Gauge Repeatability and Reproducibility with interaction
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
... iv...v... vii...xiv...1 1.1...1 1....3 1.3...4 1.4...5...7...7....9...8...8 3. GR&R...30 3..1 ANOVA...30 3.. GR&R Classical GR&R...35 3..3 Long Form Method)...36 3.3 GR&R...43 GR&R...47 4.1 GR&R...47 GR&R...56 5.1...56 5....59 v
5..1...60 5.....78...98 6.1 GR&R...98 6. GR&R...99 6.3 GR&R...10 6.3.1...10 6.3....113...15 6.1...15 6....16...18...18...18...131 B...147 vi
.1...1....6....7 3.1...31 3....33 3.3...34 3.4 Classical R&R...35 3.5...40 3.6...41 3.7 QS9000 P/T...43 3.8 QS9000 GR&R...44 3.9 GR&R...45 4.1 1...48 4. 3...51 4.3 GR&R...5 4.4 GR&R...5 4.5 GR&R ANOVA...53 4.6 GR&R...53 4.7 3...54 4.8 GR&R ANOVA...55 4.9 GR&R..55 5.1...59 5....60 5.3 vii
GR&R...61 5.4 GR&R...6 5.5 GR&R...63 5.6 GR&R...64 5.7 GR&R...65 5.8 GR&R...66 5.9 GR&R...68 5.10 GR&R...68 5.11 GR&R...70 5.1 GR&R...70 5.13 GR&R...7 5.14 GR&R...73 5.15 GR&R...74 viii
5.16 GR&R...75 5.17 GR&R...76 5.18 GR&R...77 5.19...78 5.0...79 5.1 GR&R...79 5. GR&R...80 5.3...81 5.4 GR&R...8 5.5 GR&R...8 5.6...84 5.7 GR&R...84 5.8 GR&R...84 5.9...86 ix
5.30 GR&R...86 5.31 GR&R...87 5.3...88 5.33 GR&R...88 5.34 GR&R...89 5.35...90 5.36 GR&R...91 5.37 GR&R...91 5.38...93 5.39 GR&R...93 5.40 GR&R...94 5.41...95 5.4 x
GR&R...95 5.43 GR&R...96 6.1 GR&R σ = 0.5 npk...103 6. GR&R σ = npk...104 6.3 GR&R σ =.5 npk...105 6.4 GR&R npk MAD...106 6.5 GR&R npk MSE...107 6.6 GR&R npk...107 6.7 GR&R npk...108 6.8 GR&R npk...109 6.9 10 0,3, σ =.5 10,6, 10,3,4...110 6.10 10 0,3, 10,6, 10,3,4 MAD...110 6.11 10 0,3, 10,6, 10,3,4 MSE...111 6.1 10 0,3, xi
10,6, 10,3,4...111 6.13 40 0,6, 40,3, 0,3,4 MAD...111 6.14 40 0,6, 40,3, 0,3,4 MSE...11 6.15 60 10,3, 5,6, 5,3,4 MAD...11 6.16 60 10,3, 5,6, 5,3,4 MSE...11 6.17 GR&R...113 6.18 GR&R σ = 0.5 npk...114 6.19 GR&R σ = 1 npk...115 6.0 GR&R σ = 1.5 npk...116 6.1 GR&R npk MAD...116 6. GR&R npk MSE...117 6.3 GR&R npk...118 6.4 GR&R npk...119 6.5 GR&R xii
npk...10 6.6 10 0,3, σ = 1.5 10,6, 10,3,4...11 6.7 10 0,3, 10,6, 10,3,4 MAD...11 6.8 10 0,3, 10,6, 10,3,4 MSE...1 6.9 10 0,3, 10,6, 10,3,4...1 6.30 40 0,6, 40,3, 0,3,4 MAD...13 6.31 40 0,6, 40,3, 0,3,4 MSE...13 6.3 60 10,3, 5,6, 5,3,4 MAD...13 6.33 60 10,3, 5,6, 5,3,4 MSE...14 6.34 GR&R...14 xiii
1.1...6.1...13. ACTUAL C OBSERVED C R&R...15 P P 3.1...8 3....9 3.3...38 3.3...39 3.4...46 4.1...48 4....49 5.1...57 5....57 5.3...61 5.4 GR&R...6 5.5...63 5.6 GR&R...64 5.7...65 5.8 GR&R...67 5.9...67 5.10 GR&R...69 5.11...69 5.1 GR&R...71 5.13...7 5.14 xiv
GR&R...73 5.15...74 5.16 GR&R...75 5.17...76 5.18 GR&R...77 5.19...79 5.0 GR&R...80 5.1...81 5. GR&R...83 5.3...83 5.4 GR&R...85 5.5...86 5.6 GR&R...87 5.7...88 5.8 GR&R...89 5.9...90 5.30 GR&R...9 5.31...9 5.3 GR&R...94 5.33 xv
...95 5.34 GR&R...96 6.1 GR&R...100 6. GR&R NPK...103 6.3 GR&R NPK...104 6.4 GR&R NPK...105 6.5 GR&R NPK MAD...106 6.6 GR&R NPK MSE...107 6.7 GR&R NPK...108 6.8 GR&R NPK...109 6.9 GR&R NPK...109 6.10 GR&R NPK...114 6.11 GR&R NPK...115 6.1 GR&R NPK...116 6.13 GR&R NPK MAD...117 6.14 GR&R NPK MSE...118 6.15 GR&R NPK...118 6.16 GR&R NPK...119 6.17 GR&R NPK...10 xvi
1.1 ISO International Organization for Standardization ISO Accuracy Precision ISO 9000 1994 QS 9000 QS 9000 QS 9000 1
QS 9000 1997 7 1997 1 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 5.15 99 P/T 10% Precision P/T 10% 30% P/T 30% GR&R
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
npk GR&R 1.3 1.. 3. 4. 1.. 3. 4. 4
Non-Calibration Non-Linearity Non-Stability 1.4 1.1 GR&R GR&R GR&R GR&R n p k GR&R npk GR&R 5
GR&R GR&R 1.1 6
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
Precision Calibration Confidence Level Gauge Gage Inspection Linearity Laboratory Process Capability Analysis Quality 8
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
( ) = 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 ( ( )) ± 1. 96.5.6 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 Tsai1988 10
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
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
σ ms % GR & R 100.13 = 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 1991.1 Barrentine1991 13
.1 Repeatability Calibration Stability Linearity Long Form 4 4 npk 16 10 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
. 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
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 515.0 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).17.18 16
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
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
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
< 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.31.31 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
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
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)
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
003 P/T R&R P/T. R&R Reproducibility Repeatability Montgomery and Runger1993a Classical GR&R Classical GR&R MSA 1991Barrentine1991 Burdick Larsen1997 00 Dolezal1998 n p k GR&R Classical GR&R npk 4
Classical GR&R Classical GR&R npk npk. 5
. Repeatability =.77 σ m Reproducibility=.77 σ + ( σ m) L σ σ L Mandel 197 5.15 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 Klein1991 5.15σ 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
. 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
3.1 3.1 3. GR&R 3.3 GR&R GR&R GR&R Repeatability 3.1 Repeatability 8
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 3.1 3.3 σ σ Total σ gauge part σ = σ + σ + σ 3.3 Total part repeatability reproducibility 9
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
n p 1 3.1 p n k 3.1 1 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
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 5. 6. σ P σ O σ PO σ R 7. 3.1 3. 3. 3. 3.5 3
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 3.5 3.5 3.6 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
3.7 i = 1,, K, n Y ijk = µ + Pi + O j + Rijk j = 1,, K, p 3.7 k = 1,, K, κ 3.3 3.3 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 ( ) 3.3 3.3 3.8 P O σ R σ P σ O = MS = = R ( MS MS ) P pk ( MS MS ) nk O R R 3.8 3.8 3.9 34
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 3.9 3.. 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 3.10 3.11 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 1 1 1 1 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
d B 3.1 3.13 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 3.1 3.13 σ gauge = σ repeatability + σ reproducibility 3.14 Classical R&R R Chart ANOVA 3..3 Long Form Method) QS 9000 MSA1991 Long Form 3.3 1998 36
ABC 110 A B C 3.3 6 11 3 3.3 7 1 3 8 13 13 5 678 11113 10 15 5 10 a R 10 15 R b R c a R R b c R 17 R R 19 0 3 D 3 D 4 3.5 R UCL R LCL R 13 4 4 X a 9b X 14c X X a X b X c X diff 37
3.3 38
3.3 39
18 16 16 X 16 Rp R X diff Rp 3.3 14. 3.3 EVAVR&R PVTV EV AV R&R PV 3.5 #Trials 3 4 5 6 7 8 9 10 D 3 0 0 0 0 0 0.076 0.136 0.184 0.3 D 4 3.67.568.66.089 1.970 1.88 1.815 1.761 1.716 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 3.15 3.16 K 1 40
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 15 15 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
4 R σ X -R = d R R σ * reproducibility X = d diff σ 3.19 3.19 3.6 rn rn d O R diff * reproducibility X σ σ σ + = = 3.0 3.0 3.1 * reproducibility X = + = d rn diff R O σ σ σ 3.1 3.1 R σ rn d R d rn d diff R diff O * * X X = = σ σ 3. 99 AV 5.15 ( ) ( ) ( ) = = rn d AV R diff O * X 5.15 5.15 σ σ 3.3 3.18
( 5. 15 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 3.7 3.7 QS9000 P/T P/T 10 % 10 % ~ 30 % 30 % GR&R 3.16 43
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 3.8 3.8 QS9000 GR&R GR&R & P/T 10 % 10 % ~ 30 % 30 % GR&R 3.9 3.4 44
45 3.9 GR&R Classical GR&R 16 GR&R 4
1.. 3. 1.. 3. 1.. 3. 1. ANOVA. Classical GR&R 3. Long Form GR&R 3.4 46
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
4.1 1 MEASUREMENTS PART NO. 1 X R 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 1 4 0 7 19 3 19 4 5 1 18 3 4 9 6 0 19 5 19 0 3 1 7 18 1 1 17 3 3 0 19 5 4 30 6 0 1 6 19 0.5 3.5 0.5 7.0 18.5.0 1.5 18.0 4.0 4.0 0.5 18.5 4.0 4.0 9.5 6.0 0.0 0.0 15.5 19.0 1 1 0 0 1 1 1 1 0 1 0 0 1 0 X =.3 R = 1.0 30.0 5.0 UCL=4. 0.0 15.0 LCL=0.4 1 3 4 5 6 7 8 9 10111131415161718190 4.1 4.1 4.1 4. 48
4 UCL=3.67 0 1 3 4 5 6 7 8 9 10111131415161718190 4. R 1.0 σ gauge = = = 0.887 d 1.18 0 X i X i= 1 σ Total = = 10.05 σ Total = 3.17 n 1 product σ σ Total σ gauge 10.05 0.887 9.6 product σ 3.04 4. P/T (Precision to Tolerance Ratio) P/T 49
P T 6σ gauge = = USL LSL ( 0.887) 6 60 5 = 5.3 = 0.097 < 10% 4.3 55 USL=60 LSL=5 P/T σ gauge 0.887 GR&R 100 % = 100% = 9.% σ 3.04 product σ gauge 0.887 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 4.3 50
4. 3 OPERATOR 1 OPERATOR OPERATOR 3 MEASUREMENTS MEASUREMENTS MEASUREMENTS PART NO. 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 1 X R 1 X R 1 X R 1 0 0.5 1 0 0 0.0 0 19 1 0.0 4 3 3.5 1 4 4 4.0 0 3 4 3.5 1 0 1 0.5 1 19 1 0.0 0 1.0 7 7 7.0 0 8 6 7.0 7 8 7.5 1 19 18 18.5 1 19 18 18.5 1 18 1 19.5 3 3 1.0 4 1.5 3 3.5 1 1 1.5 1 4 3.0 0 1.0 19 17 18.0 18 0 19.0 19 18 18.5 1 4 3 3.5 1 15 3 4.0 4 4 4.0 0 5 3 4.0 16 5 5.5 1 4 5 4.5 1 1 0 0.5 1 0 0 0.0 0 1 0 0.5 1 18 19 18.5 1 17 19 18.0 18 19 18.5 1 3 5 4.0 5 5 5.0 0 5 5 5.0 0 4 4 4.0 0 3 5 4.0 4 5 4.5 1 9 30 9.5 1 30 8 9.0 31.30 30.5 1 6 6 6.0 0 5 6 5.5 1 5 7 6.0 0 0 0.0 0 19 0 19.5 1 0 0 0.0 0 19 1 0.0 19 19 19.0 0 1 3.0 5 6 5.5 1 5 4 4.5 1 5 5 5.0 0 19 19 19.0 0 18 17 17.5 1 19 17 18.0 X 1 =.30 1 R = 1.00 X =.8 R = 1.5 X 3 =.60 R 3 = 1.0 4.3 GR&R 51
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. ) 4. 4.4 4.4 GR&R Repeatability Reproducibility Gauge Classical GR&R 1.00 0.1900 1.037 ANOVA with interaction 0.990-0.150 Long Form 1.018 0.3800 1.869 4.4 GR&R Classical GR&R ANOVA ANOVA ANOVA 4.5 4.5 P 95 α0.05 H 0 0 H 1 0 P H 0 5
4.5 GR&R ANOVA P Source of Sum of Degrees of Mean P-Value Variability Squares Freedom Squares Operators.6 1.31 0.75 Parts 1185.43 19 6.39 0.000 Parts*Operators 7.05 38 0.71 0.861 Error 59.50 60 0.99 Total 174.60 119 3.3 4.6 4.6 GR&R Repeatability Reproducibility Gauge Classical GR&R 1.00 0.1900 1.037 Long Form 1.018 0.3800 1.869 ANOVA without interaction 0.940 0.1030 0.945 Classical GR&R 4.7 53
4.7 3 OPERATOR 1 OPERATOR OPERATOR 3 MEASUREMENTS MEASUREMENTS MEASUREMENTS PART NO. 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 1 X R 1 X R 1 X R 8 7 7.5 1 4 6 5 6 6 6 0 5 6 5.5 1 7 6 6.5 1 4 5 4.5 1 4 3 3.5 1 5 4 4.5 1 5 5 5 0 3 4 3.5 1 0 1 3 1 18 19 18.5 1 0 1 0.5 1 18 18 18 0 6 6 6 0 8 7 7.5 1 9 8 8.5 1 9 9 9 0 7 7 7 0 8 9 8.5 1 5 5 5 0 6 6 6 0 4 3 3.5 1 18 19 18.5 1 0 0 0 0 1 1 1 0 4 3 3.5 1 1 1 1 0 3.5 1 1 3 3.5 1 0 1 3 3 3 0 4 4 4 0 5 6 5.5 1 6 5 5.5 1 4 4 4 0 5 4 4.5 1 9 30 9.5 1 30 31 30.5 1 9 9 9 0 8 8 8 0 9 9 9 0 30 31 30.5 1 7 7 7 0 5 5 5 0 6 7 6.5 1 4 5 4.5 1 5 5 5 0 4 3 3.5 1 0 3 3 3 0 4 4 4 0 31 31 31 0 9 9 9 0 30 30 30 0 0 1 0.5 1 0 1 0 0 0 0 X 1 = 4.675 1 R = 0.65 X = 4.6 R = 0.6 X 3 = 4.7 R 3 = 0.7 4.8 P 0.05 Classical GR&R 54
4.8 GR&R ANOVA P Source of Sum of Degrees of Mean P-Value Variability Squares Freedom Squares 0.16667 Operators 0.108333 0.78 Parts 1164.49 19 61.8904 0.00 Parts*Operators 90.78333 38.389035 0.00 Error 5.5 60 0.45 Total 180.99 119 180.99 4.9 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 0.57641 0.059067 0.5796 Long Form 0.575533981 0.1010997 0.6168716 ANOVA with interaction 0.6519041 0.96176903 1.16189500 Classical GR&R 55
GR&R GR&R GR&R GR&R GR&R GR&R Classical GR&R Classical GR&R 5.1 5. 5.1 5.1 5. GR&R Classical GR&R 56
5.1 5. GR&R Mandel Tsai Fruit 57
Classical GR&R GR&R Classical GR&R 3.4 5.1 k n X 1 j X 1j1 X 1j X 1jk / K j1,,, o X j n = X i1 n j1,,, o i= 1 5.1 j 5.1 5. 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 5.3 5.4 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
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 5.4 5.3 5.4 5.1 5.1 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
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 5. 5. 1A A 3A 4A 1B B 3B 4B Repeatability Reproducibility Gauge 5.. 5..1 1A 1A A 5.3 5.3 ANOVA P-value 0.05 Mean Squares 60
P-value 0.05 40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.3 5.3 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators 0.16667 0.108333 0.99067 Parts 41.45 19.1806 0.0908 Parts*Operators 795.45 38 0.9389 0.01936 Error 693.5 60 11.55833 Total 1910.59 119 5.4 1A 5.4 ANOVA ANOVA Modified Classical GR&R Modified Long Form ANOVA ANOVA 61
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 3.399755.04141 3.965476 Classical GR&R.14434988 0.059066745.14349066 Modified Classical GR&R with interaction.14434988 3.573538098 4.166557598 Long Form.13980585 0.13980585 Modified Long Form with interaction.13980585 3.153748456 3.811154457 5.4 GR&R 5.6 Repeatability Reproducibility Gage ANOVA CR&R Modified CR&R Long Form Modified Long Form 5.4 GR&R 1B 6
1B A 5.5 5.5 ANOVA P-value 0.05 Mean Squares P-value 0.05 40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.5 5.5 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators 3.1667 11.60833 0.374965 Parts 91.95833 19 4.83991 0.981744 Parts*Operators 746.1167 38 19.63465 0.03438 Error 698.5 60 11.64167 Total 1559.79 119 5.6 1B 5.6 ANOVA ANOVA 63
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 3.411989 1.9489 3.99058 Classical GR&R.171985816 0.605434141.54788877 Modified Classical GR&R with interaction.171985816 3.514471353 4.131468416 Long Form.16930388 0.41367509.08410736 Modified Long Form with interaction.16930388 3.10050639 3.784054381 Repeatability Reproducibility Gage ANOVA CR&R Modified CR&R Long Form Modified Long Form 5.6 GR&R Long Form 64
5.6 GR&R 5.6 A A A 5.7 5.7 ANOVA P-value 0.05 Mean Squares 40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.7 5.8 A 5.8 ANOVA ANOVA 5.7 65
GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators 4.55.75 0.00 Parts 11.7 19 0.615789 0.01 Parts*Operators 331.45 38 8.7368 0.00 Error 16 60 0.66667 Total 363.7 119 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 0.516397779.016597795.081665999 Classical GR&R 0.384160757 0.80567041 0.4757071 Modified Classical GR&R with interaction 0.384160757 1.3900177 1.38341071 Long Form 0.3836893 0.4156538 0.45337905 Modified Long Form with interaction 0.3836893 1.178050597 1.3895954 5.8 GR&R 5.8 66
ANOVA CR&R Modified CR&R Long Form Repeatability Reproducibility Gage Modified Long Form 5.8 GR&R B B A 5.9 5.9 ANOVA 40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.9 P-value 0.05 Mean Squares 67
P-value 0.05 5.9 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators.316667 1.158333 0.0 Parts 7.866667 19 0.414035 0.10 Parts*Operators 113.6833 38.991667 0.00 Error 16 60 0.66667 Total 139.8667 119 5.10 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 0.516397779 1.147460965 1.58305739 Classical GR&R 0.384160757 0.19196693 0.49454056 Modified Classical GR&R with interaction 0.384160757 1.3900177 1.38341071 Long Form 0.3836893 0.159336 0.415414548 Modified Long Form with interaction 0.3836893 1.178050597 1.3895954 Long Form 68
5.10GR&R 5.10 ANOVA CR&R Modified CR&R Long Form Repeatability Reproducibility Gage Modified Long Form 5.10 GR&R 3A 40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.11 3A A 5. 11 5.11 69
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 19.5167 64.75833 0.01 Parts 55.4917 19 9.07851 0.01 Parts*Operators 1555.483 38 40.93377 0.00 Error 737.5 60 1.9167 Total 974.99 119 5.1 3A 5.1 ANOVA ANOVA Modified Classical GR&R Modified Long Form ANOVA ANOVA 5.1 GR&R Repeatability Reproducibility Gauge ANOVA with interaction 3.50594738 3.8610075 5.16160785 Classical GR&R.6063898 1.461901949.69144651 Modified Classical GR&R with interaction.6063898 4.59439457 5.11868984 Long Form.57864078 1.47496075.57957959 Modified Long Form with interaction.57864078 4.0605507 4.6460756 70
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 5. 13 5.13 5.13 ANOVA P-value 0.05 Mean Squares P-value 0.05 71
40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.13 5.13 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators 109.85 54.95 0.01 Parts 169.45 19 8.917105 0.73 Parts*Operators 1615.15 38 4.50395 0.00 Error 697.5 60 11.65 Total 591.95 119 5.14 3B 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
GR&R Long Form 5.14 GR&R Repeatability Reproducibility Gauge ANOVA with interaction 3.40954544 3.96866967 5.311384 Classical GR&R.14434988 1.31435086.5134133 Modified Classical GR&R with interaction.14434988 4.71057947 5.174893735 Long Form.13980585 1.11636106.41351058 Modified Long Form with interaction.13980585 4.167356485 4.68461611 5.14 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 5. 15 5.15 5.15ANOVA P-value 73
0.05 Mean Squares P-value 0.05 40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.15 5.15 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators 1.316667 0.658333 0.09 Parts 1.16667 19 1.114035 0.00 Parts*Operators 90.68333 38.386404 0.00 Error 16 60 0.66667 Total 19.1667 119 5. 16 4A 5.16 ANOVA ANOVA Modified Classical GR&R Modified Long Form ANOVA ANOVA 74
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 0.516397779 1.00898897 1.13843031 Classical GR&R 0.384161 0.147667 0.411564 Modified Classical GR&R with interaction 0.384160757 1.40401654 1.9858301 Long Form 0.3836893 0.11618499 0.400893836 Modified Long Form with interaction 0.3836893 1.0999815 1.164334111 5. 16GR&R 5.16 ANOVA CR&R Modified CR&R Long Form Repeatability Reproducibility Gage Modified Long Form 5.16 GR&R 4B 4B A 5. 17 5.17 75
5.17 ANOVA P-value 0.05 Mean Squares P-value 0.05 40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.17 5.17 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators 1.316667 0.658333 0.09 Parts 5.966667 19 0.314035 0.31 Parts*Operators 90.68333 38.386404 0.00 Error 16 60 0.66667 Total 113.9667 119 5. 18 4B 5.18 ANOVA ANOVA 76
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 0.516397779 1.00898897 1.13843031 Classical GR&R 0.384161 0.147667 0.411564 Modified Classical GR&R with interaction 0.384160757 1.40401654 1.9858301 Long Form 0.3836893 0.11618499 0.400893836 Modified Long Form with interaction 0.3836893 1.0999815 1.164334111 5.18GR&R 5.18 ANOVA CR&R Modified CR&R Long Form Repeatability Reproducibility Gage Modified Long Form 5.18 GR&R 77
Modified Classical GR&R Modified Long Form Classical GR&R Long Form GR&R GR&R 5.. Modified Classical GR&R Modified Long Form 5.1.1 5.19 P-value 0.05 H 0 5.19 5A 6A 7A 8A 5B 6B 7B 8B 5A 5A A 5.19 5.0ANOVA 78
P-value 0.05 5.1 P-value 0.05 P-value 0.05 Mean Squares 40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.19 5.0 Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators 3.616667 38 0.095175 1 5.1 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators 40.71667 0.35833 0.093347 Parts 349.3333 19 18.38596 0.009804 Error 498.6167 98 5.08795 Total 888.6667 119 5. 5A 5. Classical GR&R 79
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.55643 0.617867.338736 Classical GR&R 1.566193853 0.8417011 1.77803936 Modified Classical GR&R with interaction 1.566193853 0.8417011 1.77803936 Long Form 1.56471845 0.704958078 1.71578389 Modified Long Form with interaction 1.56471845 0.704958078 1.71578389 ANOVA CR&R Repeatability Reproducibility Gage Modified CR&R Long Form Modified Long Form 5.0 GR&R 80
5.GR&R 5.0 5B 5B A 5.1 5.3 ANOVA P-value 0.05 5.4 P-value 0.05 P-value 0.05 Mean Squares 40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.1 5.3 Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators 3.616667 38 0.095175 1 81
5.4 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators 40.71667 0.35833 0.093347 Parts 114.1333 19 6.007018 0.77567 Error 498.6167 98 5.08795 Total 653.4667 119 5. 5 5B 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.55643 0.617867.338736 Classical GR&R 1.566193853 0.8417011 1.77803936 Modified Classical GR&R with interaction 1.566193853 0.8417011 1.77803936 Long Form 1.56471845 0.704958078 1.71578389 Modified Long Form with interaction 1.56471845 0.704958078 1.71578389 5. 5GR&R 5. 5 8
ANOVA CR&R Repeatability Reproducibility Gage Modified CR&R Long Form Modified Long Form 5. GR&R 6A 6A A 5.3 5.6 ANOVA P-value 0.05 5.7 40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.3 83
5.6 Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators 3.466667 38 0.0918 1 5.7 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators 18. 641.1 0.00 Parts 14.1333 19 6.533333 0.00 Error 18.46667 98 0.188435 Total 144.8 119 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 0.4340910 4.00848 4.06316 Classical GR&R 0.443641 4.69580661 4.71668085 Modified Classical GR&R with interaction 0.443641 4.69580661 4.71668085 Long Form 0.44718447 4.167373307 4.1908397 Modified Long Form with interaction 0.44718447 4.167373307 4.1908397 84
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 5.5 5.9 ANOVA P-value 0.05 5.30 P-value 0.05 P-value 0.05 Mean Squares 85
40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.5 5.9 Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators 6.4 38 0.16841 0.90 5.30 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators 17.67 636.1333 0.00 Parts 6.7 19 0.3563 0.16 Error 1.4 98 0.18367 Total 1300.367 119 5.31 6B 5.31 Classical GR&R Modified Classical GR&R Long Form Modified Long Form GR&R Long Form Modified Long Form 86
ANOVA Long Form Modified Long Form ANOVA Classical GR&R Modified Classical GR&R 5.31 GR&R Repeatability Reproducibility Gauge ANOVA without interaction 0.46798 3.98714 4.014504 Classical GR&R 0.4436411 4.6667888 4.6877898 Modified Classical GR&R with interaction 0.4436411 4.6667888 4.6877898 Long Form 0.44718447 4.141155994 4.164753605 Modified Long Form with interaction 0.44718447 4.141155994 4.164753605 5.31GR&R 5.6 Repeatability Reproducibility Gage ANOVA CR&R Modified CR&R Long Form Modified Long Form 5.6 GR&R 7A 7A A 5.7 87
5.3 ANOVA P-value 0.05 5.33 P-value 0.05 Mean Squares 40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.7 5.3 Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators 3.466667 38 0.0918 1 5.33 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators 18. 641.1 0.0000 Parts 456.1333 19 4.0070 0.0099 Error 650.4667 98 6.637415 Total 388.8 119 88
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.576318103 3.98657985 4.74330899 Classical GR&R 1.8617018 4.69580661 5.05138904 Modified Classical GR&R with interaction 1.8617018 4.69580661 5.05138904 Long Form 1.859417476 4.157579156 4.554437154 Modified Long Form with interaction 1.859417476 4.157579156 4.554437154 ANOVA CR&R Repeatability Reproducibility Gage Modified CR&R Long Form Modified Long Form 5.8 GR&R ANOVA Long 89
Form Modified Long Form ANOVA Classical GR&R Modified Classical GR&R 5.34GR&R 5.8 7B 7B A 5.9 5.35 ANOVA P-value 0.05 5.36 P-value 0.05 40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.9 5.35 Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators 3.466667 38 0.0918 1 90
5.36 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators 18. 641.1 0.00 Parts 8.533333 19 0.44913 1.00 Error 650.4667 98 6.637415 Total 1941. 119 P-value 0.05 Mean Squares 5. 37 7B 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.576318103 3.98657985 4.74330899 Classical GR&R 1.8617018 4.69580661 5.05138904 Modified Classical GR&R with interaction 1.8617018 4.69580661 5.05138904 Long Form 1.859417476 4.157579156 4.554437154 Modified Long Form with interaction 1.859417476 4.157579156 4.554437154 91
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 5.31 5.38 ANOVA 40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.31 9
5.38 Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators.083333 38 0.05485 1 5.39 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators 1.5 0.65 0.09 Parts 19.9667 19 10.15614 0.00 Error 17.08333 98 0.1743 Total 11.3 119 P-value 0.05 5.39 P-value 0.05 P-value Mean Squares 5.40 8A 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
5.40 GR&R Repeatability Reproducibility Gauge ANOVA without interaction 0.417516141 0.106146158 0.430797789 Classical GR&R 0.4436411 0.147666864 0.4671016 Modified Classical GR&R with interaction 0.4436411 0.147666864 0.4671016 Long Form 0.44718447 0.110809837 0.456375331 Modified Long Form with interaction 0.44718447 0.110809837 0.456375331 5.40GR&R 5.3 Repeatability Reproducibility Gage ANOVA CR&R Modified CR&R Long Form Modified Long Form 5.3 GR&R 8B 8B A 5.33 5.41 ANOVA P-value 0.05 5.4 P-value 0.05 Mean Squares 94
40 35 30 5 0 15 10 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 5.33 5.41 Source of Sum of Degrees of Mean P-value Variability Squares Freedom Squares Parts*Operators 1.183333 38 0.03114 1 5.4 GR&R Source of Sum of Degrees of Mean Variability Squares Freedom Squares P-value Operators 0.15 0.075 0.7639 Parts 8.491667 19 0.44693 0.079146 Error 17.68333 98 0.18044 Total 6.35 119 5.43 8B 5.43 Classical GR&R Modified Classical GR&R Long Form Modified Long Form ANOVA GR&R ANOVALong Form Modified Long Form 95
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 0.4478486 0 0.41670633 Classical GR&R 0.48758865 0.044300059 0.489596966 Modified Classical GR&R with interaction 0.48758865 0.044300059 0.489596966 Long Form 0.48699091 0 0.48699091 Modified Long Form with interaction 0.48699091 0 0.48699091 5.43GR&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
ANOVA Classical GR&R Long Form Modified Classical GR&R Modified Long Form 97
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
ANOVACRRLFMCRRMLF 6.1 6. GR&R GR&R GR&R Mean Absolute Deviation, MAD mean squares error, MSE Mean Ratio 6.1 6. 6.3 MAD MAD 6.1 6. 99
10000 σ P σ O σ PO σ R ANOVA Classical R&RModified Classical R&RLong Form Modified Long Form 6.1 100
6.3 θ 6.4 MSE Bias Var MSE = = = E[ θ θ ] [ E( θ ) θ ] + E[ θ E( θ )] Bias + Var( θ ) ( θ ) 6.4 6.5 6.6 6.7 MSE ( ) ( ) ( ) 6.5 6.6 6.7 6.8 6.9 6.10 1 101
6.8 6.9 6.10 GR&R 6.3 GR&R npk GR&R npk n p k GR&R GR&R 6.3.1 10 σ 0 σ 1 P σ 1 σ 0.5 PO npk GR&R 6.1 6.1 GR&R npk 6.1 ANOVA GR&R R O 10
6.1 GR&R npk σ = 0.5 npk ANOVA CRR MCRR LF MLF 0.49865686 0.5966969 0.5966969 0.5966969 0.5966969 0.499048060.579856 0.579856 0.5798560.579856 0.507347360.55047688 0.55047688 0.550476880.55047688 0.498160.53563749 0.53563749 0.535637490.53563749 0.507906890.5417631 0.5417631 0.54176310.5417631 0.499991650.578894 0.578894 0.5788940.578894 0.499044350.514879 0.514879 0.5148790.514879 6. ANOVA GR&R npk GR&R 6. GR&R npk 6. GR&R npk 6. 6.3 ANOVA MLFMCRRCRR LF npk MLFMCRRCRRLF MLF MCRR CRR LF 6. 103
ANOVA CRRLFMLF MCRR 6. GR&R npk σ = ANOVA CRR MCRR LF MLF 1.993587401.55876014.333470793 1.169778141.8561705.01075931.46078783.7647117 1.91858111.9147174.0019983811.43693831.9466473 1.1151196 1.79937984 1.9868849511.30580661.18018688 1.098494991.87474694 1.975797071.33941964.450494 1.050098641.76175615 1.98018951.3545051.48576776 1.039777611.765010508.00064461.181149079.19196077 1.05895631.966510185 6.3 GR&R npk 6.3 GR&R npk 6.3 6.4 ANOVA ANOVA MLF MCRRCRR npk MLF CRR MCRR LF MLF MCRR CRR LF 6.3 ANOVA 104
CRRLFMLF MCRR 6.3 GR&R npk σ =.5 ANOVA CRR MCRR LF MLF.434530881.81849814.593140483 1.476647503.08586714.60630791.719377309.59945644 1.486484337.1701575.57331171.687741519.549714 1.377198884.0544758.3670655 1.55614441.47158436 1.35181348.16038443.6503961.5935955.499377573 1.3047171.01598783.30181151.57839437.5013657 1.9516685.01779943.505468611.43397358.44308556 1.31017084.17758464 6.4 GR&R npk AVOVA ANOVA MLFMCRRCRRLF MLF MCRR CRR LF npk MLFMCRR LF CRR GR&R 6.4 6.5 MLF MAD ANOVA MAD 105
MCRR LF GR&R MAD npk GR&R MAD MAD 6.4 GR&R npk MAD ANOVA CRR MCRR LF MLF 0.969966491.3114118 1.0898734 1.608539190.946533 0.80610061 1.10374809 0.88357694 1.1138599690.78575596 0.87090891.8900436 0.973374546 1.6608410.867389993 0.716901041 1.1084341 0.774367793 1.1407403690.70694459 0.8089856111.7073916 0.88908458 1.67786100.815807555 0.7905144751.60876914 0.8706011 1.6436881 0.79940995 0.5679061 0.99747515 0.5656906 1.054137300.519984995 6.5 GR&R npk MAD MSE 6.5 6.6 GR&R MAD MLF MSE ANOVA MCRRLF CRRMCRR CRR npk MSE 106
6.5 GR&R npk MSE ANOVA CRR MCRR LF MLF 30 1.6858016.66711.586669.1358146 1.514348004 40 1.19874 1.734553 1.5531351 1.63373505 1.0408064 60 1.3893118.441685.063467337.075835041 1.58599917 80 0.8836077 1.639769 1.16845611 1.63616794 0.81690089 90 1.1786156.6919 1.71070853.0631085 1.073460744 10 1.130363.1997 1.64558035.00941549 1.03157339 40 0.463863 1.74607 0.5940454 1.37091 0.45014674 6.6 GR&R npk MSE 6.6 6.7 ANOVA 1 GR&R 6.6 GR&R npk ANOVA CRR MCRR LF MLF 0.9994674 1.038679 1.038678758 1.0386787581.038678758 0.9996193 1.09194 1.09194106 1.091941061.09194106 1.0093894 1.00191 1.00190751 1.001907511.00190751 0.999864 1.01455 1.01454995 1.014549951.01454995 1.0031676 1.016691 1.01669054 1.016690541.01669054 0.99999666 1.011156 1.011155698 1.0111556981.011155698 0.99961774 1.004993 1.004993115 1.0049931151.004993115 107
6.7 GR&R npk npk GR&R 1 6.6 ANOVA GR&R 6.7 6.8 ANOVA 1 MLFMCRRCRR 1 LF npk MLF MCRR 1CRR LF 1 6.7ANOVA CRRLF MLF MCRR 6.8 6.9 ANOVA 1 MLF MCRRCRR LF npk ANOVA 6.7 GR&R npk ANOVA CRR MCRR LF MLF 30 0.9967937 0.77938 1.166735397 0.6084889070.9180851 40 1.0053696 0.731039 1.13633559 0.614599050.95735861 60 1.00099919 0.716347 1.14733366 0.561075598 0.8996899 80 0.9934448 0.6519 1.109009344 0.54914750.93637347 90 0.98786485 0.66971 1.160471 0.5504930.880878076 10 0.99100947 0.6675 1.1488388 0.519863880.8850554 40 1.000311 0.590575 1.095980138 0.5946180.9835509 108
6.8 GR&R npk GR&R 1 MLF MCRR 1CRR LF 1 6.8 ANOVA CRRLFMLF MCRR 6.8 GR&R npk ANOVA CRR MCRR LF MLF 0.9970906 0.808191 1.15506881 0.656877790.96794095 1.0047477 0.764168 1.144086 0.6606597060.965340333 0.99739413 0.716746 1.0955147 0.6519983430.9908776 0.9940918 0.6916 1.098481083 0.6008058880.944905974 0.99815108 0.799936 1.11305996 0.651576660.89759157 0.9900805 0.70144 1.111718089 0.574451860.896799748 1.0004305 0.63661 1.085870469 0.5898150.98567048 6.9 GR&R npk 109
ANOVA MLF MCRR LF CRR MCRR MLF npk MSE MLF ANOVA MCRRLF CRR 6.9 10 0,3, 10,6, 10,3,4 6.9 n, p, k 6.9 10 0,3, 10,6, 10,3,4 σ =.5 n p k ANOVA CRR MCRR LF MLF 0 3.3018 1.578.5013657 1.95167.0177994 10 6.85 1.47.437517 1.345757.06739 10 3 4.348 1.634.46041 1.336607 1.985334 MAD 6.10 3 6 MAD MAD 6.10 10 0,3, 10,6, 10,3,4 MAD n p k ANOVA CRR MCRR LF MLF 0 3 0.79051 1.61 0.87060 1.643688 0.7994099 10 6 0.56181 0.989 0.6007943 1.0434069 0.5570607 10 3 4 0.8406 1.74 0.9388361 1.65687 0.859600 MSE 6.11 3 6 110
MSE 6.11 10 0,3, 10,6, 10,3,4 MSE n p k ANOVA CRR MCRR LF MLF 0 3 1.130363. 1.645580.009415 1.031573 10 6 0.5039 1.6 0.6670393 1.3506768 0.51977 10 3 4 1.95146.34 1.876559.0457867 1.1990697 6.1 n, p, k 6.1 10 0,3, 10,6, 10,3,4 n p k ANOVA CRR MCRR LF MLF 0 3 0.990081 0.70144 1.11171808 0.5744518 0.8967997 10 6 0.9904543 0.65347 1.08130 0.5967003 0.9807730 10 3 4 0.99135 0.764 1.0935155 0.5938936 0.88359 10 npk npk MAD MSE MAD 6.13 40 3 6 MAD 6.13 40 0,6, 40,3, 0,3,4 MAD n p k ANOVA CRR MCRR LF MLF 0 6 0.5679 0.997 0.56569 1.0541373 0.5199849 40 3 0.76891 1.76 0.844368 1.8507 0.7889378 0 3 4 0.79318 1.65 0.8868411 1.615064 0.849554 111
MSE 6. 3 6 MSE 6.14 40 0,6, 40,3, 0,3,4 MSE n p k ANOVA CRR MCRR LF MLF 0 6 0.46386 1.75 0.594045 1.37091 0.450146 40 3 1.096177.5 1.607059.0463688 1.005163 0 3 4 1.187473.83 1.736719.019440 1.10913 60 MAD 6.15 3 6 MAD 6.15 60 10,3, 5,6, 5,3,4 MAD n p k ANOVA CRR MCRR LF MLF 10 3 0.8709 1.89 0.9733745 1.6608 0.8673899 5 6 0.6340 0.946 0.6855015 0.988608 0.67434 5 3 4 0.9465 1.301 1.0636991 1.56699 0.94118 MSE 6.16 5,6, MSE 6.16 60 10,3, 5,6, 5,3,4 MSE n p k ANOVA CRR MCRR LF MLF 10 3 1.38931.44.0634673.075835 1.585999 5 6 0.673859 1.08 0.8671331 1.600697 0.665433 5 3 4 1.595483.55.3317433.0799189 1.45374 MSE MLF ANOVA MCRRLF CRR MCRR 11
CRR MLF MLF MCRR MLF MCRR npk ANOVA MCRR MLF CRR LF npk 6.17 GR&R 6.17 GR&R MAD MSE ANOVA 1 1 1 1 CR&R 4 4 5 5 4 MCR&R 3 3 3 LF 5 5 4 4 5 MLF 1 1 MCRR MLF 6.3. 10 σ 0 σ 1 P σ 0 σ 0.5 6.18 PO GR&R npk 6.18 6.10 ANOVA GR&R npk GR&R ANOVA R O 113
GR&R 6.18 GR&R npk σ = 0.5 npk ANOVA CRR MCRR LF MLF 0.501 0.594 0.593674 0.593670.593674 0.5008 0.58 0.57979161 0.57979160.57979161 0.5051 0.554 0.55393398 0.5539340.55393398 0.4967 0.535 0.53511911 0.53511910.53511911 0.500 0.536 0.5363757 0.5363760.5363757 0.5048 0.53 0.5303419 0.53034130.5303419 0.50 0.517 0.5177461 0.5177460.5177461 6.10 GR&R npk 6.19 GR&R npk 6.19 6.11 AVOVA LF MLF CRR MCRR LF MLF CRR MCRR npk GR&R ANOVA LF CRR MCRR 114
6.19 GR&R npk σ = 1 npk ANOVA CRR MCRR LF MLF 1.0059 1.3111 1.38168094 1.040771.07788014 1.0064 1.145 1.93645864 1.01978661 1.08669344 0.99973 1.91 1.36957986 1.01094161.07505503 0.98536 1.1783 1.66887883 0.993104111.068066403 1.00113 1.861 1.365815561 1.00803961.070845163 0.98394 1.638 1.34599918 0.99151671.055859788 0.99855 1.19 1.5598154 1.006656681.09886088 6.11 GR&R npk CRRMCRRLF MLF 6.0 GR&R npk 6.0 6.1 ANOVA LFMLFCRR MCRR 6.0 LF MLF CRR MCRR npk GR&R AVOVA GR&R 115
6.0 GR&R σ = 1.5 npk npk ANOVA CRR MCRR LF MLF 1.56 1.5704 1.641043664 1.81770451.337190739 1.57 1.475 1.5516505 1.77765771.34467601 1.504 1.5466 1.6491384 1.66335561.37898901 1.3503 1.4318 1.50399793 1.46616031.31578314 1.5115 1.5397 1.61945818 1.618411.3448419 1.344 1.5168 1.59907047 1.4485791.308893916 1.4877 1.3747 1.47735615 1.58384141.350553549 6.1 GR&R npk MAD 6.1 6.13 ANOVALF MLF MAD CRR MCRR MAD MAD 6.1 GR&R npk MAD npk ANOVA CRR MCRR LF MLF 0.75958401 0.91617 0.907307734 0.7559913890.73940996 0.63047833 0.787 0.7551001 0.6343695950.618566044 0.7481844 0.9076 0.89568659 0.7495778150.7937317 0.6441976 0.71537 0.70471593 0.68603550.607491988 0.7399671 0.895058 0.881854578 0.74074706 0.7194055 0.7369173 0.891675 0.875806857 0.740368130.71733854 0.4885735 0.55041 0.54193133 0.5057076750.48568659 116
6.13 GR&R npk MAD npk MAD MSE 6. 6.14 ANOVA LF MLF MSE CRR MCRR MSE MSE CRR MCRR MSE npk MSE 6. 6.15 ANOVA 1 GR&R npk GR&R 6. GR&R npk MSE npk ANOVA CRR MCRR LF MLF 30 1.0718347071.855315784 1.886704317 1.0851903781.0804837 40 0.7098884351.083990409 1.1078839 0.740191170.737396 60 1.035484 1.79551116 1.81368701 1.055651.044905678 80 0.6701465750.99651764 1.0154301 0.68986980.67991998 90 1.00438171.748417 1.75433666 1.01407651.004081108 10 0.9968083571.7135786 1.734907016 1.0147758191.00010774 40 0.396319730.556763583 0.57197663 0.4365934540.49771151 117
6.14 GR&R npk MSE 1 ANOVA GR&R 6.3 GR&R npk npk ANOVA CRR MCRR LF MLF 1.000415 1.037451 1.037451 1.037451 1.037451 1.0003 1.031917 1.031917 1.031917 1.031917 1.00031 1.01574 1.01574 1.01574 1.01574 0.998686 1.014048 1.014048 1.014048 1.014048 1.000099 1.014549 1.014549 1.014549 1.014549 1.001913 1.01137 1.01137 1.01137 1.01137 1.000861 1.00691 1.00691 1.00691 1.00691 6.15 GR&R npk 6.4 6.16 AVOVA 118
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 1.0059 1.311068 1.381681 1.0408 1.07788 1.00637 1.1457 1.93646 1.019787 1.086693 0.99973 1.91161 1.36958 1.01094 1.07506 0.985356 1.1785 1.66888 0.993104 1.068066 1.00119 1.86085 1.365816 1.00804 1.070845 0.98394 1.63759 1.345993 0.9915 1.05586 0.99855 1.1946 1.5598 1.006657 1.09886 6.16 GR&R npk CRRMCRRLFMLF 6.5 6.17 AVOVA 1 LF MLF CRR MCRR LF MLF CRR MCRR npk GR&R 119
1 CRR MCRR 6.5 GR&R npk npk ANOVA CRR MCRR LF MLF 1.004803 1.56345 1.31835 1.05416 1.069753 1.00174 1.178005 1.413 1.013 1.075738 1.00019 1.3743 1.99937 1.013068 1.06319 0.9880 1.14541 1.163 0.99793 1.05763 1.00093 1.31778 1.9556 1.009473 1.059586 0.987534 1.13435 1.79 0.99549 1.047115 0.99901 1.099739 1.18186 1.006707 1.080443 6.17 GR&R npk GR&R ANOVALF MLF CRR MCRR MCRR MLF MLF CRRMCRR LF 10
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, 6.6 10 0,3, 10,6, 10,3,4 σ = 1.5 n p k ANOVA CRR MCRR LF MLF 0 3 1.34418 1.517 1.599070 1.44858 1.3088939 10 6 1.4738 1.373 1.4713898 1.554593 1.3435906 10 3 4 1.604 1.501 1.5387513 1.31847 1.613557 MAD 6.7 ANOVALFMLF CRR MCRR 0,3, 10,3,4 MAD 6 MAD 6.7 10 0,3, 10,6, 10,3,4 MAD n p k ANOVA CRR MCRR LF MLF 0 3 0.7369 0.89 0.8758068 0.7403681 0.717338 10 6 0.49404 0.554 0.5468759 0.5086053 0.489787 10 3 4 0.73394 0.88 0.8734830 0.733036 0.733857 MSE 6.8 ANOVALFMLF CRR MCRR 0,3, 10,3,4 MSE 6 MSE 11
6.8 10 0,3, 10,6, 10,3,4 MSE n p k ANOVA CRR MCRR LF MLF 0 3 0.996808 1.714 1.7349070 1.0147758 1.000107 10 6 0.404803 0.56 0.5758881 0.4398748 0.433801 10 3 4 0.979533 1.658 1.6684895 0.9855001 0.9793109 6.9 npk ANOVALFMLF CRR MCRR 0,3, 10,3,4 GR&R 6 ANOVA LF 1 CRR MCRR MLF 6.9 10 0,3, 10,6, 10,3,4 n p k ANOVA CRR MCRR LF MLF 0 3 0.9875344 1.1343 1.7916 0.995486 1.0471151 10 6 0.9941906 1.09859 1.17711184 1.0043674 1.074875 10 3 4 0.980819 1.0093 1.310010 0.9854597 1.0090845 MCRR MLF MAD MSE CRR MCRR LF MLF MCRR MLF MCRRMLF ANOVA npk npk MAD MSE 1
npk 40 MAD 6.30 3 6 MAD 40,3, 0,3,4 MAD 6.30 40 0,6, 40,3, 0,3,4 MAD n p k ANOVA CRR MCRR LF MLF 0 6 0.4886 0.55 0.5419313 0.5057077 0.485686 40 3 0.7414 0.899 0.88541 0.743649 0.71958 0 3 4 0.7388 0.9 0.891960 0.743505 0.73197 MSE 6.31 3 6 MSE CRR MCRR MSE 6.31 40 0,6, 40,3, 0,3,4 MSE n p k ANOVA CRR MCRR LF MLF 0 6 0.3963 0.557 0.5719766 0.4365935 0.497711 40 3 1.0079 1.763 1.7868999 1.0393359 1.056500 0 3 4 0.99153 1.78 1.734415 1.014958 1.0057115 60 MAD 6.3 3 6 MAD 6.3 60 10,3, 5,6, 5,3,4 MAD n p k ANOVA CRR MCRR LF MLF 10 3 0.74818 0.908 0.895686 0.7495778 0.79373 5 6 0.50437 0.573 0.56789 0.53038 0.5075035 5 3 4 0.7469 0.897 0.8909565 0.7451711 0.7363786 MSE 6.33 5,6, MSE 13
6.33 60 10,3, 5,6, 5,3,4 MSE n p k ANOVA CRR MCRR LF MLF 10 3 1.0355 1.796 1.813687 1.0556 1.0449056 5 6 0.43449 0.65 0.640895 0.485465 0.481507 5 3 4 1.09657 1.775 1.7850440 1.047574 1.044919 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 1 1 1 1 1 CR&R 4 4 5 4 4 MCR&R 5 5 4 5 5 LF 3 3 MLF 3 3 1 3 GR&R MLF ANOVA LF MCRR CRR 14
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
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 6. 1. 16
. 3. 4. GR&R Classical R&R Modified Classical R&R 17
1. (1998, 6) ISO 90014.11. (1997) 58 3. (001) (MSA) 77 4. 003,11 PT GR&R 5. 1995 37939 6. 1977 315 1. 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.61-73 4. 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.97-105 18
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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.115-135. 17. 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.89-305. 18. Morchower, N.D. (1999), Two-location Gauge Evaluation, Quality Progress 3, pp.79-86 19. Mandel, J. (197), Repeatability and Reproducibility, Journal of Quality Technology, Vol.4, No., pp.74-85. 0. Tsai, P. (1988-89), Variable Gauge Repeatability and Reproducibility Study Using The Analysis Of Variance Method, Quality Engineering, 1(1), pp.107-115. 1. Vardeman, S. B. and VanValkenburg, E. S. (1999), Two-way Random-effect Analyses and Gauge R&R Studies, Technometrics 41, pp.0-11. 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, 179~185. 130
1A 1 3 1 31 1 0 1 8 7 19 19 4 3 18 17 3 34 0 1 6 6 4 3 3 31 30 6 5 5 31 0 1 7 8 6 0 1 8 7 4 5 7 9 30 5 5 3 3 8 15 6 5 4 6 6 9 4 4 1 7 8 10 3 3 6 7 0 1 11 15 6 3 3 6 7 1 0 1 6 7 5 4 13 33 4 5 0 1 14 5 5 1 6 8 15 14 7 3 15 16 16 6 7 4 3 5 6 17 8 8 5 6 1 1 18 4 4 7 7 19 3 16 0 0 16 15 0 31 19 5 4 7 8 131
1B 1 3 1 31 1 0 1 8 7 5 5 30 9 4 3 3 34 0 1 6 6 4 3 3 31 30 6 5 5 31 0 1 6 6 6 0 1 8 7 4 5 7 9 30 5 5 3 3 8 17 8 1 6 6 9 4 4 1 7 8 10 3 3 6 7 0 1 11 19 30 0 0 6 4 1 0 1 4 4 7 6 13 31 0 4 5 0 1 14 5 5 1 6 8 15 0 33 8 9 1 16 7 4 1 7 6 17 8 8 4 5 1 1 18 5 5 1 7 7 19 34 18 5 5 1 0 0 31 19 0 1 7 8 13
A 1 3 1 6 6 4 5 6 7 5 5 7 6 4 4 3 7 6 4 4 6 5 4 4 4 5 5 6 7 5 4 4 5 5 6 6 6 4 4 6 7 5 5 7 6 6 5 5 4 4 8 6 6 4 4 5 5 9 4 4 5 5 6 7 10 4 4 5 5 6 6 11 5 5 8 8 0 0 1 5 6 4 4 7 6 13 6 5 6 7 5 4 14 5 5 4 4 6 6 15 6 7 0 19 9 8 16 7 4 1 7 6 17 4 4 6 6 5 5 18 5 4 6 5 6 6 19 6 6 5 5 3 4 0 9 8 6 5 0 0 133
B 1 3 1 6 6 4 5 6 7 5 5 7 6 4 4 3 7 6 4 4 6 5 4 4 4 5 5 6 7 5 4 4 5 5 6 6 6 4 4 6 7 5 5 7 6 6 5 5 4 4 8 6 6 4 4 5 5 9 4 4 5 5 6 7 10 4 4 5 5 6 6 11 5 5 6 6 4 4 1 5 6 4 4 7 6 13 6 5 6 7 5 4 14 5 5 4 4 6 6 15 6 7 5 6 5 4 16 7 4 1 7 6 17 4 4 6 6 5 5 18 5 4 6 5 6 6 19 6 6 5 5 3 4 0 5 4 6 5 5 5 134
3A 1 3 1 31 1 0 1 8 7 18 18 3 17 16 3 34 0 1 6 6 4 3 3 34 33 4 3 5 31 0 1 6 6 6 3 14 1 0 17 18 7 9 30 5 5 15 15 8 17 8 31 3 3 3 9 19 19 16 17 3 10 3 3 30 31 14 14 11 19 30 0 0 6 4 1 0 1 4 4 7 6 13 31 0 4 5 14 14 14 5 5 1 6 8 15 0 33 8 9 1 16 7 4 1 7 6 17 34 34 4 5 16 16 18 5 5 1 6 6 19 34 18 5 5 1 0 0 31 19 0 1 5 6 135
3B 1 3 1 31 1 0 1 8 7 5 5 30 9 4 3 3 34 0 1 6 6 4 3 3 34 33 4 3 5 31 0 1 6 6 6 0 1 8 7 4 5 7 9 30 5 5 15 15 8 17 8 31 3 3 3 9 4 4 1 7 8 10 3 3 30 31 14 14 11 19 30 0 0 6 4 1 0 1 4 4 7 6 13 31 0 4 5 14 14 14 5 5 1 6 8 15 0 33 8 9 1 16 7 4 1 7 6 17 34 34 4 5 16 16 18 5 5 1 6 6 19 34 18 5 5 1 0 0 31 19 0 1 5 6 136
4A 1 3 1 6 6 4 5 6 7 5 5 7 6 4 4 3 7 6 4 4 6 5 4 4 4 5 5 6 7 5 4 4 5 5 6 6 6 4 4 6 7 5 5 7 6 6 5 5 4 4 8 6 6 4 4 5 5 9 4 4 5 5 6 7 10 4 4 5 5 6 6 11 5 5 6 6 4 4 1 5 6 4 4 7 6 13 4 3 4 5 3 14 5 5 4 4 6 6 15 6 7 5 6 5 4 16 7 4 4 5 7 6 17 4 4 6 6 5 5 18 5 4 6 5 6 6 19 6 6 5 5 3 4 0 5 4 6 5 5 5 137
4B 1 3 1 6 6 4 5 6 7 5 5 7 6 4 4 3 7 6 4 4 6 5 4 4 4 5 5 6 7 5 4 4 5 5 6 6 6 4 4 6 7 5 5 7 6 6 5 5 4 4 8 6 6 4 4 5 5 9 4 4 5 5 6 7 10 4 4 5 5 6 6 11 5 5 6 6 4 4 1 5 6 4 4 7 6 13 6 5 6 7 5 4 14 5 5 4 4 6 6 15 6 7 5 6 5 4 16 7 4 4 5 7 6 17 4 4 6 6 5 5 18 5 4 6 5 6 6 19 6 6 5 5 3 4 0 5 4 6 5 5 5 138
5A 1 3 1 31 1 5 5 5 4 7 6 5 6 5 5 3 6 5 5 4 4 4 4 5 6 4 5 4 4 5 5 5 4 5 4 4 6 31 0 5 5 5 4 7 31 3 31 31 30 31 8 6 6 5 5 5 4 9 3 0 6 5 5 5 10 7 6 6 6 5 5 11 37 1 8 8 7 8 1 5 5 4 4 3 3 13 6 5 5 4 4 4 14 5 6 5 5 4 5 15 3 6 6 5 5 16 6 6 6 5 5 5 17 36 4 9 9 8 8 18 6 5 5 4 4 4 19 30 0 4 5 3 3 0 5 6 5 5 4 5 139
5B 1 3 1 3 6 6 6 5 7 6 5 6 5 5 3 6 5 5 4 4 4 4 4 5 3 4 3 3 5 5 5 4 5 4 4 6 3 1 6 6 6 5 7 7 8 7 7 6 7 8 6 6 5 5 5 4 9 3 0 6 5 5 5 10 7 6 6 6 5 5 11 33 17 4 4 3 4 1 5 5 4 4 3 3 13 6 5 5 4 4 4 14 6 7 6 6 5 6 15 3 6 6 5 5 16 7 7 7 6 6 6 17 33 1 6 6 5 5 18 5 4 4 3 3 3 19 30 0 4 5 3 3 0 6 7 6 6 5 6 140
6A 1 3 1 31 30 5 6 3 31 30 5 6 3 3 31 31 6 6 3 3 4 31 30 6 6 3 5 31 30 6 6 3 6 30 31 6 6 3 3 7 31 31 6 6 3 3 8 30 30 6 5 9 3 3 8 7 4 4 10 30 31 5 6 3 11 30 31 5 6 3 1 35 35 30 30 7 7 13 31 31 6 5 3 3 14 30 30 6 5 3 15 30 30 6 5 3 16 31 30 6 6 3 3 17 31 30 6 6 3 3 18 30 31 5 6 19 31 31 6 6 3 0 31 31 6 6 3 141
6B 1 3 1 31 30 5 6 3 31 30 5 6 3 3 31 31 6 6 3 3 4 31 30 6 6 3 5 31 30 6 6 3 6 30 31 6 6 3 3 7 31 31 5 5 3 3 8 30 30 6 5 3 9 30 30 6 5 10 30 31 5 6 3 11 30 31 5 6 3 1 31 31 6 6 3 3 13 31 31 6 5 3 3 14 30 30 6 5 3 15 30 30 6 5 3 16 31 30 6 6 3 3 17 31 30 6 6 3 3 18 30 31 5 6 19 31 31 6 6 3 0 31 31 5 5 3 3 14
7A 1 3 1 41 31 31 38 36 30 8 30 31 3 35 31 3 9 5 6 6 31 6 4 30 6 6 6 31 6 5 6 3 8 3 6 7 3 3 3 7 3 7 41 31 31 38 36 30 8 8 30 31 3 35 31 9 9 5 6 6 31 6 10 30 6 6 6 31 6 11 6 3 8 3 1 7 3 3 3 7 3 13 41 31 31 38 36 30 14 8 30 31 3 35 31 15 9 5 6 6 31 6 16 30 6 6 6 31 6 17 6 3 8 3 18 7 3 3 3 7 3 19 41 31 31 38 36 30 0 8 30 31 3 35 31 143
7B 1 3 1 37 4 5 6 3 31 30 5 6 3 3 31 31 6 6 3 3 4 38 3 6 6 3 5 31 30 6 6 3 6 30 31 6 6 3 3 7 37 5 6 6 3 3 8 30 30 6 5 9 30 30 6 5 10 30 31 5 6 3 11 5 36 5 6 3 1 31 31 6 6 3 3 13 38 4 6 5 3 3 14 30 30 6 5 3 15 35 5 6 5 3 16 31 30 6 6 3 3 17 37 4 6 6 3 3 18 30 31 5 6 19 37 5 6 6 3 0 31 31 6 6 3 144
8A 1 3 1 6 6 6 6 6 5 6 5 5 6 6 5 3 5 5 5 4 5 4 4 3 4 3 4 4 3 5 5 5 4 5 5 4 6 9 9 9 9 9 9 7 7 8 7 7 7 7 8 5 5 5 5 5 5 9 6 5 6 5 5 5 10 6 6 6 6 6 6 11 5 4 4 4 4 4 1 4 5 4 4 4 4 13 5 4 5 4 4 4 14 6 6 6 6 5 6 15 5 6 6 5 5 5 16 6 7 7 6 6 7 17 6 6 6 6 6 6 18 4 3 4 3 3 4 19 4 5 4 5 5 4 0 6 6 6 6 6 6 145
8B 1 3 1 6 5 6 5 6 5 6 5 5 6 6 5 3 6 5 6 5 6 5 4 5 5 5 5 5 5 5 5 5 6 5 5 5 6 6 5 6 5 6 5 7 5 5 5 5 5 5 8 5 5 5 5 5 5 9 6 5 6 5 5 5 10 6 5 6 5 5 6 11 6 5 5 5 5 5 1 5 5 5 5 5 5 13 6 5 6 5 5 5 14 6 6 6 6 6 6 15 5 6 6 5 5 6 16 6 5 6 5 5 6 17 6 5 6 5 6 5 18 5 5 5 5 5 5 19 5 5 5 5 5 5 0 6 5 5 6 6 5 146
B n 3 4 5 6 7 8 9 10 d 1.18 1.693.059.36.534.704.847.907 3.078 n 11 1 13 14 15 16 17 18 19 0 d 3.173 3.58 3.336 3.407 3.47 3.53 3.588 3.640 3.689 3.735 147