Taylor Enterprses, Inc. Control Lmts for P Charts Copyrght 2017 by Taylor Enterprses, Inc., All Rghts Reserved. Control Lmts for P Charts Dr. Wayne A. Taylor Abstract: P charts are used for count data followng the Bnomal dstrbuton. However, the P chart has symmetrcal control lmts when the Bnomal dstrbuton s nonsymmetrcal. As a result, the upper control lmt can have a rate of false detecton as hgh as 1 n 11.5 ponts plotted. Ths can result n wasted resources nvestgatng false sgnals. Further, the lower control lmt has a rate of false detecton consstently above 1 n 1000. Ths makes t slow to detect mprovements n qualty. control lmts are provded that correct both these problems. The adjusted control lmts result n a P chart truly based on the assumpton of the Bnomal dstrbuton. 1.0 Introducton The average run length (ARL) of a control chart s the average number of ponts that are plotted before one goes outsde the control lmts. The ARL vares based on the sze of the shft. When a sgnfcant shft occurs the ARL should be small, approachng 1. Also of nterest s the ARL when there s no shft. Ths s the tme between false sgnals. It should be large. The ARL when there s no change wll be referred to as the false detecton tme, FDT. 1/FDT wll be referred to as the false detecton rate, FDR. Control charts based on the normal dstrbuton, such as X and I charts (Taylor 2017b), wth ±3 standard devaton control lmts, have FDRs of: 1 n 740 relatve to the upper control lmt: Φ = 0.0013499 1 3 = 1 740.80 1 n 740 relatve to the lower control lmt: Φ( 3) = 0.0013499 = 1 740.80 1 n 370 for both control lmts combned: 2 0.0013499 = 1 370. 40 Φ ( z) s the probablty of beng less than z for the normal dstrbuton (µ=0, σ=1). P charts assume pass/fal data and are based on the Bnomal dstrbuton. The Bnomal dstrbuton has 2 parameters: sample sze (n) and probablty fal (p), where: Average = np Devaton = np( 1 p) The standard control lmts for a P chart are: = ( ) UCL = np + 3 np( 1 p) LCL np 3 np 1 p As the Bnomal dstrbuton s not symmetrcal and the standard control lmts are symmetrcal, the FDRs of a P chart wll dffer from those above. 3 standard devaton control lmts are generally robust to the assumpton of normalty as most dstrbutons have a hgh percentage of values wthn three standard devatons of the average. Whle ths may be true most of the tme, for the P chart the false detecton rate can be as hgh as 1 n 11.5. Ths s an unacceptable rate. Ths artcle documents the false detecton rates for P charts and offers a soluton n the form of an adjustment to the control lmts. Revson 1: September 18, 2017 1 www.varaton.com/techlb/bref3.html
Taylor Enterprses, Inc. Control Lmts for P Charts 2.0 False Detecton Rates The P chart works dfferently than an X chart when detectng a worsenng of qualty. For an X chart, both upward and downward shfts can sgnal a shft from the target and a worsenng of qualty. The FDR relatve to a worsenng n qualty s then 1 n 370. For a P chart of falures or nonconformng unts, only an ncrease n the counts sgnals a worsenng of qualty. Therefore, t s approprate to have a 1 n 370 FDR assocated wth the UCL by tself. It would be of concern f the FDR dropped below 1 n 200, as ths represents nearly a doublng n the number of false sgnals. The lower control lmt sgnals an mprovement n qualty. The consequence of a false detecton relatve to the LCL s dfferent from that assocated wth the UCL. It s also approprate to have a 1 n 370 FDR assocated wth the LCL by tself. It would be of concern f the FDR dropped below 1 n 200. The artcle Control Lmts for U Charts, Taylor (2017a), descrbes how the FDR for U charts can be as hgh as 1 n 11.5. As shown below, the same s true for P charts. Ths s an unacceptable rate. Ths artcle determnes when the standard control lmts for a P chart can be safely used,.e., the FDR s at worse 1 n 200. Ths artcle also shows how to handle stuatons where the FDR s worse than 1 n 200. Fgures 1-5 show the FDR as a functon of the average count (n p). Formulas for the FDR are gven n Secton 4. The FDR s not a smooth curve. Ths s due to counts beng ntegers. The FDR jumps whenever the control lmts cross an nteger value. Ths creates an oscllaton pattern. As long as 10 np n-10, the FDR s better than 1 n 200. Ths s true for all sample szes, even the mnmum sample sze satsfyng these condtons of n=20. When n s 100 or more, the FDR s close to the desred 1 n 370 over a wde range. P charts wth standard control lmts should only be used when 10 np n-10. Ths corresponds to the condtons where the Bnomal dstrbuton s well approxmated by the normal dstrbuton. 1 n 370 1 n 200 Fgure 1: False Detecton Rate for the Control Lmts, n=20 Revson 1: September 18, 2017 2 www.varaton.com/techlb/bref3.html
Taylor Enterprses, Inc. Control Lmts for P Charts 1 n 370 1 n 200 Fgure 2: False Detecton Rate for the Control Lmts, n=30 1 n 370 1 n 200 Fgure 3: False Detecton Rate for the Control Lmts, n=50 1 n 370 1 n 200 Fgure 4: False Detecton Rate for the Control Lmts, n=100 Revson 1: September 18, 2017 3 www.varaton.com/techlb/bref3.html
Taylor Enterprses, Inc. Control Lmts for P Charts 1 n 370 1 n 200 Fgure 5: False Detecton Rate for the Control Lmts, n=1000 3.0 Control Lmts It s common for P charts to have an average count np < 10. STAT-10 Statstcal Technques for Trendng Data n Taylor (2017c) states the P chart s generally the best chart for pass/fal counts less than 25 but that the I chart (or Laney P chart) s generally the best chart for counts greater than 25. The nche for the P chart s when counts are less than 25. Ths ncludes counts < 10 where adjusted control lmts are needed. One approach s to cumulate data over longer perods of tme. For example, nstead of performng a weekly chart wth an average count of 2.5, perform a monthly chart wth an average count of 10. Whle ths may better control the false rejecton rate, t may also delay the detecton of a problem. Another approach takes advantage of the relatonshp between the bnomal and Posson dstrbutons. Holdng the average count constant, as n gets larger the bnomal dstrbuton converges to the Posson dstrbuton wth λ = np. Ths means the adjusted control lmts for the U chart n Taylor (2017a) may be used wth a P chart. The adjusted control lmts for the P chart are then: UCL = np + 2.782 np 1 p + 1 LCL = np 2.782 np 1 p + 1.1 These are the adjusted control lmts for the U chart wth the standard devaton of the bnomal dstrbuton substtuted for the standard devaton of the Posson dstrbuton. Fgures 6-9 show the FDR of the adjusted control lmts as a functon of the average count (np). The FDR s calculated based on the bnomal dstrbuton, so s exact. In all the charts the FDR s better than 1 n 200. However, for n=50 the FDR dverges from 1 n 370. For n=100 and above, the FDR stays n the 1 n 370 regon. For n 100 and np<10, the adjusted control lmts may be used. Revson 1: September 18, 2017 4 www.varaton.com/techlb/bref3.html
Taylor Enterprses, Inc. Control Lmts for P Charts 1 n 370 1 n 200 Fgure 6: False Detecton Rate for the Upper Control Lmt, n=30 1 n 370 1 n 200 Fgure 7: False Detecton Rate for the Upper Control Lmt, n=50 1 n 370 1 n 200 Fgure 8: False Detecton Rate for the Upper Control Lmt, n=100 Revson 1: September 18, 2017 5 www.varaton.com/techlb/bref3.html
Taylor Enterprses, Inc. Control Lmts for P Charts 1 n 370 1 n 200 Fgure 9: False Detecton Rate for the Upper Control Lmt, n=1000 For n 100 and np>n-10, usng symmetry the adjusted control lmts below can be used: LCL = np 2.782 np 1- p 1 UCL = np + 2.782 np 1- p 1.1 Ths only leaves the stuaton where n<100 and ether np<10 or np>n-10. In ths case cumulate data untl there are at least 100 samples. 4.0 P Chart Formulas In the prevous sectons t was assumed that each pont s a count F followng the bnomal dstrbuton wth parameters n and p: F ~ Bnomal n,p The bnomal dstrbuton has the followng propertes: Average = np Devaton = np( 1 p) The fact that p s used to estmate the standard devaton smplfes the chart as a separate estmate of the standard devaton s not needed and makes the chart more powerful for bnomal data. However, t makes the chart senstve to the assumpton of the bnomal dstrbuton. The bnomal dstrbuton occurs when the tems beng counted occur ndependently of each other. When counts are larger, the counts tend to vary more than the bnomal due to dependences. Ths s called overdspersed. onconformng unts may occur n runs, occur because of a problem wth a sngle cavty or nozzle, etc. For the bnomal dstrbuton, the number of falng unts F are plotted along wth the followng control lmts: Counts = 3 np( ) UCL = np + 3 np( 1 p) LCL np 1 p Revson 1: September 18, 2017 6 www.varaton.com/techlb/bref3.html
Taylor Enterprses, Inc. Control Lmts for P Charts LCL=0 whenever the lower control lmt s negatve. UCL=n whenever the upper control lmt s greater than n. The adjusted control lmts for n 100 and np<10 are: UCL = np + 2.782 np1 p + 1 ( p) LCL = np 2.782 np1 + 11. =0 for p 6.46427826534363 2.77024877318692 + 9.36600203111234 / n. n + 7.74049754637384 Ths was obtaned by settng the LCL to zero and solvng for p. The value 2.78217496688721 = 1 2 ( 3) Φ Φ was selected to have the same FDR for just the UCL as for the standard devaton control lmts. Ths s around 1 n 370. The adjusted control lmts for n 100 and np>n-10 are: UCL = np + 2.782 np 1- p 1.1 =n for 6.46427826534363 2.77024877318692 + 9.36600203111234 / n p 1 n + 7.74049754637384. LCL = np 2.782 np 1- p 1 The false detecton rates are: FDR FDR FDR Lower Upper Both 1 = Bnomal cel LCL 1n,p 1 = 1- Bnomal floor ( UCL) n,p 1 = Bnomal cel( LCL) 1n,p + 1- Bnomal floor ( UCL) n,p Cel() rounds the value up to an nteger. Floor() rounds the value down to an nteger. Bnomal() s the bnomal dstrbuton functon. A control chart of counts s referred to as an P chart. For a P chart, proportons P are plotted based on the sample szes : P F = wth F ~ Bnomal(,p ) Then the control lmts for the proportons P are: Revson 1: September 18, 2017 7 www.varaton.com/techlb/bref3.html
Taylor Enterprses, Inc. Control Lmts for P Charts UCL LCL ( ) p + 3 p 1 p p1 p = = p + 3 ( ) ( ) p 3 p 1 p p1 p = = p 3 LCL=0 whenever the lower control lmt s negatve. UCL=1 whenever the upper control lmt s greater than 1. Smlarly, the adjusted control lmts for n 100 and np<10 are: UCL = p + 2.782 LCL ( p) p1 ( p) 1 + p1 1.1 = p 2. 782 + =0 for p 6.46427826534363 2.77024877318692 + 9. 36600203111234 / ( + 7.74049754637384) The adjusted control lmts for n 100 and np>n-10 are: UCL = p + 2. 782 p 1-p 1.1. =1 for 6.46427826534363 2.77024877318692 + 9.36600203111234 / p 1. p 1-p 1 LCL = p 2. 782 5.0 Conclusons ( + 7.74049754637384) P charts are useful for pass/fal count data followng the Bnomal dstrbuton, whch s most lkely for counts below 25. For the Bnomal dstrbuton, the P chart s better than an I chart because t provdes a better estmate of the standard devaton. However, a P chart s not entrely based on the Bnomal dstrbuton because the Bnomal dstrbuton s skewed whle a P chart uses symmetrcal control lmts. For the standard upper control lmt, the rate of false detecton can be as hgh as 1 n 11.5 for low counts. It does not reman above 1 n 200 untl the average count reaches 10 or s less than n-10 The adjusted UCL control lmt fxes ths problem for n 100. It mantans the false detecton rate above 1 n 200 and averages around 1 n 370. For n<100 and ether Revson 1: September 18, 2017 8 www.varaton.com/techlb/bref3.html
Taylor Enterprses, Inc. Control Lmts for P Charts np<10 or np>n-10, cumulate data untl there are at least 100 samples. The adjusted UCL also mproves the detecton of a worsenng of qualty for larger counts, as long as the assumpton of the Posson contnues to hold. For the standard lower control lmt, the rate of false detecton s consstently above 1 n 1000. Ths makes t slow to detect mprovements n qualty. Ths can result n mssng mprovements and the assocated lessons learned. The adjusted LCL makes the chart much qucker at detectng mprovements n qualty. The adjusted control lmts result n a P chart truly based on the assumpton of the Bnomal dstrbuton. The same approach can be appled to U charts as descrbed n Taylor (2017a). 6.0 References Control Lmts for U Charts (2017a), Dr Wayne A. Taylor, Taylor Enterprses, Inc. (www.varaton.com/techlb/bref2.html). ormalzed Indvduals (I) Chart (2017b), Dr Wayne A. Taylor, Taylor Enterprses, Inc. (www.varaton.com/techlb/bref1.html). Statstcal Procedures for the Medcal Devce Industry (2017c), Dr Wayne A. Taylor, Taylor Enterprses, Inc. (http://www.varaton.com/procedures). Revson 1: September 18, 2017 9 www.varaton.com/techlb/bref3.html