCL

ˆ

ˆ%865,99

3-7 Estimator for dispersion 2sˆ

3-8 Estimator for location in location chart xˆ

3-9 Number of subgroups n

Nm

N = number of characteristic values and n = subgroup size whose value is displayed in the Subgroup

size input field in the characteristics mask

99,865% quantile of the standard normal distribution u99,865% = 3.

3.1.1.2 Stability Criteria for Shewhart ��-chart (Analysis Chart)

According to the stability evaluation of “stage 2“, the sum of control limit violations must not exceed

the variation limits of the two-sided 99% random variation range of the binomial distribution BD(P,

m). P = 100% - 99,73% = 0,27% and m = number of subgroups.

3.1.2 Analysis Chart for Variation: Shewhart s-chart

Figure 3-6: Configuration of the variation chart



The default chart type for assessing the stability of the dispersion is the s-chart, i.e. a Shewhart chart

of standard deviations.

3.1.2.1 Calculation Method for Control Limits in Shewhart s-chart

The program calculates the control limits of the s-chart based on a probability of acceptance (non-

interference probability) Pa = 99,73%. They are determined by means of the calculation method

”normal calculation” as follows.

3-10 Upper control limit of the s-chart (”normal calculation“) f

UCLf

2

%;865,99ˆ

3-11 Lower control limit of the s-chart (”normal calculation“) f

LCLf

2

%,135,0ˆ

3-12 Estimator for dispersion 2sˆ

3-13 Number of degrees of freedom 1nf

Both, the value of the 99,865% quantile of the chi-squared distribution 99,865 %,𝑓2 and the value of

the 0,135% quantile 0,135 %,𝑓2 depend on the number of degrees of freedom f = n-1. The software

uses the subgroup size n specified in the Subgroup size input field in the characteristics mask.

3.1.2.2 Stability Criterion for the Shewhart s-chart (Analysis Chart)


the variation limits of the one-sided 99% random variation range of the binomial distribution BD(P,

m) bounded above. P=0,135% and m = number of subgroups.



3.1.3 Distribution Time Models A1 and A1* – Stable/Not Stable

Figure 3-7: Distribution time model A1 and A1* - stable/not stable

Click on the box Cp/Cpk M11,6 or Pp/Ppk M11,6.to open the C-value function dialog box. It shows the

calculation method for capability indices but you may also change this method here.

Figure 3-8: C-value function dialog box

The beginning of this chapter already described the calculation methods – meaning of the notation

and formulas for the indices. The default method is M13,6.



3.1.3.1 Distribution Time Models A1 and A1* – Q-QIS

Figure 3-9: Distribution time model A1 - Q-QIS

The program Q-QIS is designed to maintain online quality control charts. In order to make some

preparations, you may calculate some online quality control charts in the program qs-STAT and save

them to the data set. If you open this file in the program Q-QIS now, the respective online quality

control chart will be imported automatically. Please find more details on how to work with quality

control charts in Q-QIS in the documentation about the program Q-QIS.

The settings for an online QCC are identical to the settings for analysis charts. For more details see

chapters 3.1.1 and 3.1.2.

Figure 3-10: Q-QIS location chart (distribution time model A1)



Figure 3-11: Q-QIS variation chart (distribution time model A1)

3.2 Distribution Time Models A2 and A2*

The distribution model selected for the characteristic is a non-normal, unimodal distribution, i.e. a

logarithmic normal distribution, a folded normal distribution or a Weibull distribution. These models

are suitable for skewed (asymmetrical) frequency distributions.

Figure 3-12: Distribution time models A2 and A2* – analysis

The software uses the Johnson average chart (or Pearson chart) in order to evaluate the stability of

the location parameter µ of the characteristic distribution. It applies the Shewhart s-chart for

evaluating the stability of the dispersion.



3.2.1 Analysis Chart for Location: Johnson Average Chart

Figure 3-13: The analysis location chart is a Johnson average chart (or Pearson chart).

The program applies the average chart as default chart type. The estimator for the location

parameter µ of the analysis average chart is the average of all subgroup averages.

3-14 Estimator for Location

m

1i

ixm

1xµ

3.2.1.1 Calculation Method for Control Limits in Johnson Chart

The calculation method for control limits is the Johnson calculation (or Pearson calculation). The

program adapts a Johnson distribution to the subgroup averages and calculates the control limits for

the probability of acceptance (non-interference probability) of Pa = 99,73%.

UCL = 99,865% quantile of the Johnson distribution adapted to the subgroup averages

LCL = 0,135% quantile of the Johnson distribution adapted to the subgroup averages

By default, the program only displays the control limit for the specified side in case of characteristics

with one specification limit only.



Figure 3-14: Example for the calculation of control limits (Test_04.dfq) based on a Johnson distribution adapted to the subgroup averages (Qlo3 = 0,135% quantile and Qup3 = 99,865% quantile)

Figure 3-15: Johnson average chart for the Test_04.dfq example (or Pearson chart)

The Johnson chart displayed in Figure 3-15 only includes an upper control limit since test data set

Test_04.dfq only contains a one-sided characteristic bounded below.



3.2.1.2 Stability Criteria of the Johnson Chart (or Pearson Chart)

The program uses “stage 2“ for the stability evaluation of the location chart by default.

The sum of control limit violations does not exceed the limits of the two-sided 99% random variation

range of the binomial distribution BD(P, m), where P = 100% - 99,73% = 0,27% and m = number of

subgroups.


The program applies the Shewhart standard deviation chart (s-chart) as default chart type.

Figure 3-16: Settings for the analysis variation chart (s-chart)

The estimator for the dispersion is the square root of the average of the subgroup variances.

3-15 Estimator for the dispersion of the s-chart 2sˆ

3.2.2.1 Calculation Method for Control Limits in the s-chart

The calculation method for control limits is based on the chi-squared distribution (“normal

calculation”) including a probability of acceptance (non-interference probability) of Pa =1-

= 99,73 %.

3-16 Lower control limit of the s-chart (“normal calculation“) f

LCLf

2

%,135,0ˆ

3-17 Upper control limit of the s-chart (“normal calculation“) f

UCLf

2

%,865,99ˆ




uses the subgroup size n specified in the Subgroup size input field in the characteristics mask.



3.2.2.2 Stability Criteria for the Shewhart s-chart (Analysis Chart)




3.3 Distribution Time Models C2,4, B and D

In case the characteristic distributions are categorized as distribution time models C2,4, B or D based

on numerical test procedures, the software selects the mixed distribution distribution model.

Technically speaking, the mixed distribution is not a probability distribution in the classical sense.

3.3.1 Analysis Chart for Location: Extended Shewhart Chart

For distribution time models C1 to C4, B and D, the software uses a Shewhart average chart with

extended control limits for the stability evaluation of the location parameter by default.

Figure 3-17: Configuration of the Shewhart average chart with extended control limits

The program applies the Shewhart average chart with extended control limits as the default chart

type.

The standard calculation of control limits is based on the acceptance probability (non-interference

probability) Pa = 99,73 %.



3.3.1.1 Calculating the External Standard Deviation ��𝑨

The random effect ANOVA resolves the total variance of a characteristic into the two variance

components (1) internal variance (random variation of individual values within the subgroup) and (2)

external variance (variation of averages between the subgroups).

Figure 3-18: Example – resolving the total variance into (1) internal and (2) external variance (Test_01.dfq)

The standard deviation determined from the external variance is used in order to calculate the range

bounded by the extended limits (see chapter 3.3.1.2).

3-19 Estimator for external variance 2

AA sˆ

3.3.1.2 Calculating Control Limits

By clicking on the Shewhart average chart to open the respective configuration (see Figure 3-17) and

selecting the Parameter button, a dialog box opens. Here you adjust the settings for the parameters

you use in the calculation of control limits.

Figure 3-19: Parameters of the Shewhart average chart with extended control limits

As the dialog box shows (extended limits), the extended control limits correspond to the 86,64%

variation range of the external variance (1,5 ��𝐴).



3-20 Upper control limit n

uUCL A

ˆˆ5,1ˆ

%865,99

3-21 Lower control limit n

uLCL A

ˆˆ5,1ˆ

%865,99

3-22 Estimator for sigma 2

i

2 sm

1s

3-23 Estimator for µ ixm

1x

3-24 Number of subgroups n

Nm

N = number of characteristic values and n = subgroup size of the single subgroups whose value is

specified in the subgroup size input field of the characteristics mask.

Note: 99,865% quantile of the standard normal distribution u99,865% = 3.

3.3.1.3 Stability Criteria of the Extended Shewhart Average Chart

The program uses “stage 2“ for the stability evaluation of the location chart by default.

The sum of control limit violations does not exceed the limits of the two-sided 99% random variation

range of the binomial distribution BD(P, m), where P = 100% - 99,73% = 0,27% and m = number of

subgroups.


The program applies the Shewhart standard deviation chart (s-chart) with extended control limits as

default chart type.



Figure 3-20: Settings for the analysis variation chart (s-chart)

The estimator for dispersion is the square root of the average of the subgroup variances.

3-25 Estimator for dispersion of s-chart 2sˆ

3.3.2.1 Calculation Method for Control Limits in the s-chart

The calculation method for control limits is based on the chi-squared distribution (“normal

calculation”) including a probability of acceptance (non-interference probability) of Pa =1-

= 99,73 %.

3-26 Lower control limit of the s-chart (”normal calculation“) f

LCLf

2

%,135,0ˆ

3-27 Upper control limit of the s-chart (”normal calculation“) f

UCLf

2

%,865,99ˆ




uses the subgroup size n specified in the Subgroup size input field of the characteristics mask.

3.3.2.2 Stability Criteria of the Shewhart s-chart






4 Requirements Here you find the adjusted requirements for capability analyses classifying a characteristic as

capable, conditionally capable or not capable.

Figure 4-1: Requirements icons

Click on the Requirements variable characteristics, Requirements attribute characteristics and

Positional tolerances Po/Pok:MPo2 icons in order to open the corresponding dialog box. The dialog

shows you the currently adjusted requirements and helps you to adjust them.

4.1 Requirements Variable Characteristics

4.1.1 Target Values / QCC Stable Tab

Figure 4-2: Requirements for characteristics classified as stable (Target values /QCC stable tab)

The displayed minimum capability indices or minimum number of values apply to characteristic

distributions whose location parameter and dispersion have been classified as stable in the analysis

chart.



4.1.1.1 Capability Indices

The program calculates the capability indices in case there are at least 125 values available. The

minimum values are the same for normally distributed and non-normal characteristic values by

default.

The description of capability indices conforms to the ISO 21 747 standard for the potential capability

index Cp and the critical capability index Cpk.

4.1.1.2 Preliminary Capability Indices

The program calculates preliminary capability indices in case the subgroup size amounts to at least N

= 10 values. More precisely, there have to be at least two subgroups respectively including five

values. Otherwise, the software does not calculate any capability index.

The description of capability indices conforms to the ISO 21 747 standard for the potential capability

index Cp and the critical capability index Cpk.

4.1.1.3 Warning Limits for Insufficient Number of Values (Limits for Conditionally Capable)

In case the number of values does not meet the limit of 50 values, the program raises the target

value for the capability index automatically.

The respective increase is based on the confidence interval for the capability index. Generally, a small

number of values leads to a wider confidence interval whereas a higher number of values involves a

narrow confidence interval.

In case there are less than 50 values available, the requirements for capability indices are raised automatically (higher minimum values for Cp and Cpk).

4.1.1.4 Confidence Intervals for the Potential Capability Index Cp

Here is the equation for calculating the confidence limits of the two-sided 95% confidence interval

for the potential capability index.

4-1 Lower confidence limit for Cp f

C

2

f% ;5,2

p

4-2 Upper confidence limit for Cp f

C

2

f% ;5,97

p

where f = N-1 degrees of freedom and N is the number of characteristic values.



4.1.1.5 Confidence Interval for the Critical Capability Index Cpk

Here is the equation for calculating the confidence limits of the two-sided 95% confidence interval

for the critical capability index Cpk.

4-3 Lower confidence limit for Cpk N9

1

1N2

CuC

2

pk

%5,97pk

4-4 Upper confidence level for Cpk N9

1

1N2

CuC

2

pk

%5,97pk

The program raises the minimum value for Cp or Cpk automatically in case the number of values does

not reach the limit of 50 values. Here is the equation for calculating the adjusted target value.

4-5 Adaptation of target value Cpk

2

1;

2

1;

1

1

2

11

2

11

ngr

n

gr

nomPKn

n

n

nC

gr

where

CPK-nom = initial minimum value for Cpk

ngr = limit for the number of values causing the increase of the Cpk target value in case the limit is not

met

n = actual number of values

Figure 4-3: Diagram of the target value adaptation for capability indices (basic target value Cpk-nom = 1,67)



4.1.2 Target Values / QCC Unstable Tab

The tab shows the minimum values for characteristics whose characteristic distribution was classified

as unstable according to the analysis quality control chart.

Figure 4-4: Target value / QCC unstable tab

A characteristic distribution classified as unstable requires more space within the tolerance

compared to a distribution classified as stable. This is due to the fluctuation of averages. For this

reason, the potential capability index for characteristic distributions classified as unstable has to

meet higher requirements.

4.1.3 AIAG Pp/Cp Requirements Tab

Figure 4-5: AIAG Pp/Cp requirements tab

The program calculates the capability indices according to AIAG. The requirements displayed here do

not affect the evaluation of capability indices regarding stability. The reason is that the evaluation

criteria (1) process capability index AIAG and (2) smallest process capability index AIAG are not

activated in the Requirements tab.



4.1.4 Requirements Tab (for Evaluating the Capability of the Characteristic)

Figure 4-6: Requirements tab

This dialog box shows which criteria are considered in order to evaluate the capability of the

characteristic.

In order to gain a capable characteristic in the overall evaluation, the following criteria apply.

The number of values is equal to or greater than the warning limit for insufficient values.

The potential capability index reaches or exceeds the minimum value.

The critical capability index reaches or exceeds the minimum value.

The proportion of automatically deleted outliers is equal to or less than 5%. In order to obtain a conditionally capable characteristic in the overall evaluation, the first two criteria have to apply and at least the third or fourth criterion.

The potential capability index reaches or exceeds the minimum value.

The critical capability index reaches or exceeds the minimum value.

The number of values does not reach the warning limit for insufficient values.

The proportion of automatically deleted outliers exceeds 5%.

In order that a characteristic is classified as not capable in the overall evaluation, at least one of the following criteria applies.

The potential capability index is less than the minimum value.



The critical capability index is less than the minimum value. There is no overall evaluation in case

The number of values is less than the value entered in the min. values input field (always 10 values in the Target values/QCC stable and Target values/QCC unstable tabs).

The number of subgroups is less than the value entered in the Min. subgroups input field (always 2 subgroups in the Target values/QCC stable and Target values/QCC unstable tabs).

Conditionally capable applies in case of too many outliers that have been deleted automatically or if the number of all characteristic values is less than 50.

4.1.5 Total Part Evaluation Tab

Figure 4-7: Total part evaluation tab

The settings displayed here refer to the grading according to the part type grading, an evaluation

system originally initiated by the Scania company.

Figure 4-7 shows the grading key and the significance of the characteristic classes. The capability

according to the party type grading is evaluated as follows:

1,00 part type grade < 2,66 capable

2,66 part type grade 4,33 conditionally capable

4,33 < part type grade not capable



4.1.5.1 Example for a Total Part Evaluation (Test_all.dfq File)

Figure 4-8: Capability evaluation of the characteristics (Test_all.dfq example)

The characteristic class for all 13 characteristics is significant and that is the reason why the

weighting = 50 for all these characteristics. Figure 4-8 shows that the first, fifth, tenth and twelfth

characteristics are not capable (= grade 6); however, the remaining characteristics are capable (grade

1).

Number of

the

characteristic

Weighting

for the

characteristic

Grade

for the

characteristic

Weighting

multiplied

by grade

1 50 6 300

2 50 1 50

3 50 1 50

4 50 1 50

5 50 6 300

6 50 1 50

7 50 1 50

8 50 1 50

9 50 1 50

10 50 6 300

11 50 1 50

12 50 6 300

13 50 1 50



Sum= 650 Sum= 1650

Part type grade =1650

650≈ 2,54

The result of the part type grading is displayed in the Part evaluation menu under Characteristics

Parts.

Figure 4-9: Part type grading for the Test_all.dfq example (Part evaluation menu – Characteristics Parts)

4.1.5.2 Execute Part Grading for Individuals Option

This option helps you to evaluate every single part. The program analyses whether the actual value of

each characteristic of the part lies within the specification. It awards points for characteristic values

outside the specification. However, this procedure takes a long computing time and is considerably

time-consuming. This is the reason why the option is not activated by default in order to avoid

unnecessarily long loading times.

The Execute part grading for individuals option is deactivated by default in order to avoid unnecessarily long loading times. Only activate the part grading for individuals temporarily when you need it.

Deactivate the option again as soon as you do not need it any longer.

Figure 4-10: Part grading for individuals – adjust the basis for evaluation

Figure 4-11: Part grading for individuals – weighting of points depending on the characteristic class



According to the characteristic class adjusted under characteristic class, the program multiplies the

number of points by the weighting.

The software only conducts the part grading for parts with all characteristic values available in the

values mask. In case of an incomplete part with several characteristic values missing, the program

does not start the part grading.

4.1.5.2.1 Example – Execute Part Grading for Individuals Option

The Test_all.dfq data set contains 13 characteristics. However, each characteristic includes a

different number of values. The second characteristic has the smallest number of 100 values. When

executing the part grading with this data set, the program only evaluates parts 1 to 100 since the

data set only includes all values of the 13 characteristic for the first 100 parts.

The part grading for individuals considers each line of the values mask to contain the characteristic values of a certain part. Please consider that all characteristic values can be clearly allocated to a part. All characteristic values displayed in one line should belong to the same part.

Figure 4-12: A line in the characteristics mask shows all the characteristics of a part

In case the program finds parts with some characteristic values missing, it will not consider these

parts. In other words, there must not be any value missing in a line.

Parts with values missing for one or several characteristics are not considered in the part grading and will not be part of the evaluation.

Figure 4-13: Test_all.dfq (German version)– Characteristic 1 of part 11 and part 66 is outside the specification



Each one of the 13 characteristics contained in the Test_all.dfq file is allocated to the characteristic

class significant. The program assigns grade six to all characteristic values beyond the specification.

So parts 11 and 66 of the Test_all.dfq file received a six since their values of characteristic 1 are

already beyond the specification.

By observing the data of the Test_all.dfq file in the values mask, you will see that only the lines of the

first 100 parts are complete. Some values are missing in the lines of part 101 to the last part. These

parts are not considered in the part grading.

4.1.6 Additional Conditions Stable Tab

You may adjust additional conditions and exceptions in order to specify more conditions for

conditionally capable characteristics as given in the Requirements tab. However, it is not

recommended to adjust further settings in order to avoid confusion among users.

By default, this evaluation strategy does not activate any additional conditions for characteristic

distributions classified as stable.

Figure 4-14: Options available in the Additional conditions stable tab



4.1.7 Additional Conditions Unstable Tab

Use this tab to adjust additional conditions and exceptions in order to specify more conditions for

conditionally capable characteristics as given in the Requirements tab. However, it is not

recommended to adjust further settings in order to avoid confusion among users.

No additional conditions for characteristic distributions classified as stable are activated.

Figure 4-15: Options available in the Additional conditions unstable tab

4.1.8 AFNOR Settings Tab

The Association Française de NORmalisation is an official French service network developing

standards, granting certifications and providing information to the specialized industrial press.

Settings adjusted in this tab refer to the evaluation of characteristics created and classified as AFNOR

E60-181 characteristic groups.



Figure 4-16: Part of the create new characteristics dialog (File menu - New)

Figure 4-17: Options in the AFNOR settings tab

4.2 Requirements Attribute Characteristics

By clicking on the Requirements attribute characteristics icon you open the dialog box displayed

below.

Figure 4-18: Evaluation setup for attribute characteristics dialog box

4.2.1 Quality Control Chart

Click the settings button to open the dialog box for analysis charts referring to attribute

characteristics.



Figure 4-19: Quality control chart – Binomial distribution tab

The binomial distribution is designed for attribute characteristics that can only take one of two

mutually exclusive values. Examples are e.g. good unit and bad unit or delivery on time and

unpunctual delivery or the like.

The Shewhart x-chart chart type is adjusted by default and calculated according to the normal

calculation calculation method, i.e. the control limits are calculated based on the random variation

range of the binomial distribution. The selected acceptance probability (non-interference

probability) amounts to Pa = 99,73 %.

4.2.1.1 Calculation of Control Limits – x-chart Binomial Distribution

First, the program calculates the average fraction nonconforming �� based on the characteristic

values. This value serves as an estimator for the true fraction nonconforming �� in the population.

4-6 Estimator for the true fraction nonconforming in the population

i

i

i

i

n

x

p

This estimator and the subgroup size specified in the characteristics mask are used in order to

determine the two variation limits xup and xlo of the two-sided 99,73% random variation range of the

binomial distribution.

4-7 Upper control limit of the x-chart 5,0 upxUCL

4-8 Lower control limit of the x-chart 5,0 loxLCL



Example

The SEITE_231_TAB. 7.4-3.DFQ data set (Buch_6_Auflage directory) leads to the following estimator.

4-9 Estimator for the fraction nonconforming 0433,010000

433

n

x

p

i

i

i

The subgroup size of n = 200 units is constant for all subgroups. In order to calculate the control

limits, you need both variation limits xlo and xup of the two-sided 99,73% random variation range of

the binomial distribution (traceable by selecting Extras – Probability distribution – Discrete

distributions – Binomial distribution – Random variation range).

Figure 4-20: Variation limits of the two-sided 99,73% random variation range BD(p=4,33%,n=200)



4-10 Lower variation limit 1lox

4-11 Upper variation limit 18upx

4-12 Upper control limit of the x-chart 5,185,0185,0 obxUCL

4-13 Lower control limit of the x-chart 5,05,015,0 unxLCL

Figure 4-21: Analysis x-chart of the SEITE_231_TAB. 7.4-3.DFQ data set

Note: Select QCC – Analysis QCC – Design 1 or tap <F8> in order to open the analysis x-chart.



4.2.1.2 Calculation of Control Limits – x-chart Poisson Distribution

Figure 4-22: Quality control chart dialog box – Poisson distribution tab

First, the program calculates the estimator for the average number of defects per reference unit (e.g.

soldering defect per circuit board or paint defects per body).

Please consider that this is about the sum of defects occurring in these units and not the sum of fractions nonconforming. This is an important difference compared to the x-chart based on the binomial distribution.

4-14 Estimator for average number of defects per unit in the population

i

i

i

i

n

x

Sumofunits

tsSumofdefecp

The program uses this estimator and the subgroup size n specified in the Subgroup size input field of

the characteristics mask in order to calculate the variation limits xup and xlo of the two-sided 99,73%

random variation range of the Poisson distribution.

Now you use the general calculation method to determine the control limits of the x-chart (Poisson

distribution).

4-15 Upper control limit of the x-chart 5,0 upxUCL

4-16 Lower control limit of the x-chart 5,0 loxLCL

Example

The SEITE_237_TAB. 7.4-6.DFQ data set (Buch_6_Auflage directory) leads to the following estimator.



4-17 Estimator for the average number of defects per unit 2,0400

80

n

x

p

i

i

i

i

Each subgroup contains n = 20 units. The average fraction nonconforming amounts to 20 units.

4-18 Average number of defects per subgroup 4202,0npˆ

The two variation limits of the two-sided 99,73% random variation range of the Poisson distribution

are as follows.

Figure 4-23: Variation limits of the two-sided 99,73% random variation range PD (µ=4, n=20)

4-19 Upper control limit of the x-chart (Poisson) 5,115,0115,0 upxUCL

4-20 Lower control limit of the x-chart (Poisson) 5,05,005,0 loxLCL

The lower control limit LCL does not apply since the number of defects x cannot be less than zero.



Figure 4-24: Analysis x-chart for the SEITE_237_TAB. 7.4-6.DFQ data set

Note: Select QCC – Analysis QCC – Design 1 or tap <F8> in order to open the analysis x-chart.

4.2.2 Requirements for Attribute Characteristics

Figure 4-25: Settings referring to the requirements for an attribute characteristic

The input values in this dialog box are applied in the overall evaluation of an attribute characteristic

whose result is contained in output field no. 15000. The Critical capability index line displays the

minimum value for Cpk. As an alternative, you may enter the maximum error percentage for each

characteristic class.

Please complete just one of these two lines, either the Critical capability index or the maximum error percentage line.

Reason: The calculation of the Cpk value directly depends on the percentage (see Chapter 4.2.2.1).

The program specifies a maximum error percentage of pmax = 31 ppm for each characteristic class by

default. This p-value refers to an excess proportion expected to amount to Cpk = 1,33 (one-sided) in

case of a normally distributed random variable. The p-value of a random variable of a Poisson

distribution is taken as the maximum permissible number of defects ppm.



In case the maximum error percentage (i.e. fraction nonconforming) exceeds 31 ppm or the number

of defects ppm exceeds 31, the overall evaluation is as follows.

Figure 4-26: Overall evaluation of the SEITE_237_TAB. 7.4-6.DFQ data set

4.2.2.1 Cpk for an Attribute Characteristic Value (Binomial)

First, the program estimates the fraction nonconforming in the population by dividing the sum of

fractions nonconforming by the sum of inspected units.

4-21 Estimator for the fraction nonconforming

i

i

i

i

n

x

p

Now it calculates the (1-p)quantile based on this estimator. The quantile refers to the inverse

distribution function of the standard normal distribution (µ=0, =1).

4-22 (1-��) quantile of the standard normal distribution )p1(Gu 1

p1

The last step is to divide the value of the (1-��) quantile by three.

4-23 Cpk equivalent for attribute characteristics 3

uC

p1

pk



Example: SEITE_231_TAB. 7.4-3.DFQ data set

4-24 Estimator for the fraction nonconforming 0433,010000

433

n

x

p

i

i

i

i

4-25 (1-��) quantile of the standard normal distribution 7136,1uu 0433.01p1

Figure 4-27: (1-��) quantile of the standard normal distribution for ��=4,33 %

4-26 Cpk equivalent for the attribute characteristic 57,03

7136,1

3

uC

p1

pk

Figure 4-28: Extract from form sheet design 3 (output field 5220 for Cpk completed manually)



4.2.2.2 Cpk Value for an Attribute Characteristic (Poisson)

First, the program estimates the average number of defects per unit in the population by dividing the

sum of defects by the sum of inspected units.

4-27 Estimator for the average number of defects per unit

i

i

i

i

n

x

p

Now it calculates the (1-p) quantile based on this estimator. The quantile refers to the inverse

distribution function of the standard normal distribution (µ=0, =1).

4-28 (1-��) quantile of the standard normal distribution )p1(Gu 1

p1

The last step is to divide the value of the (1-��) quantile by three.

4-29 Cpk equivalent for attribute characteristics 3

uC

p1

pk

The use of formula 4-29 for the calculation of the Cpk value for a random variable of a Poisson

distribution is controversial since the average number of defects per reference unit is not equal to the

average fraction nonconforming. In many cases, an alternative calculation is preferred based on the

estimator for the expected first time yield.

4-30 Expected first time yield (Poisson) peY

4-31 Y-quantile of the standard normal distribution )Y(Gu 1

Y

4-32 Critical capability index Cpk (Poisson) 3

uC Y

pk

Formula 4-32 is currently not available as an output point.



Comparison of results (Cpk) based on the SEITE_237_TAB. 7.4-6.DFQ example

Through formula 4-29 Through formula 4-32

p = 0.2 p = 0.2

1-p = 0.8 Y = e-p = 0.81873075

u1-p = 0.84162123 uY = 0.91053868

Cpk = 0.281 Cpk = 0.304

Table 4-1: Comparison of results for Cpk (Poisson)

4.3 Positional Tolerances Po/Pok:MPo2

The software includes an own calculation method for capability indices designed for positional

characteristics with a X-coordinate and a Y-coordinate.

4.3.1 Calculation Method Tab

The program enables the MPo2: max. probability ellipse calculation method for statistics by default.

The data are considered to be a two-dimensionally spread characteristic depending on two random

variables (x-coordinate and y-coordinate). A two-dimensional normal distribution, also referred to as

bivariate normal distribution, is adapted to these values.

Figure 4-29: Position values with an adapted bivariate normal distribution

Volume enclosed by the 2-

D normal distribution

= probability



4.3.1.1 Meaning of a Dispersion Ellipse

For a better illustration, imagine the 2-D normal distribution - upside down – as a jar of glass with a

volume V = 1 l. In case you pour 0,9973 l of water in the jar, the surface of the water forms a shape

similar to the ellipse displayed in Figure 4-29. The volume of 0,9973 l actually corresponds to the

probability of P = 99,73%. This probability is interpreted as follows. You expect 9973 of a total of 10

000 position values to lie within the ellipse.

Figure 4-30: Position values in the X-Y plot with a big Po and a small Pok dispersion ellipse

4.3.1.2 Calculation of the Potential Capability Index Po

The vector of the average taken from the bivariate normal distribution is moved to the target position.

The program determines the Po dispersion ellipse touching the tolerance circle at two single points.

The software calculates the other probability volume Vo of the 2-D normal distribution for the bigger dispersion ellipse.

The probability volume is transferred into a univariate standard normal distribution P = 1-(1-Vo)/2.

The program determines the P-quantile uP of the standard normal distribution.

The potential capability index is Po = uP/3.



4.3.1.3 Calculation of the Critical Capability Index Pok

The 2-D normal distribution remains within the vector of the average.

The program determines the Pok dispersion ellipse touching the tolerance ellipse at two single points.

The software calculates the other probability volume Vok of the 2-D normal distribution for the Pok dispersion ellipse.

The probability volume is transferred into a univariate standard normal distribution P = 1-(1-Vok)/2.

The program determines the P-quantile uP of the standard normal distribution.

The critical capability index is Pok = uP/3.

4.3.2 Target Values Tab

Figure 4-31: Settings for positional tolerances in the Target values tab

4.3.2.1 Minimum Values for Capability Indices

The software adjusts the same minimum value for all characteristic classes by default. At least 10

values have to be available in order that the program calculates a preliminary capability index. This

function helps you to calculate at least a preliminary capability index even for small series.

4.3.2.2 Automatic Adaptation of Target Values

In case the number of values does not reach the limit of 125 values, the minimum value for

capabilities is raised automatically. The software raises Cp to Cpk (for more details see Chapter

4.1.1.5).

4.3.2.3 Warning Limit for Insufficient Values (Conditionally Capable Limit)

In case the number of values does not reach a limit of 50 values, the evaluation result will be

conditionally capable instead of capable as soon as the number of values reaches or exceeds the

specified minimum values.



4.3.3 Requirements Tab

Figure 4-32: Settings for positional tolerances in the Requirements tab

4.3.3.1 Selection of Acceptance Criteria

The data has to meet the following criteria in order to be classified as capable in the overall

evaluation.

There are at least 50 values available.

The potential capability index is equal or greater than the minimum value for Po.

The critical capability index is equal or greater than the minimum value for Pok.

The data has to meet the following criteria in order to be classified as conditionally capable in the

overall evaluation.

The number of values does not reach the limit of 50 values.

The potential capability index is equal or greater than the minimum value for Po.

The critical capability index is equal or greater than the minimum value for Pok.

The data has to meet the following criteria in order to be classified as not capable in the overall

evaluation.

The potential capability index does not reach the minimum value for Po.

The critical capability index does not reach the minimum value for Pok.



4.3.4 Additional Conditions Tab

Figure 4-33: Options for positional tolerances in the Additional conditions tab

You may define additional conditions in this tab in order to specify more criteria for a characteristic

categorized as conditionally capable. The program does not activate any of these options by default.

Documents

qs-STAT Evaluation Strategy - Q-DAS