# Statistical Fundamentals and Control Charts

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1 Statistical Fudametals ad Cotrol Charts 1. Statistical Process Cotrol Basics Chace causes of variatio uavoidable causes of variatios Assigable causes of variatio large variatios related to machies, materials, operators or others Process i statistical cotrol the system is operatig with oly chace causes of variatio preset Process out of cotrol the system is operatig i presece of assigable causes of variatio The evetual goal of SPC is reductio or elimiatio of variability i the process by idetificatio of assigable causes 40

2 Basic Priciples A typical cotrol chart has cotrol limits set at values such that if the process is i cotrol, early all poits will lie betwee the upper cotrol limit (UCL) ad the lower cotrol limit (LCL). Out-of-Cotrol Situatios If at least oe poit plots beyod the cotrol limits, the process is out of cotrol If the poits behave i a systematic or oradom maer, the the process could be out of cotrol. Eample. I a electroic maufacturig process, the true = 1.5 ad stadard deviatio is = May samples are take with each sample of size 5. The stadard deviatio of the sample average is: If the cotrol limits are set at 3 stadard deviatios from the mea, it gives the 3- sigma cotrol limits : 41

3 UCL = (0.0671) = CL= 1.5 LCL = 1.5-3(0.0671) = The cotrol chart 4

4 The quality cotrol process Types of Process Variability Statioary behavior, ucorrelated data Statioary behavior, autocorrelated data Nostatioary behavior 99.7% of the Data approimately 99.7% of the data lies withi 3 of the mea (i.e., 99.7% of the data should lie withi the cotrol limits), 0.3% of the data ca fall outside 3 (or 0.3% of the data lies outside the cotrol limits) is the probability of a Type I error or a false alarm 43

5 Three-Sigma Limits The use of 3-sigma limits geerally gives good results i practice. Distributio of should be ormal distributio These limits are ofte referred to as actio limits Ratioal subgroups Select cosecutive uits of productio to provide a sapshot of the process. Effective at detectig process shifts. May be ieffective i detectig if the mea has wadered out-of-cotrol ad the back Select a radom sample over the etire samplig iterval. 44

6 Noradom patters ca idicate out-of-cotrol coditios Cycles, treds ad rus : all above or below the ceter lie, ru up ad ru dow Rus of 8 observatios or more could idicate out-of-cotrol. A o-radom patter eample Patter is very oradom i appearace 19 of 5 poits plot below the ceter lie, while oly 6 plot above Followig 4 th poit, 5 poits i a row icrease i magitude, a ru up There is also a uusually log ru dow begiig with 18 th poit. Widely Used Cotrol Charts for Variables: -R chart ad -S chart Moitor both the mea value of the characteristic ad the variability associated with the characteristic. 45

7 If the process mea ad stadard deviatio are kow, we ca follow the oe phase approach to set up -R or -S cotrol charts to moitor the process o - size of the sample (sometimes called a subgroup) o i - average of the observatios i the i-th sample i=1,,3, i i1 i... o is a ormally distributed variable with mea ad stadard i deviatio o 1 is the probability that will fall betwee ad Z Z Z Z o I settig up a Shewhart cotrol chart, it typically uses Z / The i values will be plotted i the cotrol chart with kow : UCL= 3 Ceter Lie = LCL= 3 o We also eed to use R or S cotrol chart to moitor the process variace as productio cotiues. This will be discussed later. If the process mea ad stadard deviatio are ot kow while we ca assume that the process follows ormal or close to ormal distributio, we eed a two-phase approach to set up Shewhart cotrol chart i Phase I ad to use the 46

8 established cotrol charts i Phase II. The most popular oes are ad R cotrol charts. I Phase I: o m is the umber of samples selected ad o is the size of each sample o - grad average or average of the averages (this value is used as the ceter lie of the cotrol chart) 1... m m o Ri - rage of the values i the ith sample R i ma{ j ij } mi{ j ij } i,ma i,mi o R - average rage for all m samples R R1 R... Rm m cotrol Limits for the chart UCL= A R Ceter Lie = LCL= A R A is foud i Appedi VI for various values of. Cotrol Limits for the R chart UCL= D 4 R Ceter Lie = R LCL= D 3 R 47

9 D 3 ad D 4 are foud i Appedi VI for various values of. The above cotrol charts are based o the followig ubiased estimator of the process stadard R deviatio : ˆ as discussed i Chapter 3. d Sice A R d R, so A d 3. Its value ca be foud i Appedi VI for various values of. 48

10 Eample R 5 R i i UCL= D 4 R =(.114)(0.351)= Ceter Lie = R =0.351 LCL= D 3 R =(0)(0.351)=0 UCL= 5 i i A R = (0.5777) (0.351)= Ceter Lie = = LCL= A R = (0.5777) (0.351)=

11 ad R charts Cotrol limit for S Charts o S is a ubiased estimator of o S is NOT a ubiased estimator of o S is a ubiased estimator of o The stadard deviatio of S is c 4 1 c 4 o If a stadard deviatio is give, the cotrol limits for the S chart are: UCL= c 3 1 c c 3 1 c B Ceter Lie = c 4 LCL= c 3 1 c c 3 1 c B B 5, B 6, ad c 4 are foud i the Appedi for various values of. 50

12 o If a stadard deviatio is ot give, use a average sample stadard deviatio, ad the cotrol chart will be: chart whe usig S S 1 m m S i i 1 UCL= B 4 S Ceter Lie = S LCL= S The upper ad lower cotrol limits for the chart are give as UCL= A3 S Ceter Lie = LCL= A3 S S where A 3 is foud i the Appedi. ca be estimated by ˆ. c Eample 5. B

13 S 5 i1 5 5 i 5 s i i UCL= 3 A S = (1.47) (0.0094)= Ceter Lie = = LCL= 3 A S = (1.47) (0.0094)= For S chart UCL= B 4 S =(.089) (0.0094)= Ceter Lie = S = LCL= B 3 S =(0) (0.0094)=0 5

14 3. The Shewhart Cotrol Chart for Idividual Measuremets Sample size is 1 Every uit is aalyzed The productio rate is very slow Repeat measuremets o the process differ oly because of laboratory or aalysis error. X ad Movig Rage Charts The movig rage (MR) is defied as: MRi i MR i i i 1, ad MR 1 m The X chart is the plot of the idividual observatios. The cotrol limits: UCL= MR 3 d Ceter Lie = LCL= MR 3 d The cotrol limits o the movig rage chart are: UCL= D 4 MR m Eample 5. Ceter Lie = M R LCL=0 53

15 MR UCL= = ( ) = d 1.18 Ceter Lie = = MR LCL= = ( ) = 3.57 d 1.18 UCL= D 4 MR =3.67(0.576) = Ceter Lie = M R = LCL=0 54

16 Iterpretatio of the Charts o X charts ca be iterpreted similar to charts. MR charts caot be iterpreted the same as or R charts. o Sice the MR chart plots data that are correlated with oe aother, the lookig for patters o the chart does ot make sese. o MR chart caot really supply useful iformatio about process variability. o More emphasis should be placed o iterpretatio of the X chart. 4. Cotrol Limits, Natural Tolerace Limits ad Specificatio Limits Cotrol limits are fuctios of the atural variability of the process Natural tolerace limits represet the atural variability of the process (usually set at 3-sigma from the mea) Specificatio limits are determied by developers/desigers. There is o mathematical relatioship betwee cotrol limits ad specificatio limits. Do ot plot specificatio limits o the charts 55

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