Introduction. Part-to-Part and Reproducibility Variation. Technical Support Document Number of Parts and Operators for a Gage R&R Study.

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1 Tchnical upport Documnt Introduction Th primary purpos of a gag study is to analyz how much variation in th data is du to th masurmnt systm so that you know whthr th masurmnt systm is capabl of assssing procss prformanc. In a typical masurmnt systm study, a gag is usd to obtain rpatd masurmnts on slctd parts by svral oprators. Two componnts of masurmnt systm variability ar frquntly gnratd in such studis: rpatability and rproducibility. Rpatability rprsnts th variability whn th gag is usd to masur th sam part by th sam oprator. Rproducibility rfrs to th variability from diffrnt oprators masuring th sam part. Thus, masurmnt systm studis ar oftn rfrrd to as gag rpatability and rproducibility studis, or gag R&R studis. Anothr sourc of variability in th data is th actual variation btwn parts producd by th procss, calld th part-to-part variation. To dtrmin whthr th masurmnt systm can distinguish on part from anothr, you nd to hav good stimats for part-to-part variation, rproducibility, and rpatability. Usrs oftn want to know how many parts, oprators, and rplicats thy nd for a gag R&R study and whthr th widly accptd practic of using 10 parts, 3 oprators, and 2 or 3 rplicats (AIAG, 2003; Raffaldi and Ramsir, 2000; Tsai, 1988) is sufficint to obtain prcis stimats for part-to-part variation, rproducibility, and rpatability. To addrss ths qustions, w prformd th following studis: Part-to-part and rproducibility variation - W prformd a simulation to valuat how many parts ar ndd to obtain prcis stimats of part-to-part variation with a givn numbr of oprators. Ths rquirmnts for th numbr of parts also apply to th numbr of oprators, which affcts th prcision of th rproducibility stimat. Howvr, in a typical gag study, th numbr of parts is far mor critical than th numbr of oprators bcaus vry oftn most of th variation rsults from diffrncs btwn parts rathr than diffrncs btwn oprators. Rpatability - W prformd xact calculations to valuat how th numbr of parts, th numbr of oprators, and th numbr of rplicats affct th prcision of th stimat for rpatability. pcifically, w usd a formula to calculat th lowr and uppr bounds of th stimatd rpatability standard dviation ovr th tru valu and xamind how th dgrs of frdom affct th margin of rror. Part-to-Part and Rproducibility Variation Mthod Bcaus thr is no xact formula to calculat th confidnc intrval for th part-to-part standard dviation, w prformd a simulation to stimat th intrval. To focus th simulation on how th numbr of parts affcts th prcision of th stimatd part-to-part variation, w xamind th ratio of th stimatd confidnc intrval for th standard dviation of th parts ovr th tru standard dviation of th parts. As th numbr of parts incrass, th intrval bcoms narrowr. W thn idntifid th numbr of parts such that th margin of rror for th ratio is 10% or 20%. Th intrval for th 10% margin of rror is (0.9, 1.1), and for th 20% margin of rror is (0.8, 1.2). Knowldgbas ID 2642: Pag 1

2 Tchnical upport Documnt A gag R&R study assums that th k th masurmnt of th i th part by th j th oprator, dnotd as following modl:, fits th Whr, and,,, and ar indpndntly normally distributd with man 0, and variancs of,,, and. Hr,,, and rprsnt parts, oprators, parts x oprators, and rror trms. Lt r b th ratio of th total gag standard dviation ovr th total procss standard dviation. Thn, Typically, th following rul is usd to dtrmin whthr a masurmnt systm is accptabl: r 0.1 (10%): Accptabl 0.1 < r 0.3: Marginal 0.3 < r: Unaccptabl W choos r = 0.1 (accptabl), r = 0.25 (marginal), and r = 0.35 (unaccptabl) to dfin th thr rgions. For th purposs of th simulation, w assum that th rpatability varianc quals th rproducibility varianc, which givs: W us and 1,, and to gnrat th obsrvations, and assum that 3 oprators masur ach part twic to valuat how th numbr of parts affcts th stimat for th standard dviation of th parts. For ach numbr of parts, r, and, w prformd th simulation as follows: 1. Gnrat 5000 sampls using th modl abov. 2. Estimat th standard dviation of th parts, and calculat th ratio of th stimatd standard dviation ovr th tru standard dviation for ach of th 5000 sampls. 3. ort th 5000 ratios in incrasing ordr. Of th 5000 sortd ratios, th 125 th and 4875 th ratios rprsnt th lowr and uppr bounds of th intrval at th 95% confidnc lvl, and th 250 th and 4750 th ratios rprsnt th lowr and uppr bounds of th intrval at th 90% confidnc lvl. 4. Examin th intrvals to idntify th numbr of parts such that th margin of rror is 10% or 20%. Th intrval for th 10% margin of rror is (0.9, 1.1). Th intrval for th 20% margin of rror is (0.8, 1.2). Knowldgbas ID 2642: Pag 2

3 Tchnical upport Documnt Rsults Th rsults in Tabls 1-6 show th simulation rsults at ach confidnc lvl for diffrnt numbrs of parts, with ach tabl corrsponding to a spcific combination of valus for r and. Ovrall, ths rsults show that: Using 10 parts, 3 oprators, and 2 rplicats, th ratio of th 90% confidnc intrval ovr th tru standard dviation is about (0.61, 1.37) with 35% to 40% margin of rror. At 95% confidnc, th intrval is about (0.55, 1.45) with 45% margin of rror. Thrfor, 10 parts ar not nough to produc a prcis stimat for th part-to-part variation componnt. You nd approximatly 35 parts to hav a 90% confidnc of stimating th part-to-part variation within 20% of th tru valu. You nd approximatly 135 parts to hav a 90% confidnc of stimating th part-to-part variation within 10% of th tru valu. Not that this summary of th rsults is not spcific to a particular combination of r and Th rows corrsponding to th bulltd rsults abov ar highlightd in Tabls 1, 2, 3, 4, 5, and 6 blow. Bcaus oprator variation, which rprsnts rproducibility, is stimatd similarly to part-to-part variation in th modl, ths rsults can also apply to th numbr of oprators. Thrfor, if you suspct a larg variation among oprators, you should us mor than 3 oprators. You can us th rsults for th rquird numbr of parts to slct th rquird numbr of oprators at ach lvl of prcision. Tabl 1: Accptabl gag (r = 0.1),, tru part stdv = Ratio of stimatd confidnc intrval for part stdv/tru part stdv Numbr of parts 95% Confidnc 90% Confidnc 3 ( , ) ( , ) 5 ( , ) ( , ) 10 ( , ) ( , ) 15 ( , ) ( , ) 20 ( , ) ( , ) 25 ( , ) ( , ) 30 ( , ) ( , ) 35 ( , ) ( , ) 50 ( , ) ( , ) Knowldgbas ID 2642: Pag 3

4 Tchnical upport Documnt 100 ( , ) ( , ) 135 ( , ) ( , ) 140 ( , ) ( , ) Tabl 2: Accptabl gag (r =0.1),, tru part stdv = Ratio of stimatd confidnc intrval for part stdv/tru part stdv Numbr of parts 95% Confidnc 90% Confidnc 5 ( , ) ( , ) 10 ( , ) ( , ) 15 ( , ) ( , ) 35 ( , ) ( , ) 40 ( , ) ( , ) 135 ( , ) ( , ) 140 ( , ) ( , ) 145 ( , ) ( , ) 150 ( , ) ( , ) Knowldgbas ID 2642: Pag 4

5 Tchnical upport Documnt Tabl 3: Marginal gag (r = 0.25),, tru part stdv = Ratio of stimatd confidnc intrval for part stdv/tru part stdv Numbr of parts 95% Confidnc 90% Confidnc 30 ( , ) ( , ) 35 ( , ) ( , ) 40 ( , ) ( , ) 135 ( , ) ( , ) 140 ( , ) ( , ) 145 ( , ) ( , ) 150 ( , ) ( , ) Tabl 4: Marginal gag (r = 0.25),, tru part stdv = Ratio of stimatd confidnc intrval for part stdv/tru part stdv Numbr of parts 95% Confidnc 90% Confidnc 30 ( , ) ( , ) 35 ( , ) ( , ) 40 ( , ) ( , ) 135 ( , ) ( , ) 140 ( , ) ( , ) 145 ( , ) ( , ) Knowldgbas ID 2642: Pag 5

6 Tchnical upport Documnt Tabl 5: Unaccptabl gag (r = 0.35),, tru part stdv = Ratio of stimatd confidnc intrval for part stdv/tru part stdv Numbr of parts 95% Confidnc 90% Confidnc 30 ( , ) ( , ) 35 ( , ) ( , ) 40 ( , ) ( , ) 135 ( , ) ( , ) 140 ( , ) ( , ) 145 ( , ) ( , ) Tabl 6: Unaccptabl gag (r = 0.35),, tru part stdv = Ratio of stimatd confidnc intrval for part stdv/tru part stdv Numbr of parts 95% Confidnc 90% Confidnc 30 ( , ) ( , ) 35 ( , ) ( , ) 40 ( , ) ( , ) 135 ( , ) ( , ) 140 ( , ) ( , ) 145 ( , ) ( , ) Knowldgbas ID 2642: Pag 6

7 Tchnical upport Documnt Rpatability Mthod Unlik th confidnc intrvals for th part-to-part standard dviation, which ar basd on an approximation, th ratio of th stimatd rpatability standard dviation ovr its tru valu follows a chi-squar distribution. Thrfor, w can calculat th lowr and uppr bounds of th ratio associatd with 95% and 90% probabilitis, and thn valuat how both bounds approach 1 as th numbr of parts, numbr of oprators, and th numbr of rplicats incras. Using th sam notation dfind in th prcding study, th rpatability varianc is stimatd by 2 ( Y ijk Y ij. ) 2 / IJ ( K 1) 2 IJ ( K 1) Thn, follows a chi-squar distribution with IJ(K-1) dgrs of frdom (df), whr I is th 2 numbr of parts, J is th numbr of oprators, and K is th numbr of rplicats. Basd on this rsult, th ratio of th stimatd standard dviation ovr its tru valu satisfis th following probability quation: Probability 2 df, / 2. df 2 df,1( / 2) df 1. whr df = IJ(K-1) = numbr of parts * numbr of oprators * (numbr of rplicats 1). If th numbr of rplicats quals 2, th dgrs of frdom qual th numbr of parts tims th numbr of oprators. Using this formula, for ach givn valu of th dgrs of frdom, w calculat th lowr and uppr bounds of th ratio at probabilitis of 95% and 90%. W thn idntify th dgrs of frdom such that th stimatd standard dviation is within 10% and 20% of its tru valu. Th corrsponding intrval is (0.9, 1.1) for th 10% margin of rror, and (0.8, 1.2) for th 20% margin of rror. Rsults Th graph in Figur 1 shows th lowr and uppr bounds of th ratio of frdom, with th dgrs of frdom ranging from 1 to 200. at 95% probability vrsus th dgrs Knowldgbas ID 2642: Pag 7

8 Ratio Tchnical upport Documnt Lowr and Uppr Bounds of th Ratio at 95% Probability Variabl Lowr Uppr Dgrs of Frdom Figur 1: Lowr and uppr bounds of at 95% probability vrsus dgrs of frdom (1 to 200) Notic that that th intrval formd by th lowr and uppr bounds narrows as th dgrs of frdom incras. Th width of th intrval dcrass dramatically as th dgrs of frdom incras from 1 to 50. W can s this mor clarly in th nlargd graph shown in Figur 2, which displays th rsults for dgrs of frdom from 1 to 50. Knowldgbas ID 2642: Pag 8

9 Ratio Tchnical upport Documnt Lowr and Uppr Bounds of th Ratio at 95% Probability Variabl Lowr Uppr Dgrs of Frdom Figur 2: Lowr and uppr bounds of at 95% probability vrsus dgrs of frdom (1 to 50) As shown in Figur 2, whn th dgrs of frdom ar lss than 10, th intrval is widr than (0.57, 1.43). As th dgrs of frdom incras, th intrval bcoms narrowr, as shown by th valus summarizd in Tabl 7 blow. Tabl 7: Dgrs of frdom and lowr/uppr bounds at 95% probability Dgrs of frdom Intrval formd by lowr and uppr bounds 5 (0.41, 1.60) 10 (0.57, 1.43) 15 (0.65, 1.35) 20 (0.69, 1.31) 25 (0.72, 1.28) 30 (0.75, 1.25) 35 (0.77, 1.23) Knowldgbas ID 2642: Pag 9

10 Ratio Tchnical upport Documnt Dgrs of frdom Intrval formd by lowr and uppr bounds 40 (0.78, 1.22) 50 (0.80, 1.20) Thrfor, to hav a 95% probability that th standard dviation stimat for rpatability is within a 20% margin of rror, w nd about 50 dgrs of frdom. This mans that th Num of Parts * Num of Oprators should b clos to 50. If w want to dcras th margin of rror to 10%, which corrsponds to th intrval (0.9, 1.1), w nd about 170 dgrs of frdom (s Figur 1). If w rduc th probability from 95% to 90%, th dgrs of frdom rquirmnts dcras accordingly. Th graphs in Figurs 3 and 4 show th rsults for th sam calculations prformd at 90% probability. Lowr and Uppr Bounds of th Ratio at 90% Probability 2.0 Variabl Lowr Uppr Dgrs of Frdom Figur 3: Lowr and uppr bounds of at 90% probability vrsus dgrs of frdom (1 to 200) As xpctd, th rsults in Figur 3 show that th dgrs of frdom rquird to produc a givn intrval ar gnrally smallr at 90% probability than thy wr at 95% probability in Figur 1. Th nlargd graph in Figur 4 blow mor clarly shows th intrvals for dgrs of frdom from 1 to 50. Knowldgbas ID 2642: Pag 10

11 Ratio Tchnical upport Documnt Lowr and Uppr Bounds of th Ratio at 90% Probability Variabl Lowr Uppr Dgrs of Frdom Figur 4: Lowr and uppr bounds of at 90% probability vrsus dgrs of frdom (1 to 50) As shown in Figur 4, whn th dgrs of frdom ar lss than 10, th intrval is widr than (0.63, 1.35). As th dgrs of frdom incras, th intrval bcoms narrowr, as indicatd by th valus in Tabl 8 blow. Tabl 8: Dgrs of frdom and lowr and uppr bounds at 90% probability Dgrs of frdom Intrval formd by lowr and uppr bounds 5 (0.48, 1.49) 10 (0.63, 1.35) 15 (0.70, 1.29) 20 (0.74, 1.25) 25 (0.76, 1.23) 30 (0.79, 1.21) 35 (0.80, 1.19) 40 (0.81, 1.18) Knowldgbas ID 2642: Pag 11

12 Tchnical upport Documnt Thrfor, at 90% probability, you nd about 35 dgrs of frdom to obtain a 20% margin of rror for th standard dviation stimat of rpatability. Rcall that th dgrs of frdom qual th Numbr of Parts * Numbr of Oprators * (Numbr of Rplicats 1). Thrfor, th typical rcommndation of 10 parts, 3 oprators, and 2 rplicats provids dgrs of frdom (30) that ar clos to this rquirmnt. To obtain a 10% margin of rror at 90% probability, you nd about 135 dgrs of frdom (s Figur 3). Rfrncs Burdick, R.K., Borror, C. M., and Montgomry, D.C. (2005). Dsign and analysis of gaug R&R studis: Making dcisions with confidnc intrvals in random and mixd ANOVA modls. Philadlphia, PA: ocity for Industrial Applid Mathmatics (IAM). Automotiv Industry Action Group (AIAG) (2003). Masurmnt systms analysis (MA) manual (3rd dition). outhfild, MI: Chryslr, Ford, Gnral Motors upplir Quality Rquirmnts Task Forc. Montgomry, D.C. (2000). Dsign and analysis of xprimnts. Nw York, NY: Wily. Montgomry, D.C., and Rungr, G.C. (1993 a). Gag capability and dsignd xprimnts. Part I: Basic mthods. Quality Enginring, 6 (1993/1994), Montgomry, D.C., and Rungr, G.C. (1993 b). Gag capability analysis and dsignd xprimnts. Part II: Exprimntal dsign modls and varianc componnt stimation. Quality Enginring, 6 (1993/1994), Raffaldi, J. and Ramsir,. (2000). 5 ways to vrify your gags. Quality Magazin, 39 (3), Tsai, P. (1988). Variabl gag rpatability and rproducibility study using th analysis of varianc mthod. Quality Enginring, 1(1), Vardman,.B. and VanValknburg, E.. (1999). Two-way random-ffcts analyss and gag R&R studis. Tchnomtrics, 41 (3), Knowldgbas ID 2642: Pag 12