This document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC.

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1 SPC Formulas ad Tables 1 This documet cotais a collectio of formulas ad costats useful for SPC chart costructio. It assumes you are already familiar with SPC. Termiology Geerally, a bar draw over a symbol meas the average of all the values i a sample or group of values. Thus x is the average of the x values i a sample. sample size, the umber of values i a sample x - average of measuremets i a sample x - average of averages (of all samples) R the rage of a sample (differece betwee largest ad smallest values) R - the average of rages S sample stadard deviatio S - the average of the stadard deviatios of the samples - estimate of the process stadard deviatio (proouced sigma hat ) - stadard error; stadard deviatio of sample meas (proouced sigma e ) CL ceter lie UCL upper cotrol limit LCL lower cotrol limit Cotrol Charts for Variables x ad R Charts Calculatig x " i1 x i x ad R R max(x 1, x 2,...x ) " mi(x 1,x 2,...x ) Estimatig populatio stadard deviatio R Estimatig stadard deviatio of sample meas R #

2 SPC Formulas ad Tables 2 Calculatig Ceter Lie ad Cotrol Limits CL x x UCL x x + A 2 " R LCL x x " A 2 # R CL R R UCL R R" D 4 UCL R R" D Table of costats A 2 D D x ad S Charts Calculatig x ad S x " i1 x i S # i1 (x i " x) 2 "1 Estimatig populatio stadard deviatio S

3 SPC Formulas ad Tables Estimatig stadard deviatio of sample meas S # Calculatig Ceter Lie ad Cotrol Limits CL x x UCL x x + A " S CL S S UCL S S" B 4 Table of costats LCL x x " A # S UCL S S" B c 4 A B B For values of greater tha 25, the followig may be used:

4 SPC Formulas ad Tables 4 4" ( #1) 4" # B 1" # 2# ( "1) A " B 4 1+ " 2" ( #1) X (idividuals) ad Movig Rage (MR) Charts I this special case is iterpreted as 2 whe cosiderig the rage, but as 1 whe cosiderig the umber of values i each sample. Thus, D, ad D 4 are chose based o 2. Whe calculatig the samplig distributio, however, 1 is used. Calculatig the Movig Rage MR x i " x i"1 (the positive differece betwee oe value ad the previous) Estimatig populatio stadard deviatio R (where 1.128, sice the sample size is 2) Estimatig stadard deviatio of sample meas MR # MR 1.128# 1 MR Calculatig Ceter Lie ad Cotrol Limits CL x x UCL x x + " MR x " MR LCL x x " # MR x " 2.660# MR CL MR MR where MR k " MR ad k is the umber of samples take. k #1 i1 UCL MR MR" D 4 MR".267 LCL MR MR" D 0

5 SPC Formulas ad Tables 5 Cotrol Charts for Attributes Chart for proportio defective - p Charts Requires costat sample size. Proportio defective p D CL p UCL p + LCL p " p(1" p) p"(1 p) Chart for umber defective p Chart Allows for variable sample size with varyig cotrol limits. Number defective p CL p UCL p + p(1" p) LCL p " p(1" p) Chart for umber of defects/ocoformaces per samplig uit c Chart (The subtlety here is that i c ad u charts, a sigle uit may have more tha oe defect, while with p ad p charts, a item is either defective or ot.) CL c UCL c + c

6 SPC Formulas ad Tables 6 LCL c " c Chart for umber of defects/ocoformaces per samplig uit - u Charts Allows for variable size of samplig uit with variable cotrol limits. u x CL u UCL u + LCL u " u u Sesitizig Rules for Cotrol Charts Normally, a sigle poit outside the cotrol limits is cosidered to sigal a out of cotrol process. Uder some circumstaces, however, such as while workig to establish statistical cotrol, it is desirable to employ sesitizig rules which make it more likely that a small chage i mea or variability will be detected. The so-called Wester Electric Compay or WECO rules iclude: Oe poit outside the cotrol limits Two of three cosecutive poits outside the two sigma zoe (o the same side of ceter) Four of five cosecutive poits outside the oe sigma zoe (o the same side of ceter) Eight cosecutive poits o the same side of the ceter lie Additioal commoly used sesitizig rules: Six cosecutive poits steadily icreasig or decreasig Fiftee cosecutive poits iside the oe sigma zoe Fourtee cosecutive poits alteratig up ad dow Eight cosecutive poits outside the oe sigma zoe A clearly o-radom patter

7 SPC Formulas ad Tables 7 Process Capability Aalysis Capability Idices USL LSL C p 6 " C pk is the lesser of: C pu USL µ " C pl µ LSL " where if µ ad are ot kow, they ca be replaced with their estimates ˆµ ad ˆ, e.g. x ad R or S (While the foregoig is believed to be correct ad accurate, it is provided with o warraty whatever.)

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

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