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1 Gauge Repeatability and Reproducibility with interaction

2 ISO 9000 Statistical Process Control, SPC QS 9000 Gauge Repeatibility and Reproducibility, GR&R QS 9000 Classical GR&R Long Form Classical GR&R Long Form MSE ANOVA npk npk iv

3 ... iv...v... vii...xiv GR&R ANOVA GR&R Classical GR&R Long Form Method) GR&R...43 GR&R GR&R...47 GR&R v

4 GR&R GR&R GR&R B vi

5 Classical R&R QS9000 P/T QS9000 GR&R GR&R GR&R GR&R GR&R ANOVA GR&R GR&R ANOVA GR&R vii

6 GR&R GR&R GR&R GR&R GR&R GR&R GR&R GR&R GR&R GR&R GR&R GR&R GR&R...74 viii

7 5.16 GR&R GR&R GR&R GR&R GR&R GR&R GR&R GR&R GR&R ix

8 5.30 GR&R GR&R GR&R GR&R GR&R GR&R GR&R GR&R x

9 GR&R GR&R GR&R σ = 0.5 npk GR&R σ = npk GR&R σ =.5 npk GR&R npk MAD GR&R npk MSE GR&R npk GR&R npk GR&R npk ,3, σ =.5 10,6, 10,3, ,3, 10,6, 10,3,4 MAD ,3, 10,6, 10,3,4 MSE ,3, xi

10 10,6, 10,3, ,6, 40,3, 0,3,4 MAD ,6, 40,3, 0,3,4 MSE ,3, 5,6, 5,3,4 MAD ,3, 5,6, 5,3,4 MSE GR&R GR&R σ = 0.5 npk GR&R σ = 1 npk GR&R σ = 1.5 npk GR&R npk MAD GR&R npk MSE GR&R npk GR&R npk GR&R xii

11 npk ,3, σ = ,6, 10,3, ,3, 10,6, 10,3,4 MAD ,3, 10,6, 10,3,4 MSE ,3, 10,6, 10,3, ,6, 40,3, 0,3,4 MAD ,6, 40,3, 0,3,4 MSE ,3, 5,6, 5,3,4 MAD ,3, 5,6, 5,3,4 MSE GR&R...14 xiii

12 ACTUAL C OBSERVED C R&R...15 P P GR&R GR&R GR&R GR&R GR&R xiv

13 GR&R GR&R GR&R GR&R GR&R GR&R GR&R GR&R GR&R GR&R xv

14 GR&R GR&R GR&R NPK GR&R NPK GR&R NPK GR&R NPK MAD GR&R NPK MSE GR&R NPK GR&R NPK GR&R NPK GR&R NPK GR&R NPK GR&R NPK GR&R NPK MAD GR&R NPK MSE GR&R NPK GR&R NPK GR&R NPK...10 xvi

15 1.1 ISO International Organization for Standardization ISO Accuracy Precision ISO QS 9000 QS 9000 QS

16 QS QS 9000 QS 9000 QS 9000 Supplier Quality Assurance Manual Q101 Quality System Standard North American Operations Targets for Excellence QS 9000 Measurement Systems Analysis, MSA QS9000 ISO 9000 Gauge Variation Study Gauge Repeatability and ReproducibilityGR&R GR&R QS9000 MSA MSA GR&R P/T Precision-to-Tolerance 5.15 Tolerance P/T 10% Precision P/T 10% 30% P/T 30% GR&R

17 n p k GR&R Mandel 197 ANOVAAnalysis of Variance GR&R Montgomery and Runger 1993a Classical GR&R Studies GR&R MSA 1991 Long Form P/T 1. GR&R GR&R ANOVAClassical GR&R Long Form Classical GR&R Long Form GR&R GR&R n p k GR&R 3

18 npk GR&R

19 Non-Calibration Non-Linearity Non-Stability GR&R GR&R GR&R GR&R n p k GR&R npk GR&R 5

20 GR&R GR&R 1.1 6

21 GR&R GR&R GR&R Measurement Measuring InstrumentGage Gauge Measurement Value Measuring System True Value Reference Value Measurement Error Accuracy of measurement Bias 7

22 Precision Calibration Confidence Level Gauge Gage Inspection Linearity Laboratory Process Capability Analysis Quality 8

23 Repeatability Reproducibility Gauge Repeatability and ReproducibilityGR&R Stability Traceability Test. Mandel 197 Repeatability Laboratory Reproducibility Analysis of VarianceANOVA.1. σ σ L MS W MS L 9

24 ( ) = σ E MS W.1 E ( MS ) L σ + nσ =. L 95 Variance Components.3.4 σ = MS W.3 L = / L W ( MS MS ) n σ.4 m σ σ m σ m σ m ( ) ( ) σ + σ m ( ( )) ( ) σ + σ m σ + σ m 95% L L ( ( )) ± L Repeatability = 1.96 σ m or.77 σ m.5 Reproducibility = 1.96 ( σ + ( σ m) ) or.77 σ + ( σ m) L.6 L Tsai

25 Tsai Mandel197 Tsai Tsai ANOVA Interaction po Normal Distribution ANOVA σ O σ P σ PO σ e.7.8 Repeatability = σ e.7 Reproducibility = σ O + σ PO.8 ANOVA σ PO Repeatability = σ e.9 11

26 Reproducibility = σ O.10 GR&R 5.15 Repeatability + Reproducibility Tolerance < 10%.11 Tolerance.11 McNeese and Klein 1991 Product Variation Measurement Variation Ford Capability of Precision.1 CpTolerance / 6σ.1 Tolerance σ Accurancy Precision GR&R.13 1

27 σ ms % GR & R = σ total σ ms σ repeatability + σ reproducibility σ total σ repeatability + σ reproducibility + σ part σ part AIAG (Automotive Industry Action Group) Measurement System Analysis James and Avenue Barrentine

28 .1 Repeatability Calibration Stability Linearity Long Form 4 4 npk GR&R GR&R GR&R observed processed variation actual process variation GR&R Cp CpTolerance/6σ.14 Tolerance σ. Actual C Observed P C R&R ActualC P P ObservedC C. P p 14

29 . Actual C Observed P C R&R P 0.5~0.6 %R&R %R&R 1. R&R R&R. R&R 3. Actual C pk Observed C pk 4. %R&R 30 0% C R&R p σ o σ σ A R& R 15

30 o A R& R σ = σ + σ.15 tolerance C PO = 6σ tolerance 5.15σ R& R 100 C PA = % R & R = 6σ tolerance A Tolerance.15 O 1 C PA =.16 1 % R & R 6 6 C PO Wheeler GR&R GR&R Interclass Correlation CoefficientDiscrimination Ratio Montgomery and Runger1993a Tsai ANOVA Classical Gauge Repeatability and Reproducibility Study (X bar/r chart) (X bar chart) (R chart)

31 R o = R j= 1 j O j1,,o O.17 R σ repeatability =.18 d R j j R d R d.19.0 X R X = X j min X j j j max.19 R X σ reproducib ility =.0 d X j max X j j X j min X j j X j d σ gauge = σ repeatability + σ reproducibility.1 Classical GR&R R R Chart GR&R P/T (Precision to Tolerance Ratio) P T 6σ gauge = Tolerance. Tolerance. P/T 10 17

32 P/T σ product σ total σ total σ σ product gauge.3.4 σ gauge GR&R = 100 σ product.3 σ gauge GR&R = 100 σ total.4 Montgomery and Runger1993b 1993a Factorial Design Model Nested Design Model Modified ANOVA ANOVA.5 X ijk i = 1,, K, n = µ + Pi + O j + POij + Rijk j = 1,, K, p.5 k = 1,, K, k X ijk P i O j (PO) ij R ijk unbiased σ σ O σ PO σ R P 18

33 E E E E ( MS R ) = σ R ( MS PO ) = σ R + kσ PO ( MS O ) = σ R + kσ PO + nkσ O ( MS ) = σ + kσ + pkσ P R PO P.6 σ R σ PO σ P σ O = MS = = = R ( MS MS ) ( MS MS ) P k pk ( MS MS ) nk O PO PO PO R.7 repeatability = σ σ R = MS R.8 σ reproducibility = σ O + σ PO.9 = ( MS + ( n 1) MS nms ) nk O PO R.8.9 σ gauge = σ repeatability + σ reproducibility.30 = ( MS + ( n 1) MS + n( k 1) MS ) nk O PO R 19

34 < 0 σ po σ reproducibility (Full Model) Modified ANOVA.31 E E E ( MS R ) = σ R ( MS O ) = σ R + nkσ O ( MS ) = σ + pkσ P R P σ R σ P σ O = MS = = R ( MS MS ) P pk ( MS MS ) nk O R R.3 reproducibility = σ σ O.33 = ( MS MS ) nk O R.3.33 σ gauge = σ repeatability + σ reproducibility.34 = ( MS + ( nk 1) MS ) nk O R Hoguet and Hartman1994 Miles Short Method 0

35 Phil Lunney Floyd and Laurent1995 SPC Measurement Error Studies Measurement Error RatioMER MER = 6 S d Tolerance.35 S d Tolerance.35 Jaynes 1997 (Learning Curve).1 ( Max) ( Max) 10 ~3 GR&R=15% GR&R=0% 10 ~3 GR&R=15% GR&R=0% 1

36 AIAG (MSA).1 GR&R BurdickLarsen1997 Montgomery and Runger1993b 3 GR&R P/T (Precision to Tolerance Ratio).36 6 γ P/T = 100% Tolerance.36 γ Fruit1997 P/T Dolezal1998 BurdickLarsen1997 (Mixed-Effect Model) (Fixed Effect) (Random Effect)

37 Morchower1999 (Location Effect) (Sequence Effect) ANOVA GR&R Long Form GR&R.37 P T σ gauge 5.15 = 100% Tolerance.37 P/T Precision-to-Toleranceσ gauge Tolerance P/T 10% P/T 10%0% P/T 0%30% P/T 30% Burdick Larsen 3

38 003 P/T R&R P/T. R&R Reproducibility Repeatability Montgomery and Runger1993a Classical GR&R Classical GR&R MSA 1991Barrentine1991 Burdick Larsen Dolezal1998 n p k GR&R Classical GR&R npk 4

39 Classical GR&R Classical GR&R npk npk. 5

40 . Repeatability =.77 σ m Reproducibility=.77 σ + ( σ m) L σ σ L Mandel Repeatability + Reproducibility Tolerance Repeatability = Reproducibility = σ e σ O + σ PO Tsai1988 ms % GR & R 100 = σ σ total σ σ ms total McNeese and Klein σ R& R 100 % R & R = Tolerance σ R&R Barrentine 1991 Tolerance σ gauge GR&R = 100 σ product σ gauge GR&R = 100 σ total σ σ σ gauge product total Montgomery and Runger 1993b 6

41 . GR&R = 5.15 x R d GR & R % R & R = 100 Tolerance Tolerance d R Hoguet and Hartman 1994a 1994b 1994c MER = 6 S d Tolerance MER Floyd and S d Laurent1995 Tolerance Precision to Tolerance Ratio 6 γ 100% Tolerance γ Burdick Larsen1997 Tolerance 7

42 GR&R 3.3 GR&R GR&R GR&R Repeatability 3.1 Repeatability 8

43 Reproducibility 3. C A B Reproducility 3. σ = σ + σ 3.1 gauge repeatability reproducibility σ gauge σ σ repeatability reproducibilityy σ = σ + σ 3. Total part gauge σ σ Total σ gauge part σ = σ + σ + σ 3.3 Total part repeatability reproducibility 9

44 Gauge Repeatability and Reproducibility GR&R (3.1) Variance Components 3. GR&R ANOVA Classical GR&R Long Form 3..1 ANOVA Montgomery and Runger1993b ANOVA Random Effect Random Model 30

45 n p p n k p y 111 y 11 y 11 y 1 y 1p1 y 1p y 11k y 1k y 1pk y 11 y 1 y 1 y y p1 y p y 1k y k y pk y ij1 y ij y ijk y n11 y n1 y n1 y n y np1 y np y n1k y nk y npk 3.4 Y ijk i = 1,, K, n = µ + Pi + O j + POij + Rijk j = 1,, K, p 3.4 k = 1,, K, k µ P i 31

46 O j (PO) ij R ijk 1. P i NID0,σ P. O j NID0,σ O 3. PO ij NID0,σ PO 4. R ijk NID0,σ R σ P σ O σ PO σ R

47 3. Source of Variability Sum of Squares Degrees of Freedom Mean Squares Expected Mean Squares SSp n p n1 MSp E( MS P ) = σ R + kσ PO + pkσ P Parts SSo n o p1 MSo E( MS O ) = σ R + kσ PO + nkσ O Operators np o SSpo Parts*Operators n1p1 E MS PO = σ R + kσ PO SS R n R npk1 MS R E( MS ) R Error R = σ Total SS T npk1 σ R σ PO σ P σ O = MS = = = R ( MS MS ) ( MS MS ) P k pk ( MS MS ) nk O PO PO PO R σ σ σ repeatability reproducibility gauge = σ = = σ R = σ = repeatability = MS ( MS + ( n 1) MS nms ) nk ( MS + ( n 1) MS + n( k 1) MS ) nk O O + σ O R PO + σ reproducibility PO PO R R 3.6 σ po < 0 σ reproducibility 3. 33

48 3.7 i = 1,, K, n Y ijk = µ + Pi + O j + Rijk j = 1,, K, p 3.7 k = 1,, K, κ Source of Variability Sum of Squares Degrees of Freedom Mean Squares E = σ R + Parts E MS R Operators O = σ + nkσ n R npknp SS R MS R E( MS ) R Error 1 R = σ Total SS T npk1 Expected Mean Squares SSp n p n1 MSp ( MS P ) pkσ SSo n o p1 MSo ( ) P O σ R σ P σ O = MS = = R ( MS MS ) P pk ( MS MS ) nk O R R

49 σ σ σ repeatability reproducibility gauge = = σ = σ = = MS ( MS MS ) repeatability nk ( MS + ( nk 1) MS ) nk O R = σ O O R + σ R reproducibility R GR&R Classical GR&R Montgomery and Runger1993a Classical Gauge Repeatability and Reproducibility Study 3.4 Classical R&R R R d d d GR&R X o R R j = j= 1 O j1,, O O 3.10 R σ 3.11 repeatabil ity = d 3.4 Classical R&R 1 O x 111 x 11 X 11 R11 x 11 x 1 X 1 R1 x 1o1 x 1o X 1 o 1 x 11 x 1 X 1 R1 x 1 x X R x o1 x o X o R o n x n11 x n1 X n1 x n1 x n X n x no1 x no X no R X 1 1 X R o R o X R o R j 35

50 d B R X max X X 3.1 = j min j j j R X σ reproducib ility = 3.13 d X j max X j j X j min X j j X j d σ gauge = σ repeatability + σ reproducibility 3.14 Classical R&R R Chart ANOVA 3..3 Long Form Method) QS 9000 MSA1991 Long Form

51 ABC 110 A B C a R R b R c a R R b c R 17 R R D 3 D R UCL R LCL R X a 9b X 14c X X a X b X c X diff 37

52 3.3 38

53 3.3 39

54 X 16 Rp R X diff Rp EVAVR&R PVTV EV AV R&R PV 3.5 #Trials D D Barrentine1991 K1 K o σ repeatability R R j = O j1 j=1 O O X -R R σ repeatability = 3.15 d Repeatability5.15 σ 3.16 repeatability K 1 40

55 5.15 R K 1 R d 5.15 K 1 = 3.17 d σ reproducibility = X d diff X diff X max X min σ reproducibility d * X 16 d * d σ reproducibility = d * diff K 5.15X diff 5.15 σ reproducib ility = = K X diff * d 5.15 K = 3.18 * d Barrentine1991 ANOVA Expected Mean Square 3.6 σ + R orσ P σ R + rnσ O σ σ σ R P O 41

56 4 R σ X -R = d R R σ * reproducibility X = d diff σ rn rn d O R diff * reproducibility X σ σ σ + = = * reproducibility X = + = d rn diff R O σ σ σ R σ rn d R d rn d diff R diff O * * X X = = σ σ AV 5.15 ( ) ( ) ( ) = = rn d AV R diff O * X σ σ

57 ( σ R ) ( AV ) ( K X diff ) = 3.4 rn AV ( ) ( EV ) K X diff = 3.5 rn EV5.15σ R 3.3 GR&R 3. QS9000 GR&R P/T 99 GR&R P/T 3.15 P T 5.15σ gauge = 100% Tolerance 3.15 σ = σ + σ Tolerance P/T gauge repeatability reproducibility QS9000 P/T P/T 10 % 10 % ~ 30 % 30 % GR&R

58 σ gauge GR&R 100 σ Total σ = σ + σ σ = σ + σ + σ gauge repeatability reproducibility Total part repeatability reproducibility σ part P/T QS9000 GR&R GR&R & P/T 10 % 10 % ~ 30 % 30 % GR&R

59 GR&R Classical GR&R 16 GR&R 4

60 ANOVA. Classical GR&R 3. Long Form GR&R

61 GR&R GR&R Classical GR&R ANOVA Classical GR&R ANOVA GR&R GR&R Classical GR&R 4.1 GR&R Montgomery and Runger Classical Gauge Repeatability and Reproducibility (X bar/r chart) σ = σ + σ Total product gauge σ Total σ σ product X bar/r chart (X bar chart) gauge 47

62 4.1 1 MEASUREMENTS PART NO. 1 X R X =.3 R = UCL= LCL=

63 4 UCL= R 1.0 σ gauge = = = d X i X i= 1 σ Total = = σ Total = 3.17 n 1 product σ σ Total σ gauge product σ P/T (Precision to Tolerance Ratio) P/T 49

64 P T 6σ gauge = = USL LSL ( 0.887) = 5.3 = < 10% USL=60 LSL=5 P/T σ gauge GR&R 100 % = 100% = 9.% σ 3.04 product σ gauge GR&R 100 % = 100% = 7.9% σ 3.17 total P/T GR&R Montgomery and Runger 4. GR&R ANOVA Classical GR&R Long Form

65 4. 3 OPERATOR 1 OPERATOR OPERATOR 3 MEASUREMENTS MEASUREMENTS MEASUREMENTS PART NO X R 1 X R 1 X R X 1 =.30 1 R = 1.00 X =.8 R = 1.5 X 3 =.60 R 3 = GR&R 51

66 4.3 GR&R Repeatability Reproducibility Classical GR&R σ repeatabil ity = R d σ reproducib ility = R d X ANOVA with interaction σ repeatability = MS σ MS ( n ) R reproducibility = O + 1 ( MS nms ) nk PO R 15 rn Long Form EV/5.15 ( ) ( EV ) AV 5.15 = K X diff ( 5. ) GR&R Repeatability Reproducibility Gauge Classical GR&R ANOVA with interaction Long Form GR&R Classical GR&R ANOVA ANOVA ANOVA P 95 α0.05 H 0 0 H 1 0 P H 0 5

67 4.5 GR&R ANOVA P Source of Sum of Degrees of Mean P-Value Variability Squares Freedom Squares Operators Parts Parts*Operators Error Total GR&R Repeatability Reproducibility Gauge Classical GR&R Long Form ANOVA without interaction Classical GR&R

68 4.7 3 OPERATOR 1 OPERATOR OPERATOR 3 MEASUREMENTS MEASUREMENTS MEASUREMENTS PART NO X R 1 X R 1 X R X 1 = R = 0.65 X = 4.6 R = 0.6 X 3 = 4.7 R 3 = P 0.05 Classical GR&R 54

69 4.8 GR&R ANOVA P Source of Sum of Degrees of Mean P-Value Variability Squares Freedom Squares Operators Parts Parts*Operators Error Total Classical GR&RLong Form ANOVA Classical GR&RLong Form ANOVA Classical GR&R Long Form ANOVA Classical GR&R Long Form 4.9 GR&R Repeatability Reproducibility Gauge Classical GR&R Long Form ANOVA with interaction Classical GR&R 55

70 GR&R GR&R GR&R GR&R GR&R GR&R Classical GR&R Classical GR&R GR&R Classical GR&R 56

71 GR&R Mandel Tsai Fruit 57

72 Classical GR&R GR&R Classical GR&R k n X 1 j X 1j1 X 1j X 1jk / K j1,,, o X j n = X i1 n j1,,, o i= j X max = max X 1, X, L, X o X min = min X 1, X, L, X o R diff = R = X max X min 5. X Classical GR&R X 1 j X 1j1 X 1j X 1jk / K j1,,, o max X 1 j = max( X 11, X 1, L X 1o ) min X 1 j = min( X 11, X 1, L X 1o ) j j 58

73 max j1,,, o 5.3 R = X 1 j X 1 j j j 1 min R n = R i = 1 i n i=1,,, n Reproducibility Modified Classical GR&R σ reproducib ility = R d Modified ( ) ( ) Long Form EV AV = K R rn Classical GR&R 5. Montgomery and Runger1993a 4. Excel 1 59

74 3 4 ANOVA with interaction model ANOVA without interaction modelclassical GR&Rwithout interaction modelmodified Classical GR&R with interaction modellong Formwithout interaction model Modified Long Formwith interaction model GR&R GR&R A A 3A 4A 1B B 3B 4B Repeatability Reproducibility Gauge A 1A A ANOVA P-value 0.05 Mean Squares 60

75 P-value GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Parts*Operators Error Total A 5.4 ANOVA ANOVA Modified Classical GR&R Modified Long Form ANOVA ANOVA 61

76 ANOVA Long Form Modified Long Form ANOVA Modified Classical GR&RClassical GR&R Long Form ANOVA Classical GR&R Long Form 5.4 GR&R Repeatability Reproducibility Gauge ANOVA with interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction GR&R 5.6 Repeatability Reproducibility Gage ANOVA CR&R Modified CR&R Long Form Modified Long Form 5.4 GR&R 1B 6

77 1B A ANOVA P-value 0.05 Mean Squares P-value GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Parts*Operators Error Total B 5.6 ANOVA ANOVA 63

78 Modified Classical GR&R Modified Long Form ANOVA ANOVA ANOVA Modified Long Form ANOVA Modified Classical GR&RClassical GR&R Long Form ANOVA Classical GR&R 5.6 GR&R Repeatability Reproducibility Gauge ANOVA with interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction Repeatability Reproducibility Gage ANOVA CR&R Modified CR&R Long Form Modified Long Form 5.6 GR&R Long Form 64

79 5.6 GR&R 5.6 A A A ANOVA P-value 0.05 Mean Squares A 5.8 ANOVA ANOVA

80 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Parts*Operators Error Total Modified Classical GR&R Modified Long Form ANOVA ANOVA ANOVA 5.7 ANOVA Modified Classical GR&R ANOVA Modified Long FormClassical GR&R Long Form ANOVA Classical GR&R Long Form 5.8 GR&R Repeatability Reproducibility Gauge ANOVA with interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction GR&R

81 ANOVA CR&R Modified CR&R Long Form Repeatability Reproducibility Gage Modified Long Form 5.8 GR&R B B A ANOVA P-value 0.05 Mean Squares 67

82 P-value GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Parts*Operators Error Total B 5.10 ANOVA ANOVA Modified Classical GR&R Modified Long Form ANOVA ANOVA Modified Long Form ANOVA Modified Classical GR&R Classical GR&R Long Form ANOVA Classical GR&R 5.10 GR&R Repeatability Reproducibility Gauge ANOVA with interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction Long Form 68

83 5.10GR&R 5.10 ANOVA CR&R Modified CR&R Long Form Repeatability Reproducibility Gage Modified Long Form 5.10 GR&R 3A A A

84 5.11ANOVA P-value 0.05 Mean Squares 5.11 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Parts*Operators Error Total A 5.1 ANOVA ANOVA Modified Classical GR&R Modified Long Form ANOVA ANOVA 5.1 GR&R Repeatability Reproducibility Gauge ANOVA with interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction

85 Modified Classical GR&R ANOVA Modified Long FormClassical GR&R Long Form ANOVA Classical GR&R Long Form 5.1 GR&R 5.1 Repeatability Reproducibility Gage ANOVA CR&R Modified CR&R Long Form Modified Long Form 5.1 GR&R 3B 3B A ANOVA P-value 0.05 Mean Squares P-value

86 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Parts*Operators Error Total B 5.14 ANOVA ANOVA Modified Classical GR&R Modified Long Form ANOVA ANOVA Modified Classical GR&R ANOVA Modified Long FormClassical GR&R Long Form ANOVA Classical 7

87 GR&R Long Form 5.14 GR&R Repeatability Reproducibility Gauge ANOVA with interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction GR&R 5.14 Repeatability Reproducibility Gage ANOVA CR&R Modified CR&R Long Form Modified Long Form 5.14 GR&R 4A 4A A ANOVA P-value 73

88 0.05 Mean Squares P-value GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Parts*Operators Error Total A 5.16 ANOVA ANOVA Modified Classical GR&R Modified Long Form ANOVA ANOVA 74

89 ANOVA Modified Long Form ANOVA Modified Classical GR&RClassical GR&R Long Form ANOVA Classical GR&R Long Form 5.16 GR&R Repeatability Reproducibility Gauge ANOVA with interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction GR&R 5.16 ANOVA CR&R Modified CR&R Long Form Repeatability Reproducibility Gage Modified Long Form 5.16 GR&R 4B 4B A

90 5.17 ANOVA P-value 0.05 Mean Squares P-value GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Parts*Operators Error Total B 5.18 ANOVA ANOVA 76

91 Modified Classical GR&R Modified Long Form ANOVA ANOVA ANOVA Modified Long Form ANOVA Modified Classical GR&RClassical GR&R Long Form ANOVA Classical GR&R Long Form 5.18 GR&R Repeatability Reproducibility Gauge ANOVA with interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction GR&R 5.18 ANOVA CR&R Modified CR&R Long Form Repeatability Reproducibility Gage Modified Long Form 5.18 GR&R 77

92 Modified Classical GR&R Modified Long Form Classical GR&R Long Form GR&R GR&R 5.. Modified Classical GR&R Modified Long Form P-value 0.05 H A 6A 7A 8A 5B 6B 7B 8B 5A 5A A ANOVA 78

93 P-value P-value 0.05 P-value 0.05 Mean Squares Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Error Total A 5. Classical GR&R 79

94 Modified Classical GR&R Long Form Modified Long Form ANOVA GR&R ANOVA GR&R Long Form Modified Long Form ANOVA Classical GR&R Modified Classical GR&R ANOVA GR&R Classical GR&R Modified Classical GR&R ANOVA 5. GR&R Repeatability Reproducibility Gauge ANOVA without interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction ANOVA CR&R Repeatability Reproducibility Gage Modified CR&R Long Form Modified Long Form 5.0 GR&R 80

95 5.GR&R 5.0 5B 5B A ANOVA P-value P-value 0.05 P-value 0.05 Mean Squares Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators

96 5.4 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Error Total B 5. 5 Classical GR&R Modified Classical GR&R Long Form Modified Long Form ANOVA GR&R ANOVA GR&R Long Form Modified Long Form ANOVA Classical GR&R Modified Classical GR&R ANOVA GR&R Classical GR&R Modified Classical GR&R ANOVA 5.5 GR&R Repeatability Reproducibility Gauge ANOVA without interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction GR&R

97 ANOVA CR&R Repeatability Reproducibility Gage Modified CR&R Long Form Modified Long Form 5. GR&R 6A 6A A ANOVA P-value

98 5.6 Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Error Total P-value 0.05 Mean Squares 5.8 6A 5.8 Classical GR&R Modified Classical GR&R Long Form Modified Long Form GR&R Long Form Modified Long Form 5.8 GR&R Repeatability Reproducibility Gauge ANOVA without interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction

99 ANOVA Long Form Modified Long Form ANOVA Classical GR&R Modified Classical GR&R 5. 8GR&R 5.4 Repeatability Reproducibility Gage ANOVA CR&R Modified CR&R Long Form Modified Long Form 5.4 GR&R 6B 6B A ANOVA P-value P-value 0.05 P-value 0.05 Mean Squares 85

100 Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Error Total B 5.31 Classical GR&R Modified Classical GR&R Long Form Modified Long Form GR&R Long Form Modified Long Form 86

101 ANOVA Long Form Modified Long Form ANOVA Classical GR&R Modified Classical GR&R 5.31 GR&R Repeatability Reproducibility Gauge ANOVA without interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction GR&R 5.6 Repeatability Reproducibility Gage ANOVA CR&R Modified CR&R Long Form Modified Long Form 5.6 GR&R 7A 7A A

102 5.3 ANOVA P-value P-value 0.05 Mean Squares Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Error Total

103 5.34 7A 5.34 Classical GR&R Modified Classical GR&R Long Form Modified Long Form ANOVA GR&R ANOVA GR&R Long Form Modified Long Form 5.34 GR&R Repeatability Reproducibility Gauge ANOVA without interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction ANOVA CR&R Repeatability Reproducibility Gage Modified CR&R Long Form Modified Long Form 5.8 GR&R ANOVA Long 89

104 Form Modified Long Form ANOVA Classical GR&R Modified Classical GR&R 5.34GR&R 5.8 7B 7B A ANOVA P-value P-value Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators

105 5.36 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Error Total P-value 0.05 Mean Squares B 5.37 Classical GR&R Modified Classical GR&R Long Form Modified Long Form ANOVA GR&R ANOVA GR&R Long Form Modified Long Form ANOVA Long Form Modified Long Form ANOVA Classical GR&R Modified Classical GR&R 5.37 GR&R Repeatability Reproducibility Gauge ANOVA without interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction

106 5.37GR&R 5.30 ANOVA CR&R Repeatability Reproducibility Gage Modified CR&R Long Form Modified Long Form 5.30 GR&R 8A 8A A ANOVA

107 5.38 Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Error Total P-value P-value 0.05 P-value Mean Squares A 5.40 Classical GR&R Modified Classical GR&R Long Form Modified Long Form GR&R Long Form Modified Long Form ANOVA Long Form Modified Long Form ANOVA Classical GR&R Modified Classical GR&R 8A 93

108 5.40 GR&R Repeatability Reproducibility Gauge ANOVA without interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction GR&R 5.3 Repeatability Reproducibility Gage ANOVA CR&R Modified CR&R Long Form Modified Long Form 5.3 GR&R 8B 8B A ANOVA P-value P-value 0.05 Mean Squares 94

109 Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators Parts Error Total B 5.43 Classical GR&R Modified Classical GR&R Long Form Modified Long Form ANOVA GR&R ANOVALong Form Modified Long Form 95

110 5.4 Long Form Modified Long Form ANOVA Classical GR&R Modified Classical GR&R 8B 5.43 GR&R Repeatability Reproducibility Gauge ANOVA without interaction Classical GR&R Modified Classical GR&R with interaction Long Form Modified Long Form with interaction GR&R 5.34 ANOVA CR&R Repeatability Reproducibility Gage Modified CR&R Long Form Modified Long Form 5.34 GR&R Modified Classical GR&R Modified Long Form 96

111 ANOVA Classical GR&R Long Form Modified Classical GR&R Modified Long Form 97

112 6.1 GR&R 5.1, Classical GR&R Long Form 5. GR&R Modified Classical GR&R Modified Long Form ANOVA GR&R n p k GR&R ANOVAClassical GR&RCRR Long FormLF Modified Classical GR&RMCRR Modified Long FormMLF.5 GR&R n p k σ σ P σ σ PO R O GR&R 98

113 ANOVACRRLFMCRRMLF GR&R GR&R GR&R Mean Absolute Deviation, MAD mean squares error, MSE Mean Ratio MAD MAD

114 10000 σ P σ O σ PO σ R ANOVA Classical R&RModified Classical R&RLong Form Modified Long Form

115 6.3 θ 6.4 MSE Bias Var MSE = = = E[ θ θ ] [ E( θ ) θ ] + E[ θ E( θ )] Bias + Var( θ ) ( θ ) MSE ( ) ( ) ( )

116 GR&R 6.3 GR&R npk GR&R npk n p k GR&R GR&R σ 0 σ 1 P σ 1 σ 0.5 PO npk GR&R GR&R npk 6.1 ANOVA GR&R R O 10

117 6.1 GR&R npk σ = 0.5 npk ANOVA CRR MCRR LF MLF ANOVA GR&R npk GR&R 6. GR&R npk 6. GR&R npk ANOVA MLFMCRRCRR LF npk MLFMCRRCRRLF MLF MCRR CRR LF

118 ANOVA CRRLFMLF MCRR 6. GR&R npk σ = ANOVA CRR MCRR LF MLF GR&R npk 6.3 GR&R npk ANOVA ANOVA MLF MCRRCRR npk MLF CRR MCRR LF MLF MCRR CRR LF 6.3 ANOVA 104

119 CRRLFMLF MCRR 6.3 GR&R npk σ =.5 ANOVA CRR MCRR LF MLF GR&R npk AVOVA ANOVA MLFMCRRCRRLF MLF MCRR CRR LF npk MLFMCRR LF CRR GR&R MLF MAD ANOVA MAD 105

120 MCRR LF GR&R MAD npk GR&R MAD MAD 6.4 GR&R npk MAD ANOVA CRR MCRR LF MLF GR&R npk MAD MSE GR&R MAD MLF MSE ANOVA MCRRLF CRRMCRR CRR npk MSE 106

121 6.5 GR&R npk MSE ANOVA CRR MCRR LF MLF GR&R npk MSE ANOVA 1 GR&R 6.6 GR&R npk ANOVA CRR MCRR LF MLF

122 6.7 GR&R npk npk GR&R ANOVA GR&R ANOVA 1 MLFMCRRCRR 1 LF npk MLF MCRR 1CRR LF 1 6.7ANOVA CRRLF MLF MCRR ANOVA 1 MLF MCRRCRR LF npk ANOVA 6.7 GR&R npk ANOVA CRR MCRR LF MLF

123 6.8 GR&R npk GR&R 1 MLF MCRR 1CRR LF ANOVA CRRLFMLF MCRR 6.8 GR&R npk ANOVA CRR MCRR LF MLF GR&R npk 109

124 ANOVA MLF MCRR LF CRR MCRR MLF npk MSE MLF ANOVA MCRRLF CRR ,3, 10,6, 10,3,4 6.9 n, p, k ,3, 10,6, 10,3,4 σ =.5 n p k ANOVA CRR MCRR LF MLF MAD MAD MAD ,3, 10,6, 10,3,4 MAD n p k ANOVA CRR MCRR LF MLF MSE

125 MSE ,3, 10,6, 10,3,4 MSE n p k ANOVA CRR MCRR LF MLF n, p, k ,3, 10,6, 10,3,4 n p k ANOVA CRR MCRR LF MLF npk npk MAD MSE MAD MAD ,6, 40,3, 0,3,4 MAD n p k ANOVA CRR MCRR LF MLF

126 MSE MSE ,6, 40,3, 0,3,4 MSE n p k ANOVA CRR MCRR LF MLF MAD MAD ,3, 5,6, 5,3,4 MAD n p k ANOVA CRR MCRR LF MLF MSE ,6, MSE ,3, 5,6, 5,3,4 MSE n p k ANOVA CRR MCRR LF MLF MSE MLF ANOVA MCRRLF CRR MCRR 11

127 CRR MLF MLF MCRR MLF MCRR npk ANOVA MCRR MLF CRR LF npk 6.17 GR&R 6.17 GR&R MAD MSE ANOVA CR&R MCR&R LF MLF 1 1 MCRR MLF σ 0 σ 1 P σ 0 σ PO GR&R npk ANOVA GR&R npk GR&R ANOVA R O 113

128 GR&R 6.18 GR&R npk σ = 0.5 npk ANOVA CRR MCRR LF MLF GR&R npk 6.19 GR&R npk AVOVA LF MLF CRR MCRR LF MLF CRR MCRR npk GR&R ANOVA LF CRR MCRR 114

129 6.19 GR&R npk σ = 1 npk ANOVA CRR MCRR LF MLF GR&R npk CRRMCRRLF MLF 6.0 GR&R npk ANOVA LFMLFCRR MCRR 6.0 LF MLF CRR MCRR npk GR&R AVOVA GR&R 115

130 6.0 GR&R σ = 1.5 npk npk ANOVA CRR MCRR LF MLF GR&R npk MAD ANOVALF MLF MAD CRR MCRR MAD MAD 6.1 GR&R npk MAD npk ANOVA CRR MCRR LF MLF

131 6.13 GR&R npk MAD npk MAD MSE ANOVA LF MLF MSE CRR MCRR MSE MSE CRR MCRR MSE npk MSE ANOVA 1 GR&R npk GR&R 6. GR&R npk MSE npk ANOVA CRR MCRR LF MLF

132 6.14 GR&R npk MSE 1 ANOVA GR&R 6.3 GR&R npk npk ANOVA CRR MCRR LF MLF GR&R npk AVOVA 118

133 1 LF MLF CRR MCRR LF MLF CRR MCRR npk GR&R 1 ANOVA GR&R 6.4 GR&R npk npk ANOVA CRR MCRR LF MLF GR&R npk CRRMCRRLFMLF AVOVA 1 LF MLF CRR MCRR LF MLF CRR MCRR npk GR&R 119

134 1 CRR MCRR 6.5 GR&R npk npk ANOVA CRR MCRR LF MLF GR&R npk GR&R ANOVALF MLF CRR MCRR MCRR MLF MLF CRRMCRR LF 10

135 n p k 6.6 npk 10 0,3,10,6, 10,3,4 6.6 npk 0,3, 10,3,4 GR&R 6 ANOVA LF CRR MCRR MLF MLF 10,6, ,3, 10,6, 10,3,4 σ = 1.5 n p k ANOVA CRR MCRR LF MLF MAD 6.7 ANOVALFMLF CRR MCRR 0,3, 10,3,4 MAD 6 MAD ,3, 10,6, 10,3,4 MAD n p k ANOVA CRR MCRR LF MLF MSE 6.8 ANOVALFMLF CRR MCRR 0,3, 10,3,4 MSE 6 MSE 11

136 ,3, 10,6, 10,3,4 MSE n p k ANOVA CRR MCRR LF MLF npk ANOVALFMLF CRR MCRR 0,3, 10,3,4 GR&R 6 ANOVA LF 1 CRR MCRR MLF ,3, 10,6, 10,3,4 n p k ANOVA CRR MCRR LF MLF MCRR MLF MAD MSE CRR MCRR LF MLF MCRR MLF MCRRMLF ANOVA npk npk MAD MSE 1

137 npk 40 MAD MAD 40,3, 0,3,4 MAD ,6, 40,3, 0,3,4 MAD n p k ANOVA CRR MCRR LF MLF MSE MSE CRR MCRR MSE ,6, 40,3, 0,3,4 MSE n p k ANOVA CRR MCRR LF MLF MAD MAD ,3, 5,6, 5,3,4 MAD n p k ANOVA CRR MCRR LF MLF MSE ,6, MSE 13

138 ,3, 5,6, 5,3,4 MSE n p k ANOVA CRR MCRR LF MLF ANOVA MCRR CRRMLF LF GR&R ANOVA MLF LF ANOVA CRR MCRR npk GR&R npk 6.34 GR&R 6.34 GR&R MAD MSE ANOVA CR&R MCR&R LF 3 3 MLF GR&R MLF ANOVA LF MCRR CRR 14

139 6.1 GR&R AVOVA AVONA Classical R&R Long Form Classical R&R Long Form GR&R npk n p kgr&r 1. Classical R&R Long Form Modified Classical R&R Modified Long Form Modified Classical R&R Modified Long Form npk Classical R&R Long Form Modified Classical R&R 15

140 Modified Long Form Modified Long Form ANOVA npk npk n p k ANOVA Classical R&R Modified Classical R&R Long Form Modified Long Form ANOVA npk n pk Modified Long Form ANOVA ANOVA ANOVA

141 GR&R Classical R&R Modified Classical R&R 17

142 1. (1998, 6) ISO (1997) (001) (MSA) ,11 PT GR&R AIAG Editing Group (1991), Measurement Systems Analysis-Reference Manual.. Barrantine, L. B. (1991), Concepts for R&R Studies, ASQC Quality Press, Milwaukee, WI. 3. Burdick, R. K. and Larsen, G. A. (1997), Confidence Intervals on Measures of Variability in R&R Studies, Journal of Quality Technology, Vol.9, No.3, pp Burdick, R. K.; Larsen, G. A. and Allen, A. Elizabeth (00), Comparing Variability of Two Measurement Process Using R&R Studies, Journal of Quality Technology, Vol.34, No.1, pp

143 5. Dolezal, K. K. and Burdick, R. K. and Birch, N. J. (1998), Analysis of a Two-Factor R&R Study with Fixed Operators, Journal of Quality Technology, Vol.30, No., pp Floyd, D. A. and Laurent, C. J. ( ), Gauging: An Underestimated Consideration In The Application Of Statistical Process Control, Quality Engineering, 8(1), pp Fruit, Robert (1997), The New Approach to Gage R&R, Manufacturing Engineering, Vol.119, No.1, pp Hoguet, Robert G. (1994a), A simplified gage method for test reliability., Ceramics. Engineering. Science. Proc. 15(3), Hoguet, Robert G. and Dennis M. Hartman (1994b), The use of gage R&R studies and their relationship to the testing of ceramic materials, Ceramics. Engineering. Science. Proc. 15(1), Hoguet, Robert G. and Dennis M. Hartman1994c, Gage R&R for ceramics, American Ceramics Society Bulletin, Vol. 73, No.5, Jaynes, K. R. (1997), Reviewing Gage R&R, Manufacturing Engineering, Vol.118, No., pp James, Paul D. and Finderne Avenue (1991), Graphical display of gage R&R data, ASQC Quality Congress Transactions, Milwaukee, McNeese, William H. and Robert A. Klein (1991), Measurement System Sampling and Process Capability Quality Engineering, Vol. 4, No. 1 pp Montgomery D. C. (1991), Introduction to Statistical Quality Control, nd ed., John Wiley & Sons, New York,. 15. Montgomery D. C. (001), Design and Analysis of Experiments, Ed., John Wiley & Sons, New York, pp.66-69, pp

144 16. Montgomery, D. C. and Runger, G. C. (1993a), Gauge Capability Analysis and Designed Experiments. Part I : Basic Methods, Quality Engineering, Vol.6, No.1, pp Montgomery, D. C. and Runger, G. C. (1993b), Gauge Capability Analysis and Designed Experiments. Part II : Experimental Design Models and Variance Component Estimation, Quality Engineering, Vol.6, No., pp Morchower, N.D. (1999), Two-location Gauge Evaluation, Quality Progress 3, pp Mandel, J. (197), Repeatability and Reproducibility, Journal of Quality Technology, Vol.4, No., pp Tsai, P. ( ), Variable Gauge Repeatability and Reproducibility Study Using The Analysis Of Variance Method, Quality Engineering, 1(1), pp Vardeman, S. B. and VanValkenburg, E. S. (1999), Two-way Random-effect Analyses and Gauge R&R Studies, Technometrics 41, pp Wheeler, D. J. and Lyday, R. W.1989, Evaluating the Measurement Process, nd ed., SPC Press, Inc., Knoxville, TN,. 3. Wheeler, Donald. J.,199, Problems with gauge R&R studies, ASQC Quality Congress Transactions, Nashville, 179~

145 1A

146 1B

147 A

148 B

149 3A

150 3B

151 4A

152 4B

153 5A

154 5B

155 6A

156 6B

157 7A

158 7B

159 8A

160 8B

161 B n d n d