Control Chart - hstory Control Chart Developed n 920 s By Dr. Walter A. Shewhart 2 Process n control A phenomenon s sad to be controlled when, through the use of past experence, we can predct, at least wthn lmts, how the phenomenon may be expected to vary n the future. Walter A. Shewhart, 93 A producton process s sad to be n control when the qualty characterstcs of a product are subject only to random varaton (or common cause varaton) that s varaton n process performance due to normal or nherent nteracton among process components (people, machne, materal, envronment, and methods). 3 4
Process out-of-control Control Chart A producton process s sad to be out-of-control when the qualty characterstc of a product are subject also to varaton due to assgnable causes that s varaton n process performance due to events that are not part of the normal process and represents sudden or persstent abnormal changes to one or more of the process components. Control charts are useful to establsh when a process has endured a meanngful modfcaton; control charts separate the two types of varaton n a product qualty characterstc. 5 6 Control Chart Control Chart All control charts have three basc components: a centerlne that represents the mean value for the ncontrol process two horzontal lnes, called the upper control lmt (UCL) and the lower control lmt (LCL) that defne the lmts of common varaton causes. performance data plotted over tme. Common Cause Specal Cause In the process (normal nose) Outsde the process (extraordnary) 7 8
Control Chart Control Chart Control charts let you know what your processes can do: you can set achevable goals. They represent the voce of the process. Control charts provde the evdence of stablty that justfes predctng process performance. Control charts help to separate sgnal from nose, so that you can recognze a process change when t occurs. Control charts dentfy unusual events. They pnpont fxable problems and potental process mprovements. 9 0 Control Chart - assumptons Control Chart 3 sgma The two mportant assumptons are: The measurement-functon (e.g. the mean), that s used to montor the process parameter, follows a normal dstrbuton. In practce, f your data seem very far from meetng ths assumpton, try to transform them. Measurements are ndependent of each other. Control lmts on a control chart are commonly drawn at 3- sgma from the center lne because 3-sgma lmts are a good balance pont between two types of errors: Type I or alpha errors occur when a pont falls outsde the control lmts even though no specal cause s operatng. Type II or beta errors occur when you mss a specal cause because the chart sn't senstve enough to detect t. 2
Control Chart 3 sgma Instabltes and Out-of-Control Stuatons All process control s vulnerable to these two types of errors. The reason that 3-sgma control lmts balance the rsk of error s that, for normally dstrbuted data, data ponts wll fall nsde 3- sgma lmts 99.73% of the tme when a process s n control. The lmts are chosen so that t s lkely that unusual causes of varaton wll be detected. -4-3 -2-99.73% 0 2 3 4 To test for nstabltes n processes, we examne control charts for nstances and patterns that sgnal non-random behavor. Values fallng outsde the control lmts and unusual patterns wthn the runnng record suggest that assgnable causes exst. "n control" mples that all ponts are between the control lmts and they form a random pattern. 3 4 Instabltes and Out-of-Control Stuatons Instabltes and Out-of-Control Stuatons Test : A sngle pont falls outsde the 3-sgma control lmts. Test 2: At least two of three successve values fall on the same sde of, and more than two sgma unts away from, the center lne. Test 3: At least four out of fve successve values fall on the same sde of, and more than one sgma unt away from, the center lne. Test 4: At least eght successve values fall on the same sde of the center lne. 5 6
Instabltes and Out-of-Control Stuatons Control Chart Is there a Best Control Chart? 7 8 Varable and attrbute data Two broad classes of control charts: varable data, whch s contnuous attrbute data, whch s dscrete Choce of what control chart to use should be based on knowng the rght assumptons! Use the correct formulas for the knd of control chart selected! Varable and attrbute data It s mportant to understand the dstncton between varables data and attrbutes data Because control lmts for attrbutes data are often computed n ways qute dfferent from control lmts for varables data. Unless you have a clear understandng of the dstnctons between the two knds of data, you can easly fall vctm to napproprate control chartng methods. 9 20
Varable and attrbute data Varable and attrbute data Varables data (sometmes called measurement data) are usually measurements of contnuous phenomena. Examples: measurements of length, weght, volume and speed. Software examples: elapsed tme, effort expended, years of experence, memory utlzaton and cost of rework. Attrbutes data occur when nformaton s recorded only about whether an tem conforms or fals to conform to a specfed crteron or set of crtera. Attrbutes data almost always orgnate as counts. Examples: the number of defects found, the number of source statements of a gven type, the number of lnes of comments n a module of n lnes, the number of people wth certan sklls or experence on a project or team, and the percent of projects usng formal code nspectons. 2 22 Control Chart selecton Types of Varable Control Chart X-bar chart R chart s chart Indvdual chart Movng Range chart 23 24
Types of Varable Control Chart Types of Varable Control Chart X-bar chart: based on the average of a subgroup. Subgroups of 2 to 30 samples may be used when computng the control lmts for the X-bar chart when based on the range. R chart: takes nto account the range of a subgroup. Subgroup szes may be as small as 2 or as large as 30. Indvdual chart: dsplays each value. A subgroup sze s used to compute the lmts, wth value of 2 beng most common, although the subgroup sze may be as large as 30. Movng Range chart: takes nto account the movng range of a process. It s used to control varablty of processes whch do not form natural subgroups. S chart: takes nto account the standard devaton of a subgroup. There s no lmt to the subgroup sze. 25 26 Notaton for Varable Control charts Notaton and Values n: sze of the sample (collecton of observatons, sometmes called a subgroup) chosen at a pont n tme R = range of the values n the -th sample R = max( X ) mn( X ) m: number of samples selected R= average range for all m samples x x = average of the observatons n the -th sample (where =, 2,..., m) = grand average or average of the averages (ths value s used as the center lne of the control chart) 27 µ s the true process mean, usually unknown but t can be estmated by averagng a large number (for example 20) of samples mean obtaned when the process s n control σ s the true process standard devaton, usually unknown but t can be estmated from a large sample of data collected whle the process s n control 28
X-bar charts X-bar charts Let { X X m} X =,..., be the set of observatons The process mean s changed durng the observaton perod? 29 dvded nto samples. For each sample { x x } X =,..., x + x2 + L+ xn X = n and the mean average n X + X 2 + L+ X X = m m compute the average 30 X-bar charts R charts X s normally dstrbuted wth mean, µ, and standard devaton, σ =σ n. (Central Lmt Theorem) x / Lower Control Lmt: Center Lne: X X x 2 3σ X A R Is the dsperson of the values observed n the samples due to the presence of exceptonal causes? Upper Control Lmt: X x 2 + 3 σ X + A R s a constant based on the subgroup sze. A 2 3 32
R charts R charts Let { X X m} X =,..., be the set of observaton dvded nto samples. Lower Control Lmt: 3σ D R R R 3 For each sample X the range s Center Lne: R R = max( X ) mn( X ) Upper Control Lmt: + 3σ D R R R 4 and the range average s R + R2 + L+ R R = m m D3 and D4 are constants based on the subgroup sze. 33 34 X-bar charts and R charts for a process out-of-control X-bar charts and R charts X-bar chart s typcally used n conjuncton wth R chart. In fact, snce the sample range s used to construct the X-bar chart, t s essental to examne an R chart frst (to be sure that the process varaton s stable). 35 36
X-bar charts and R charts X-bar charts and R charts: example It s mportant to construct and nterpret an R chart before the X-bar chart. As an example, suppose you msure those 4 software module szes each month for 6 months. Our example collected data looks lke the followng: When the R chart ndcates that process varaton s n control, analyze the X-bar chart otherwse X-bar chart are not meanngful 37 38 X-bar charts and R charts: example s charts Process varablty can be controlled by ether a R chart or a Standard Devaton chart (s chart) dependng on how the populaton standard devaton s estmated. S chart s used to determne whether the standard devaton has changed. 39 40
s charts X-bar charts and s charts Let { X X m} X =,..., dvded nto samples. For each sample and the average s X S 2 S = m be the set of observatons the standard devaton s = n j= ( x j n m S = x ) 2 4 Paremeters for X-bar char Paremeters for s char { Lower Control Lmt: Center Lne: X Upper Control Lmt: { Lower Control Lmt: Center Lne: S Upper Control Lmt: X x 3 3σ X A S X x 3 + 3 σ X + A S S S 3 3σ B S S S 4 + 3σ B S 42 XmR: Indvdual and Movng Range charts XmR chart An ndvdual chart s equvalent to X-bar but reported to sngle observaton, not to samples. Used when the nature of the process s such that t s dffcult or mpossble to group measurements nto subgroups Ths occurs frequently n low volume producton and n stuatons n whch contnuously varyng quanttes wthn the process are process-related varables. The soluton s to artfcally create subgroups from the data and then calculate the range of each subgroup. Ths s done by creatng rollng groups (most often pars) of data through tme and usng the pars to determne the range R. The resultng ranges are called movng ranges. 43 44
XmR chart The movng range s defned as MR = x x whch s the absolute value of the delta between two consecutve data ponts. 45 Paremeters for Indvdual char Paremeters for Movng Range char XmR chart { Lower Control Lmt: Center Lne: X Upper Control Lmt: X x 2 3σ X E MR X x 2 + 3 σ X + E MR [ MR] { Lower Control Lmt: max 0, MR 3σ = max Center Lne: MR Upper Control Lmt: R + 3σ D MR M MR 4 [ 0,D 3 MR] 46 XmR chart: example XmR chart: example The followng table contans a set of sample data, for the KLOC generated each month for one software module. 47 48
Attrbute Control Chart Attrbute Control Chart Attrbute control charts arse when tems are compared wth some standard and then they are classfed as to whether they meet the standard or not. The control chart s used to determne f the rate of non-conformng products s stable and detect when a devaton from stablty has occurred. The argument can be made that a LCL should not exst, snce rates of nonconformng product outsde the LCL s n fact a good thng; we WANT low rates of nonconformng product. However, f we treat these LCL volatons as smply another search for an assgnable cause, we may learn from the drop n nonconformtes rate and possbly permanently mprove the process. 49 50 Types of Attrbute Control Chart Types of Attrbute Control Chart p chart npchart c chart u chart p chart: a chart of the percent defectve n each sample set. The sample sze may vary. np chart: a chart of the number defectve n each sample set. The samples have the same sze. c chart: a chart of the number of defects per unt n each sample set u chart: a chart of the average number of defects n each sample set 5 52
p charts p charts Suppose y s the number of defectve unts n a random sample of sze n. We assume that y s a bnomal random varable wth unknown parameter p. Let k be the number of samples. The fracton: pˆ = y / n plotted on the chart. The mean sample proporton s for each sample s pˆ j j= p = k p ( p) The varance of the statstc p s n k 53 54 Upper Control Lmt: p charts Usng p to estmate the process proporton defectve p the center lne and upper and lower control lmts for the p chart are: p( p) Lower Control Lmt: p 3σ p = p 3 n Center Lne: p p+ 3σ = p+ 3 p p( p) n p charts: example The varable defect no. contans the number of defectve unts, and the varable lot sample sze contans the lot sample sze. 55 56
p charts: example np charts 57 It can be used when the sample are of equale sze, n. In the same way of the p chart: Lower Control Lmt: Center Lne: np Upper Control Lmt: np 3σ = np 3 p k pˆ j j= np = k np( p) n and np( p) np + 3σ p = np + 3 or 0 f UCL<0 n 58 np charts: example np charts: example The varable defect no. contans the number of defectve unts, and the lot sample sze s 500. 59 60
c charts c charts c charts are used to chart count of defects where the area of opportunty for a defect s constant. Es: defects per 000 feet of base materal n a roll of plastc flm produced. The area of opportunty could be a physcal area (defect/000ft), a product such as scratches per montor, an amount of tme such as broken spndles per day or any combnaton of these area of opportuntes. The Posson probablty dstrbuton provdes a good model for the probablty dstrbuton for the number c of defects. If c possesses a Posson probablty dstrbuton wth parameter λ, then E(c) = λ and σ = λ. Observe c over a reasonably large number, k, of equally spaced ponts n tme and use c, the average value c of to estmate λ. c 6 62 c charts The average value of c s: c k c = = k c charts: example Assume that the followng table contans defect data for one of the system desgn document. So the center lne and upper and lower control lmts for the c chart are: Lower Control Lmt: c 3σ c = c 3 c Center Lne: c Upper Control Lmt: c + 3 σ c = c + 3 c 63 64
c charts: example u charts If the area of opportunty s not constant, use the u chart nstead of the c chart. 65 u Let be the number of defect over the -th area of opportunty then the average number of defect s u k = = k u a = a 66 u charts u charts: example The center lne and upper and lower control lmts for the u chart are: Assume that the followng table contans defect data for the system desgn documents of 8 sofware applcatons Lower Control Lmt: u 3 u / a Center Lne: u Upper Control Lmt: u + 3 u / a 67 68
u charts: example Control Chart - conclusons Frst Step: Determne what type of data you are workng wth. Second Step: Determne what type of control chart can be used wth your data set. Thrd Step: Calculate the average and the control lmts. Fourth Step: Detect Instabltes and Out-of-Control Stuatons 69 70