- Email: [email protected]

Quality management involves a great deal of statistics and rather than interrupt the flow of this Workbook we have taken the choice to put many of the detailed statistical tables and charts in Appendices. These are for reference only and most readers will not need the detail given here.

“By nature, all men long to know.”

Aristotle - Metaphysics Appendices

349

Appendix 1 - Standard SPC formulae for variables and attributes charts X and R chart

Median chart ( X and R )

R chart

R chart

• Centre line:

• Centre line:

Sum of R samples Number of R samples • Upper Control Limit:

Sum of R samples Number of R samples • Upper Control Limit:

R=

UCLR = D4 × R • Lower Control Limit:

LCLR = D3 × R

R=

UCL R = D4 × R • Lower Control Limit:

LCL R = D3 × R

X chart

X chart

• Centre line:

• Centre line:

X=

Sum of X samples Number of samples

• Upper Control Limit:

UCL X = X + A 2 × R • Lower Control Limit: LCL X = X − A 2 × R

UCL X = X + A 2 × R

• Lower Control Limit:

LCL X = X − A 2 × R

Estimate of σ:

Estimate of σ: σX =

Sum of X samples Number of samples • Upper Control Limit X=

R d2

σX =

X and s chart

R d2

Individuals chart (X and MR)

s chart

MR chart

• Centre line:

• Centre line:

Sum of s values s= Number of samples • Upper Control Limit:

UCLs = B4 × s • Lower Control Limit:

LCL s = B3 × s

Sum of R samples Number of R samples • Upper Control Limit: R=

UCLR = D4 × R • Lower Control Limit:

LCLR = D3 × R

X chart

X chart

• Centre line:

• Centre line:

Sum of X samples Number of samples • Upper Control Limit: X=

UCL

X

= X + A3 × s

• Lower Control Limit:

LCL

X

= X − A3 × s

Estimate of σ: σX =

350

s c4

X=

Sum of X samples Number of samples

• Upper Control Limit UCL X = X + E 2 × R

• Lower Control Limit: LCL X = X − E2 × R

Estimate of σ: σX =

R d2

Appendices

Proportion nonconforming (p chart) • Centre line:

p=

Number nonconforming (np chart) • Centre line:

Total number of nonconforming items Total number of products inspected

• Upper Control Limit:

• Upper Control Limit:

(

p 1−p UCLp = p + 3 ×

)

n

n = Average sample size • Lower Control Limit:

(

p 1−p LCLp = p − 3 ×

n × p = Average number of nonconforming items in a sample of constant size ‘n’.

UCL np = np + 3 × np 1 − np n • Lower Control Limit:

LCLnp = np − 3 × np 1 − np n

)

n

Nonconformities per unit (u chart) • Centre line:

Number of nonconformities (c chart) • Centre line:

Total number of nonconformities u= Total number of products inspected • Upper Control Limit:

UCL u = u + 3 ×

c = Average number of nonconformities found in a set of samples. • Upper Control Limit:

u

UCL c = c + 3 × c

n

• Lower Control Limit:

n = Average sample size

LCL c = c − 3 c

• Lower Control Limit:

LCL u = u −

u n

Standard formulae for all the types of variables and attributes control charts The constants referred to (D4, D3, A2, d2 etc.) in the formulae are given in Appendix 2 for various sample sizes.

Appendices

351

Appendix 2 - Standard SPC constants for variables charts X and R chart

Sample size

A2

D3

D4

d2

2 3 4 5 6 7 8 9 10

1.880 1.023 0.729 0.577 0.483 0.419 0.373 0.337 0.308

0.076 0.136 0.184 0.223

3.267 2.574 2.282 2.114 2.004 1.924 1.864 1.816 1.777

1.128 1.693 2.059 2.326 2.534 2.704 2.847 2.970 3.078

Sample size

A3

B3

B4

c4

2 3 4 5 6 7 8 9 10

2.659 1.954 1.628 1.427 1.287 1.182 1.099 1.032 0.975

0.030 0.118 0.185 0.239 0.284

3.267 2.568 2.266 2.089 1.970 1.882 1.815 1.761 1.716

0.798 0.886 0.921 0.940 0.952 0.959 0.965 0.969 0.973

X and s chart

Median chart ( X and R ) Sample size

A2

D3

D4

d2

2 3 4 5 6 7 8 9 10

1.880 1.187 0.796 0.691 0.548 0.508 0.433 0.412 0.362

0.076 0.136 0.184 0.223

3.267 2.574 2.282 2.114 2.004 1.924 1.864 1.816 1.777

1.128 1.693 2.059 2.326 2.534 2.704 2.847 2.970 3.078

Individuals chart (X and MR) Sample size

E2

D3

D4

d2

2 3 4 5 6 7 8 9 10

2.660 1.772 1.457 1.290 1.184 1.109 1.054 1.010 0.975

0.076 0.136 0.184 0.223

3.267 2.574 2.282 2.114 2.004 1.924 1.864 1.816 1.777

1.128 1.693 2.059 2.326 2.534 2.704 2.847 2.970 3.078

352

Appendices

Appendices

X R

∑X

5

4

X 3

2

1

Date

Time

Shift

Part Name:

Feature:

Specification:

Sampling frequency:

Machine:

Operator:

Notes:

Alarms: Point outside the control limits (UCL or LCL). Run of 7 points (up or down). Run of 7 points above or below the centre line. More than 2/3 of points in the middle 1/3 area. More than 1/3 of points in the outer 2/3 area. Other non-random pattern.

Appendix 3 - Sample standard SPC chart for variables ( X and R)

353

Appendix 4 - Pz table for % nonconforming Estimating the % nonconforming Pz = the proportion of process output beyond a single specification limit that is ‘z’ standard deviation units away from the process average (for a process that is in statistical control and is normally distributed).

T-

T+

σ

Knowing the Pz allows locating X to minimise the out-of-specification output.

σ

Pz

Note: This is only for one side of the curve.

Pz

Z

z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.5 4.0 354

0.00 0.5000 0.4602 0.4207 0.3821 0.3446 0.3085 0.2743 0.2420 0.2119 0.1841 0.1587 0.1357 0.1151 0.0968 0.0808 0.0668 0.0548 0.0446 0.0359 0.0287 0.0228 0.0179 0.0139 0.0107 0.0082 0.0062 0.0047 0.0035 0.0026 0.0019 0.00135 0.00023 0.00003

0.01 0.4960 0.4562 0.4168 0.3783 0.3409 0.3050 0.2709 0.2389 0.2090 0.1814 0.1562 0.1335 0.1131 0.0951 0.0793 0.0655 0.0537 0.0436 0.0351 0.0281 0.0222 0.0174 0.0136 0.0104 0.0080 0.0060 0.0045 0.0034 0.0025 0.0018

0.02 0.4920 0.4522 0.4129 0.3745 0.3372 0.3015 0.2676 0.2358 0.2061 0.1788 0.1539 0.1314 0.1112 0.0934 0.0778 0.0643 0.0526 0.0427 0.0344 0.0274 0.0217 0.0170 0.0132 0.0102 0.0078 0.0059 0.0044 0.0033 0.0024 0.0018

0.03 0.4880 0.4483 0.4090 0.3707 0.3336 0.2981 0.2643 0.2327 0.2033 0.1762 0.1515 0.1292 0.1093 0.0918 0.0764 0.0630 0.0516 0.0418 0.0336 0.0268 0.0212 0.0166 0.0129 0.0099 0.0075 0.0057 0.0043 0.0032 0.0023 0.0017

0.04 0.4840 0.4443 0.4052 0.3669 0.3300 0.2946 0.2611 0.2297 0.2005 0.1736 0.1492 0.1271 0.1075 0.0901 0.0749 0.0618 0.0505 0.0409 0.0329 0.0262 0.0207 0.0162 0.0125 0.0096 0.0073 0.0055 0.0041 0.0031 0.0023 0.0016

Z

0.05 0.4801 0.4404 0.4013 0.3632 0.3264 0.2912 0.2578 0.2266 0.1977 0.1711 0.1469 0.1251 0.1056 0.0885 0.0735 0.0606 0.0495 0.0401 0.0322 0.0256 0.0202 0.0158 0.0122 0.0094 0.0071 0.0054 0.0040 0.0030 0.0022 0.0016

0.06 0.4761 0.4364 0.3974 0.3594 0.3228 0.2877 0.2546 0.2236 0.1949 0.1685 0.1446 0.1230 0.1038 0.0869 0.0721 0.0594 0.0485 0.0392 0.0314 0.0250 0.0197 0.0154 0.0119 0.0091 0.0069 0.0052 0.0039 0.0029 0.0021 0.0015

0.07 0.4721 0.4325 0.3936 0.3557 0.3192 0.2843 0.2514 0.2206 0.1922 0.1660 0.1423 0.1210 0.1020 0.0853 0.0708 0.0582 0.0475 0.0384 0.0307 0.0244 0.0192 0.0150 0.0116 0.0089 0.0068 0.0051 0.0038 0.0028 0.0021 0.0015

0.08 0.4681 0.4286 0.3897 0.3520 0.3156 0.2810 0.2483 0.2177 0.1894 0.1635 0.1401 0.1190 0.1003 0.0838 0.0694 0.0571 0.0465 0.0375 0.0301 0.0239 0.0188 0.0146 0.0113 0.0087 0.0066 0.0049 0.0037 0.0027 0.0020 0.0014

0.09 0.4641 0.4247 0.3859 0.3483 0.3121 0.2776 0.2451 0.2148 0.1867 0.1611 0.1379 0.1170 0.0985 0.0823 0.0681 0.0559 0.0455 0.0367 0.0294 0.0233 0.0183 0.0143 0.0110 0.0084 0.0064 0.0048 0.0036 0.0026 0.0020 0.0014

Appendices

Appendix 5 - Precision and accuracy They are different Precision and accuracy are very different things as the diagram on the upper right shows. If you can imagine the diagram to be the results of a shooter then you can easily see that precision is about how close your shots are together and accuracy is about hitting the target. • Highly precise shooters will always have

good grouping but will not necessarily hit the target. • Highly accurate shooters will always hit

the target but will not necessarily hit the centre. Precision measures the scatter of measurements or results and is affected mainly by common causes, i.e. variations that are characteristic of the limitations of the process. These cannot be removed without management action.

It is possible to consider meeting specification and being in control in the same way and this is shown on the lower right. When a process is in control then it is the equivalent of being precise and it is affected mainly by common causes that relate to the limitations of the process and can only be removed by management action on the process, e.g. better machine, better tooling, etc. When a process is in specification then it is the equivalent of being accurate and it is affected mainly by special causes that can be removed by the process operator, e.g. material changes, setting changes etc. It is best to be highly precise and highly accurate just as it is best to be in control and in specification.

High precision Precision

Precision and accuracy are not the same thing Precision is about being consistent and having low variation or bias in the results. Accuracy is about consistently achieving the desired result. The ideal is to have both high precision and high accuracy.

Specification Out of specification In specification

Accuracy measures the location of the measurements and is affected mainly by special causes, i.e. variations that can be assigned specific causes. These can be removed by the process operator.

Low precision

• Tip - As a shooter, I first seek high

precision or good grouping (remove the common causes such as poor support, good trigger control) and then adjust the sights to give high accuracy. Good grouping but poor location (high precision with low accuracy) can be adjusted by the sights to give good grouping with high accuracy.

Appendices

Meeting specification and being in control are not the same thing As with precision and accuracy, being in control and in specification are not the same thing. The ideal is to be in control and in specification.

355

Appendix 6 - Measurement systems analysis

Assessing the quality of products relies on measurements but no measurement can ever be exactly correct and there is an ‘uncertainty’ associated with every measurement. This ‘uncertainty’ tells us about the overall quality of the measurement and the measurement process. • Tip - Uncertainty is not the same thing

as ‘error’ because error (wherever it comes from) can be known or estimated and corrected for. Measurement processes are the same as any other process and they will suffer from the same type of variation in results as parts. This means that a measurement is never a single point but a distribution as shown on the upper right. When measurement and the spread of uncertainty is entirely inside or outside the desired limit then the result is either definitely good or definitely bad. The problem area is when the measurement is close to the limit and the spread of uncertainty extends across the limit. In these cases, it is impossible to absolutely decide if the result is good or bad, i.e. without a knowledge of the uncertainty then we might accept ‘bad’ parts or take action when it is not necessary or reject ‘good’ parts and take no action when it is necessary.

The two approaches Controlling the measurement process is a vital and developing area of quality management and there are two basic approaches: • The ‘uncertainty’ approach: This is based

on ISO/IEC Guide 98-3:2008 ‘Guide to the expression of uncertainty in measurement’ (also known as GUM). This approach provides information on the sources of uncertainty and how they can be treated and combined to give an overall uncertainty for a given measurement. An excellent introduction to this approach is given in ‘A Beginner's Guide to Uncertainty of Measurement’ by Stephanie Bell (available free from www.npl.co.uk/publications/guides/). • The ‘measurement systems

analysis’ (MSA) approach: This is based

356

largely on the AIAG publication ‘Measurement Systems Analysis’ (see box on opposite page). This approach provides more understanding of the actual measurement process and attempts to allocate the variations in the process to allow variation reduction and measurement improvement.

Measurements tell us about the quality of the product. Uncertainty and measurement systems analysis tells us about the quality of the measurement.

For plastics processors, we believe that the measurement systems analysis approach provides a more practical method to Upper Control Limit (UCL)

Measurement

There is no right answer

Uncertainty limits

Definitely bad

Uncertain if good or bad

Measurement Definitely good

Measurement variations can affect decisions If we do not know the size of the measurement variation then when a measurement is close to the limit we may take action when it is not needed or, alternatively, take no action when action is actually needed.

Total measured variation 2 2 (σ 2total = σ product + σ GRR )

2 σproduct

2 σGRR

Measurement system variation

Product variation What we actually want to measure

Operator variation (reproducibility)

Operator

Equipment variation (repeatability)

Operator and part interaction

The total measured variation We really want to know about part variation (so we can measure and control the process) but we will always have additional variations due to the measurement system. This variation must be quantified to have confidence in the measurement.

Appendices

analyse and improve the measurement process.

the same equipment get the same result? This is the Equipment Variation (EV).

• Tip - The AIAG approach is required by

A high EV value will indicate a need for equipment maintenance or redesign.

most large automotive manufacturers.

Sources of variation Appendix 5 discussed the concepts of precision and accuracy in processes in general terms but this is equally applicable to measurement processes and the sources of variation in a measurement can be divided into those affecting accuracy and those affecting precision.

Precision is the result of random variation and it is not possible to simply ‘correct’ for these variations. The combined estimate of the measurement system reproducibility and repeatability is referred to as the Gauge Repeatability and Reproducibility (or GRR) and the variance (σ 2 ) of the GRR can be represented by:

Accuracy (location)

σ 2G RR = σ 2rep ea tability + σ 2reprod ucibility

Accuracy is being close to the real or reference value, i.e. in a measurement system it is being ‘on target’. Accuracy will be affected by:

σ repeatability = the standard deviation for

• Bias: This is when the measured value is

reproducibility.

different from a reference value. Bias can be corrected by calibration traceable to national and international standards. • Linearity: This is how the bias varies

over the measurement range, i.e. is the bias constant over the measurement range? Linearity can be accounted for by calibration. Accuracy is generally the result of systematic errors and can be corrected for by suitable corrections to the measured values. It is also possible to correct for low accuracy (however caused) by improving areas such as equipment quality, equipment maintenance, equipment calibration and environmental consistency.

Precision (spread) Precision is how close individual measurements are to each other, i.e. in a measurement system it is being ‘consistent’. Precision will be affected by:

A high AV value will indicate a need for operator training or equipment redesign to make measuring easier. • Repeatability (equipment variation):

This is the variation between multiple measurements when using a single operator or appraiser, i.e. does the same operator measuring the same part with

Appendices

It is even better if the resolution of the measuring equipment is 1/10 of the process variation.

where: repeatability. σ reproducibility = the standard deviation for

Total measurement system variation As for processes, it is possible to define the short-term measurement system capability by: σ 2capability = σ 2bias / linearity + σ 2G RR

where:

MSA assumes a measurement system that is stable and in control. As with processes, this can be checked using control charts for stability and control.

σ bias/ linearity = the standard deviation for

any uncorrected bias or linearity. It is also possible to define the long-term measurement system performance by: σ 2p erfo rm an ce = σ 2ca pab ility + σ 2s tability + σ 2con s iste ncy

where: σ stability = the standard deviation for

variations in stability. σ consistency = the standard deviation for

variations in consistency.

Stability = the change in bias/ linearity with time. Consistency = the change in repeatability with time.

• Reproducibility (appraiser variation):

This is the variation in measurements from the same measuring equipment when using different operators or appraisers, i.e. do two operators measuring the same part with the same equipment get the same result? This is the Appraiser Variation (AV).

The resolution of measuring equipment should be at least 1/10 of the range to be measured.

All you ever wanted to know The AIAG Measurement Systems Analysis (MSA) Reference Manual (4th edition 2010) is a treasure trove of information on good practice and procedures in measurement systems analysis (‘Measurement Systems Analysis’, ISBN: 978-1-60534211-5). Get a copy from AIAG (www.aiag.org). Some of the procedures are slightly contentious (see Appendix 7) but overall it is an excellent text to begin with. My original copy of this (2nd edition from 1995) is well worn. Be advised of changes in the K1, K2 and K3 values between the 3rd and the 4th editions which many calculators and spreadsheets on the Internet do not seem to have updated.

357

Appendix 7 - Evaluating measurement systems The gauge repeatability and reproducibility (GRR) study Before conducting a gauge R&R study it is necessary to assess the accuracy elements of the measurement system. This involves assessing the bias and linearity of the measurement system. Chapter III Section B of the AIAG MSA guide (see Appendix 6) gives details of how to assess bias, linearity and stability using graphical, numerical and control chart methods. This must be done before carrying out the gauge R&R study to remove accuracy (location) errors. The reader is referred to the AIAG guide for details of these.

Decide on the GRR method There are 3 different methods of carrying out a gauge study: • ANOVA method: This is the preferred method because it is the most flexible and gives the most

information but it requires good software, e.g. Minitab™, and an experienced user to interpret the output data. This method is not considered further because not all companies have both of these. • Average and range method: This is the most common method as it can be done manually or using a simple

spreadsheet. This method allows estimation of repeatability and reproducibility but not operator to part interaction. This is the method that is followed in this Workbook. • Range method: This does not provide as much information as the average and range or ANOVA methods

and it is generally used only to check that the GRR has not changed from a previous study.

Carry out the study The standard study involves 3 appraisers (operators) who take 3 readings on a set of 10 sample parts using the same gauge. The samples should represent the whole range of the process variation, i.e. do not choose samples that are all on specification, it is best to have some at the lower and higher limits so that the part variation is effectively represented. 1. Each sample is numbered and the sample number hidden from the appraisers so that they do not know which sample they are measuring. 2. Each appraiser (A, B and C) measures the 10 parts x 3 trials, i.e. 30 measurements, in a random order and the results are recorded by part number and appraiser. 3. For each appraiser, calculate the global average from the complete set of 30 results and record this as X appraiser for the relevant appraiser. 4. Calculate X D IF F from: X D IF F = M ax X a p pra ise r − M in X ap p raiser

5. For each set of samples and each appraiser calculate the range of the measured values and then calculate the average range ( R appraiser ) for each appraiser. 6. Calculate the global R from: R=

R appraiser 1 + R appraiser 2 + R appraiser 3 Number of appraisers = 3

7. Calculate UCLR from: UCL R = R × D 4 where D4 = 3.267 for 2 sets of reading and 2.574 for 3 sets of readings (see Appendix 2).

8. If any readings have a range > UCLR then discard these readings and repeat using the same appraiser and sample. Recalculate the UCLR using the new readings. 9. For each sample calculate the average of all the appraisers readings ( X part ) and then calculate Rp from: R p = M a x X p a r t − M in X p a rt .

10. You should now have 3 values, i.e. X D IF F , R and Rp.

358

Appendices

Analysing the results The conduct of the study is relatively straightforward but the analysis of the results is still contentious and there are three basic methods. These result from criticisms of the AIAG on the basis that it uses ratios which are not mathematically valid. The three methods are described below with brief comments.

1. AIAG method1

2. Wheeler and Ermer methods2,3

Repeatability (EV):

Wheeler and Ermer both agree with the AIAG calculations for EV, AV, GRR and TV as estimators of the variations (Wheeler states that they are not the only ones and are not always unbiased but they do provide reasonable estimates).

This measures the equipment variation: EV = R × K1 where K1 varies with the number of sets of readings: Trials 2 3

K1 0.8862 0.5908

Reproducibility (AV): This measures the appraiser variation: AV =

(X

DIFF

× K2

)

2

− ( EV

2

) (nr) (default to 0 if -ve)

where: n = number of appraisers (3 in the example). r = number of parts (10 in the example). K2 varies with the number of appraisers: Appraisers 2 3

K2 0.7071 0.5231

Gauge R&R (GRR) GRR = EV 2 + AV 2

However at the assessment stage, where the %’s are calculated, the AIAG simply says ‘The sum of each percent consumed by each factor will not equal 100%’ and no explanation is given for this. The reason, as rightly pointed out by both Wheeler and Ermer, is that the %GRR etc. are not percentages but are trigonometric functions and should be calculated from: %EV = ( EV

2

%AV = ( AV

2

TV 2

)

) TV 2 2 %GRR = (GRR ) TV2 2 %PV = ( PV ) TV 2 These correct proportions will add up to 100% and then represent the actual contribution of each factor to the total variation.

Part variation

3. Notes:

This measures the total part variation: PV = R p × K 3 where K3 varies with the number of parts:

1.Considerable caution should be used with the AIAG method due to the effective use of %’s of the standard deviations (σ) rather than %’s of the variances (σ2).

Number of parts 9 10

K3 0.3249 0.3146

Total variation This is the total variation of the study:

TV =

GRR 2 + PV 2

ndc (no. of distinct categories)

(

ndc = 1.41 PV

)

GRR should be > 5.

The AIAG method then calculates: • %EV = (EV/TV)% • %AV = (AV/TV)% • %GRR = (GRR/TV)% • %PV = (PV/TV)%

The GRR is assessed by: • %GRR < 10% - OK. • 10% < %GRR < 30% - Possibly OK. • % GRR > 30% - Needs improvement.

Appendices

2.Be especially careful about condemning a gauge on the basis of the 10%, 10%-30% and >30% rules. 3.If in doubt then read the papers and form your own opinion but remember that your customers may simply want an AIAG report whether it is correct or not - this is the power of the market. 4.Be aware of changes made in AIAG MSA between the 3rd and the 4th editions - the Kn values changed to reflect changes in the methods. Many of the spreadsheets available on the Internet do not reflect these changes. Check the constants before using any external spreadsheet. 5.Practitioners agree that ANOVA is best but there are still people who use average and range. •1. AIAG. 2010. ‘Measurement Systems Analysis’. AIAG. •2. Wheeler, D. 2006. ‘An Honest Gauge R&R Study’. www.spcpress.com/pdf/DJW189.pdf. •3. Ermer, D. 2006. ‘Improved Gage R&R Measurement Studies’. Quality Progress. March 2006.

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