Variatio due to Assigable Causes Variatio mostly due to Commo Causes Variatio due to Assigable Causes Outlie Basic Terms MEC-13 Cotrol Charts Types of Cotrol Charts with their purpose Creatig Cotrol Charts Readigs PMBoK, page 238. Rita, pages 305-308. Reid, Ch 6. Krajewski. Ch 6. Lecture Notes. Class Notes. 1 Causes of Variatio Commo Causes (aka Radom Var): Radom causes that caot be idetified Uavoidable Causes like fair wear ad tear of machiery, weather chages etc cause slight differeces i process variables like diameter, weight, service time, temperature, etc. Assigable Causes: Causes that ca be idetified ad elimiated Typical causes are poor employee traiig, wor tool, machie eedig repairs, etc. 16.20 16.10 16.00 15.90 15.80 15.70 15.60 UCL CL LCL 0 5 10 15 20 25 30 2 1
Types of Data or Characteristics Attribute or Discrete Data/Characteristics: Data or characteristics that has a discrete value, caot be measured ad ca oly be couted, like good or rotte apples, Yes or No aswers, umber of people, umber of broke cookies i a packet, etc Variable or Cotiuous Data: Data or characteristics that ca be measured ad has a cotiuum of values, like height, weight, volume etc 3 Defects & Defectives Defect (or No-Coformity): A o-coformig quality characteristic, or a o-coformity, o a item Examples: Number of scratches o a tile, umber of complaits received, umber of bacteria i a petri dish, umber of baracles o the bottom of a boat, umber of pait blemishes o auto body observed for a x umber of cars, umber of imperfectios i bod paper by area ispected ad umber of imperfectios etc Item geerally Useable Charts used: c & u Charts Defective (or No-Coformig Uit): A item havig oe or more defects Examples: a carto with a x umber of broke eggs, a packet with a x umber of broke tiles; umber of samples out of 20 with x,y,z umber of ocoformig cables, umber of packets with x,y,z umber of ocoformig floppy disks whe testig a sample of 200 for 25 trials etc Item geerally Not Useable Charts used: & p Charts 4 2
Cotrol Charts Cotrol Charts: are graphic displays of process data over time ad agaist established cotrol limits, which has a ceterlie that assists i detectig a tred or ru of plotted values toward the either cotrol limit, or chage of dispersio either side of the ceterlie. i short, determie whether or ot a process is stable or has predictable performace 5 Cotrol Charts i Project Maagemet Although used most frequetly to track repetitive activities required for producig maufactured lots, cotrol charts may also be used o projects to moitor various types of output variables, like the cost ad schedule variaces, volume, ad frequecy of scope chages, or other maagemet results to help determie if the project maagemet processes are i cotrol The project maager ad appropriate stakeholders may use the statistically calculated cotrol limits to idetify the poits at which corrective actio will be take to prevet uatural performace The corrective actio typically seeks to maitai the atural stability of a stable ad capable process, withi the UCL ad LCL 6 3
Upper & Lower Specificatio Limits (USL & LSL) Maximum ad Miimum, allowable or acceptable, performace values agreed betwee the buyer ad the seller, ad made part of the agreemet Also kow as the Voice of the Customer There may be pealties associated with exceedig the ULS or fallig below the LSL 7 Upper & Lower Cotrol Limits (UCL & LCL) UCL ad LCL are differet from the USL ad LSL Also kow as the Voice of the Process Determied usig stadard statistical calculatios ad priciples to ultimately establish the atural capability for a stable process For repetitive processes, the cotrol limits are geerally set at + 3 stadard deviatios aroud a process mea that has bee set at 0 stadard deviatio A process is cosidered out of cotrol whe: a data poit exceeds a cotrol limit; seve cosecutive plot poits are above the mea; or seve cosecutive plot poits are below the mea. 8 4
Basic Types of Variatios Ru or Shift: Describes the situatio whe seve or more cosecutive poits occur o oe side of the ceter lie Idicates that a special cause has iflueced the process Poits o the ceter lie do't cout; they either break the strig, or add to it. Tred: Describes the situatio whe 6-7 cosecutive poits occur i the same directio towards the UCL or LCL Idicates that a special cause is actig o the process to cause a tred. Flat lie segmets do't cout, either to break a tred, or to cout towards it 9 Stadard Deviatio (SD or σ) A measure that is used to quatify the amout of Variatio or Dispersio of a set of data values Low SD Data poits ted to be close to the mea (also called the expected value) of the set High SD Data poits spread out over a wider rage of values Example: Data Set: 2,4,4,4,5,5,7,9 Mea = (2+4+4+4+5+5+7+9)/8 = 40/8 = 5 Total Variace = (5-2) 2 +(5-4) 2 +(5-4) 2 +(5-4) 2 +(5-5) 2 +(5-5) 2 +(5-7) 2 +(5-9) 2 = 32 Mea Variace = 32/8 = 4 = σ 2 SD = σ = 4 = 2 10 5
11 Normal Distributio & The Bell Curve 68.2% 95.4% 1σ 2σ 3σ 99.7% 0.3% LCL UCL 68.2% 95.4% 99.7% Source: https://www.piterest.com/pi/68891069276899922/ Normal Distributio must lie withi Specificatio Limits 1/2 Say a deliverable is to maufactured at 100 gm withi + 10 gm ie USL=110 & LSL=90. For the productio process to support the Specificatio Limits (SLs), the etire process must lie withi the SLs. Otherwise may deliverables will ot meet the specificatios ad will be rejected by ow QC ad the customer. I this example, clearly the process, which is producig at σ = 6 gm, is icapable of meetig the SLs, because CLs are outside the SLs LCL LSL Ideal USL UCL σ = 6 gm 80 85 90 95 100 105 110 115 120 Source: https://www.piterest.com/pi/68891069276899922/ 12 6
Normal Distributio must lie withi Specificatio Limits 1/2 This process, which is producig at σ =3 gm, is capable of meetig the Specificatio Limits; the CLs lie withi the SLs LSL LCL Ideal UCL USL σ = 3 gm 80 85 90 95 100 105 110 115 120 Source: https://www.piterest.com/pi/68891069276899922/ 13 Normal Distributio Shift & Dispersio SHIFT x=50 x=70 x=90 Shift: Process shifts either side of Mea. x chages, σ same x +3σ Dispersio: Dispersio of process across the Mea chages. x same, σ chages +3σ +3σ DISPERSION 14 7
Typical Cotrol Chart 16.30 15 16.20 16.10 16.00 15.93 15.90 15.80 15.70 15.60 16.03 15.98 15.88 15.83 TREND. Assigable Cause. Do RCA < LCL. Assigable Cause. Do RCA USL 16.20 UCL 16.08 LCL 15.78 LSL 15.70 RUN. Assigable Cause. Do RCA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 +3σ +2σ +σ 0 -σ -2σ -3σ Which Chart for What? Variable or Cotiuous Data Type of Data Attribute or Discrete Data Shift Shift or Dispersio Defects* Defects or Defectives >10 x-bar or x-mr Charts 1<<10 (i.e. 2-9) = 1 Dispersio Costat Size Varyig Size Defectives** Costat Size Varyig Size x-bar S Chart x-bar R Chart x-mr Chart R-bar Chart c Chart u Chart p Chart p Chart *Defects per Uit ** Uits Defective due to Oe or More Defects 16 8
x-bar-s & x-bar-r Charts 1/4 Used whe x=observatio; S = Stadard Deviatio there is a shift i the cetral tedecy, i.e. the process mea, ad more tha oe observatio are available per time period (sample) x-bar R used whe sample size is 2-9 ad ad x-bar S whe sample size > 10 Example 1: Masory blocks arrive at the project site by truckloads ad are placed ito a pile. A sample of 5 blocks is take from the pile ad tested for breakig stregth (i toe). 6 truckloads have so far arrived. Develop x-bar S ad x-bar R charts for 3-sigma process if the testig observatios are as follows: /Pile # Block 1 Block 2 Block 3 Block 4 Block 5 1 0.96 0.80 1.00 0.92 0.96 2 1.20 1.00 1.10 1.10 1.00 3 0.80 0.80 0.80 0.80 0.80 4 0.90 0.90 0.90 1.00 1.00 5 0.70 1.00 1.10 1.20 0.70 6 0.96 1.00 0.80 1.00 1.00 17 x-bar-s & x-bar-r Charts 2/4 Chart Limits for x-bar S: = x ± zσ x = x ± z σ where: x = mea of sample meas, i.e. the populatio mea z = Quality stadard, eg 3 sigma or more, or degree of assurace/cofidece σ x = stadard deviatio of σ sample meas = σ = stadard deviatio of populatio = sample size Chart Limits for x-bar R: x = x ± A 2 R where: = mea of sample meas, i.e. the populatio mea = Costat depedet o A 2 = Mea of Rages R = sample size 18 9
Breakig Stregth (toe) Breakig Stregth (toe) x-bar-s & x-bar-r Charts 3/4 19 Pile ( #) Block 1 Block 2 Block 3 Block 4 Block 5 1 0.96 0.80 1.00 0.92 0.96 2 1.20 1.00 1.10 1.10 1.00 3 0.80 0.80 0.80 0.80 0.80 4 1.10 1.00 0.90 1.00 1.00 5 0.80 1.00 1.10 0.90 0.70 6 0.96 0.90 0.80 1.00 0.90 Mea Here: = 0.937* x-bar S Chart x z = 3 =5 σ = 0.1184436 σ x = 0.1184436/ 5= 0.053 (usig the σ formula ), or 0.078** Chart Limits = x ± zσ x = 0.937* + 3 x 0. 053(or) 0.937* + 3 x 0.078** = 1.10, 0.78 toe (or) 1.17, 0.70 toe Mea x σ Rage R 0.928 0.077 0.200 1.080 0.084 0.200 0.800 0.000 0.000 1.000 0.071 0.200 0.900 0.158 0.400 0.912 0.076 0.200 0.937* 0.078** 0.200 Here: x-bar R Chart x = 0.937* A 2 = 0.577 (from the tables) R = 0.200ꜙ = 5 Chart Limits = നx + A 2 തR =0.937* + 0.577 x 0.200ꜙ = 1.05, 0.82 toe x-bar-s & x-bar-r Charts 4/4 1.20 1.10 1.10 x-bar S Chart x-bar R Chart 1.00 0.90 0.80 0.70 0.60 1.20 1.10 1.00 0.90 0.80 0.94 0.78 0 1 2 3 4 5 6 7 (Truckload) No 1.05 0.94 0.82 The sample size was 5 for which the right chart was x-bar R. The two charts were draw o the same sample size to brig out the differeces which ca be clearly oticed i UCL ad LCL 0.70 0.60 0 1 2 3 4 5 6 7 (Truckload) No 20 10
Rage (toe) R-Charts 1/2 R = Rage Used whe there is a shift i the dispersio, or variability, of data, as idicated by the samples Rages Example 2: Same as the for the x-bar S ad x-bar R charts, i Example 1. Develop the R Chart for the process /Pile # Block 1 Block 2 Block 3 Block 4 Block 5 1 0.96 0.80 1.00 0.92 0.96 2 1.20 1.00 1.10 1.10 1.00 3 0.80 0.80 0.80 0.80 0.80 4 0.90 0.90 0.90 1.00 1.00 5 0.70 1.00 1.10 1.20 0.70 6 0.96 1.00 0.80 1.00 1.00 21 R-Charts 2/2 22 Chart Limits for R Chart: CL = R UCL = D 4 R LCL = D 3 R Takig the same example of blocks: Pile ( #) Block 1 Block 2 Block 3 Block 4 Block 5 1 0.96 0.80 1.00 0.92 0.96 2 1.20 1.00 1.10 1.10 1.00 3 0.80 0.80 0.80 0.80 0.80 4 0.90 0.90 0.90 1.00 1.00 5 0.70 1.00 1.10 1.20 0.70 6 0.96 1.00 0.80 1.00 1.00 Chart Limits for R Chart: CL = R = 0.200 D 4 = 2.114, D 3 = 0.000 (tables) UCL = 2.114 x 0.200 = 0.423 LCL = 0.000 x 0.200 = 0.000 where: R D 3, D 4 0.60 0.50 0.40 0.30 0.20 0.10 0.00 = mea of samples Rages = Costats depedet o sample size 0.423 0.200 0.000 Rage R 0.200 0.200 0.000 0.010 0.500 0.200 Mea R= 0.200 0 1 2 3 4 5 6 7 No (Truckload No) 11
Rage (toe) Breakig Stregth (toe) x-bar Chart ad the respective R-Chart used i Uiso - 1/2 23 For ay process, the x-bar chart ad the R-chart are always see i combo, to fully appreciate how the process is goig. I this example, the relevat chart was x-bar R (the sample size beig 4) which will ow be compared with the R-chart (Examples 1 & 2 put together) 1.20 Process SHIFTING, 1.10 1.05 alteratig either side of the CL; 1.00 0.94 process out of x-bar R Chart 0.90 cotrol at oe poit Process SHIFTed to UCL without chage i DISP R Chart 0.80 0.70 0.82 0.60 0 1 2 3 4 5 6 7 (Truckload) No 0.60 0.50 0.40 0.30 0.20 0.10 0.00 Process SHIFTed to LCL, with zero DISP 0.401 0.200 0.000 Process SHIFTed ear CL, with reduced DISP 0 1 2 3 4 5 6 7 (Truckload) No No idea how the DISPERSION is doig, i.e. the status of the std dev Process just below CL, with DISP out of cot No idea if the process is SHIFTING o ay side of the mea Truckloads 1,2 & 6 preseted a steady DISP (std dev). DISP Zero/Lower i the 3 rd ad 4 th ; DISP out of cot i 5 th x-bar Chart ad the respective R-Chart used i Uiso - 2/2 3 24 1.2 1.0 SHIFT 6 0.8 0.6 0 1 2 3 4 5 6 7 0.6 DISPERSION 0.4 0.2 0.0 0 1 2 3 4 5 6 7 4 5 x=0.91, R=0.2 x=0.90, R=0.4 1 x=0.93, R=0.2 x=0.80, R=0.0 2 x=1.08, R=0.2 x=1.00, R=0.2 Pile ( #) Mea x Rage R 1 0.93 0.2 2 1.08 0.2 3 0.80 0.0 4 1.00 0.2 5 0.90 0.4 6 0.91 0.2 Mea 0.94 0.2 12
# Weight (gm) x-bar-s Aother Example - 1/2 Example 3: O a productio lie, a sample of 10 items is take after every hour ad the items weighed. The results of last 9 ispectios are tabulated. Develop a x-bar S chart if the plat is producig a 3-sigma quality level. Weight (gm) Item 1 Item 2 Item 3 Item 4 Item 5 Item 6 Item 7 Item 8 Item 9 Item 10 1 80.30 86.90 108.00 80.30 86.90 108.00 80.30 86.90 108.00 86.90 2 99.40 89.50 96.40 99.40 89.50 96.40 99.40 89.50 96.40 89.50 3 95.10 95.90 85.30 95.10 95.90 85.30 95.10 95.90 85.30 95.90 4 99.00 123.90 100.60 99.00 123.90 100.60 99.00 123.90 100.60 123.90 5 97.10 98.60 107.70 97.10 98.60 107.70 97.10 98.60 107.70 98.60 6 97.40 105.50 104.50 97.40 105.50 104.50 97.40 105.50 104.50 105.50 7 97.90 106.00 95.60 97.90 106.00 95.60 97.90 106.00 95.60 106.00 8 81.60 99.90 101.10 81.60 99.90 101.10 81.60 99.90 101.10 99.90 9 90.80 90.10 95.10 90.80 90.10 95.10 90.80 90.10 95.10 90.10 UCL, LCL = x ± zσ x = x ± z σ = 97.6 + 3 x 9.1 = 106.23, 88.97 gm 10 Mea (x) 91.25 94.54 92.48 109.44 100.88 102.77 100.45 94.77 91.81 Mea of Meas ( x ) = 97.60 σ = 9.1 25 x-bar-s Aother Example - 2/2 26 UCL, LCL = x ± zσ x = x ± z σ = 97.6 + 3 x 9.1 = 106.23, 88.97 gm 10 115 110 105 106.23 100 97.60 95 90 88.97 85 80 0 1 2 3 4 5 6 7 8 9 10 Hourly Ispectio No 13
3 Sigma Costats for x-bar & R-Charts X-bar Chart Costats R Chart Costats Size () A2 A3 D3 D4 1 2 1.880 2.659 0.000 3.267 3 1.023 1.954 0.000 2.574 4 0.729 1.628 0.000 2.282 5 0.577 1.427 0.000 2.114 6 0.483 1.287 0.000 2.004 7 0.419 1.182 0.076 1.924 8 0.373 1.099 0.136 1.864 9 0.337 1.032 0.184 1.816 10 0.308 0.975 0.223 1.777 11 0.285 0.927 0.256 1.744 12 0.266 0.886 0.283 1.717 13 0.249 0.850 0.307 1.693 14 0.235 0.817 0.328 1.672 15 0.223 0.789 0.347 1.653 16 0.212 0.763 0.363 1.637 17 0.203 0.739 0.378 1.622 18 0.194 0.718 0.391 1.608 19 0.187 0.698 0.403 1.597 20 0.180 0.680 0.415 1.585 21 0.173 0.663 0.425 1.575 22 0.167 0.647 0.434 1.566 23 0.162 0.633 0.443 1.557 24 0.157 0.619 0.451 1.548 25 0.153 0.606 0.459 1.541 27 Differece betwee p/p ad c/u Charts p/p Charts deal with Defectives (ocoformat items) c/u Charts deal with Defects (ocoformities) eed sample size ad umber of eed umber of defects oly defectives to be compared Example Situatio 1: 2 items fail QC because of 4 faults o oe & 5 o the other p-chart There are 2 defectives (o-coformat items) Situatio 2: A item fails QC because of 4 faults c-chart There are 4 defects (o-coformities) Defects Defectives Costat Size of or Sub-Group u p Variable Size of or Sub-Group c p 28 14
p & p Charts Used whe: p=proportio, p=umber proportio there is a requiremet to moitor ad cotrol the umber, or proportio, of defectives (defective uits) i a sample, as follows: Defective or ot Defective Good or Bad Broke or Not Broke p Chart used whe the sample size is varyig; therefore, proportio of defectives is cosidered p Chart is used whe the sample size is costat; so umber of defectives, rather tha their proportio, is cosidered Examples: a carto with a x umber of broke eggs, a packet with a x umber of broke tiles; umber of samples out of 20 with x,y,z umber of ocoformig cables, umber of packets with x,y,z umber of ocoformig floppy disks whe testig a samples of 200 for 25 trials etc 29 c & u Charts 1/2 Used whe there is a requiremet to moitor ad cotrol the umber of defects (actual or proportioal) i a item or i a measure, as follows: Number of dets (actual or proportioal) per item Number of complaits (actual or proportioal) per uit of time (hour, moth, year etc) Number of tears (actual or proportioal) per uit of area (square foot, square meter) Number of voids (actual or proportioal) per ispectio uit i ijectio moldig or castig processes Number of discrete compoets (actual or proportioal) that must be resoldered per prited circuit board Number of product returs (actual or proportioal) per day Examples: Actual or proportioal umber of:- bacteria i a petri dish, baracles o the bottom of a boat, complaits from the customers, blemishes o auto body observed for 30 samples, imperfectios i bod paper by area ispected ad umber of imperfectios 30 15
c & u Charts 2/2 c Chart used whe the sample size is costat; therefore, actual umber of defects is cosidered u Chart is used whe the sample size is varyig; so proportio of defects, rather tha their actual umber, is cosidered 31 c-chart 1/2 32 Example 4: A teacher wishes to moitor ad cotrol the class attedace. He records the umber of absetees over a academic week ad the result is Mo: 5, Tue:3, Wed:3, Thu:2, Fri:4. The umber of studets o roll o these five days remaied 40. Develop a cotrol chart. Explaatio: This is a case of defects (absetees) i a uit (class). The sample size (class stregth is fixed). Correct Chart is c Day Class Stregth i.e. Size () Number of Absetees i.e. Defaults/ Defects (d) Mo 40 6 Tue 40 4 Wed 40 5 Thu 40 3 Fri 40 5 Cot Limits = തc ± z തc where ഥc = CL & mea umber of defects z = Quality stadard, eg 3 sigma or more, or degree of cofidece 16
Day Size () Defaults/ Defects (d) Prop Defaults (d/) Number of Defects (Absetees ) c-chart 2/2 Prop Defaul ters (d/) M 40 6 0.092 T 40 4 0.080 Not W 40 5 0.050 Reqd Th 40 3 0.069 F 40 5 0.090 200 20 തc 23/5 = 4.6 absetees per day LCL UCL -2.0 10.0-2.0 10.0-2.0 10.0-2.0 10.0-2.0 10.0 12 10 8 6 4 2 11.03 4.60 33 Cot Limits = തc ± z തc 0 0 Mo Tue Wed Thu Fri No (Day) Here, the SD തc is costat LCL &UCL for all observatios are also costat, givig straight lies തc = 23 = 4.6, z=3 5 UCL, LCL = 4.6 + 3 4.6 = 11.03 11, -1.83 0 u-chart 1/2 34 Example 5: A teacher wishes to moitor ad cotrol the class attedace. He records the umber of absetees over a academic week ad the result is Mo: 5, Tue:3, Wed:3, Thu:2, Fri:4. The umber of studets o roll o these five days were 40, 42, 42, 38, 38. Develop a cotrol chart. Explaatio: This is a case of defects (absetees) i a uit (class). The sample size (class stregth is variable). Correct Chart is u Day Class Stregth i.e. Size () Number of Absetees i.e. Defaults/ Defects (d) Mo 40 5 Tue 42 3 Wed 42 3 Thu 38 2 Fri 38 4 Cot Limits = ഥu ± z where ഥu = CL & mea proportioal umber of defects z = Quality stadard, eg 3 sigma or more, or degree of cofidece ഥu 17
Day Size () Defaults/ Defects (d) Prop Defaults (d/) Proportioal Defects (Absetees) u-chart 2/2 35 Prop Defaul ters (d/) LCL UCL M 40 6 0.092-0.006 0.166 T 42 3 0.080-0.005 0.165 W 42 4 0.050-0.005 0.165 Th 38 3 0.069-0.004 0.164 F 38 4 0.090-0.005 0.165 200 20 ഥu 20/200 =0.10 0.30 0.25 0.20 0.15 0.10 0.05 0.250 0.100 0.243 0.243 0.258 0.258 Cot Limits = ഥu ± z ഥu 0.00 0 Mo Tue Wed Thu Fri No (Day) I each case (for each day), the sample size varies. the stadard deviatio ad hece LCL &UCL for each case vary, ad form wiggly lies ഥu, p-chart 1/2 Example 6: A HOD wishes to moitor & cotrol the class attedace i his Dept. He selects the 8 BBA classes ad observes the umber of absetees over the period of time. The results are BBA-1: 20, BBA-2: 22, BBA-3: 28, BBA-4: 22, BBA-5: 27, BBA-6: 20, BBA-7: 18 & BBA-8: 21. Assumig that the stregth of each class is 40, develop a cotrol chart. Explaatio: This is a case of defectives (classes) with defects (absetees). The sample size (class stregth) is fixed Correct Chart is p where Cot Limits = ഥp ± zσ p = ഥp ± z ഥp(1 ഥp) ഥp = CL & mea of sample proportio defectives = total defectives total observatios z = stadard deviatio of sample meas, eg 3 sigma or more, or degree of cofidece σ p = Std Dev of the Average Proportio Defective = = mea sample size ഥp(1 ഥp) 36 18
Actual Number of Defects p-chart 2/2 (Class) Size () Defec tive BBA- Defects (D) 1 40 20 2 40 22 3 40 28 4 40 22 5 40 27 6 40 20 7 40 18 8 40 21 320 178 ഥp 178/320 = 0.556 (Prop Defects D/) LCL UCL Not Reqd 12.82 31.68 12.82 31.68 12.82 31.68 12.82 31.68 12.82 31.68 12.82 31.68 12.82 31.68 12.82 31.68 35 30 25 20 15 10 5 31.68 22.25 12.82 0 0 BBA-1 BBA-2 BBA-3 BBA-4 BBA-5 BBA-6 BBA-7 BBA-8 Defective No (BBA Class)) Cot Limits =ഥp ± zσ p =ഥp ± z ഥp(1 ഥp) 37 I each case (for each class), sample size is the same. the SD ഥp 1 ഥp, ad hece LCL &UCL for each case are also costat, givig straight lies UCL, LCL = 40x0.556+3 40x0.556(1 0.556) = 31.68 32, 12.82 13 p-chart 1/2 Example 7: A HOD wishes to moitor & cotrol the class attedace i his Dept. He selects the 8 BBA classes ad observes the umber of absetees over the period of time. The results are BBA-1: 20, BBA-2: 22, BBA-3: 28, BBA-4: 22, BBA-5: 27, BBA-6: 20, BBA-7: 18 & BBA-8: 21. The class stregth was 40, 42, 36, 44, 41, 35, 44 & 43 respectively. Explaatio: This is a case of defectives (classes) with defects (absetees). The sample size (class stregth) is variable Correct Chart is p. ഥp(1 ഥp) where ഥp = CL & mea of sample proportio defectives = total defectives total observatios z = stadard deviatio of sample meas, eg 3 sigma or more, or degree of cofidece σ p = Std Dev of the Average Proportio Defective = = mea sample size Cot Limits = ഥp ± zσ p = ഥp ± z ഥp(1 ഥp) 38 19
Proportioal Number of Defects p-chart 2/2 Defe Defects ctive (Class) BBA- Size () (D) 1 40 20 2 42 22 3 36 28 4 44 22 5 41 27 6 35 20 7 44 18 8 43 21 325 178 ഥp 178/325 = 0.56 Prop Defects (D/) LCL UCL 0.14-0.10 0.30 0.10-0.10 0.30 0.06-0.11 0.31 0.10-0.10 0.30 0.06-0.12 0.32 0.15-0.10 0.30 0.17-0.11 0.31 0.11-0.11 0.31 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.55 0.78 0.78 0.80 0.77 0.78 0.80 0.77 0.78 0.31 0.32 0.30 0.32 0.31 0.30 0.32 0.32 0.00 0 BBA-1 BBA-2 BBA-3 BBA-4 BBA-5 BBA-6 BBA-7 BBA-8 Defective No (BBA Class) Cot Limits = ഥp ± zσ p = ഥp ± z ഥp(1 ഥp) 39 I each case (for each class), the sample size varies. the SD hece LCL & UCL for each case vary, ad form wiggly lies ഥp(1 ഥp), ad c-chart 1/2 Example 8: 100 workers are trasported to the project site daily i buses. The umber of workers missig the buses i the last 10 days has bee observed to be 9,8,5,7,9,8,9,4,9,12. Develop a 3-Sigma c-chart for the defaultig workers Day Number of Workers i.e. Size () Those who missed the buses i.e. Defaults/ Defects (d) 1 100 9 2 100 8 3 100 5 4 100 7 5 100 9 6 100 8 7 100 9 8 100 4 9 100 9 10 100 12 Cot Limits = തc ± z തc where ഥc = CL & mea umber of defects z = Quality stadard, eg 3 sigma or more, or degree of cofidece 40 20
Number of Defaulters c-chart 2/2 Defaults/ Day Defects 1 9 2 8 3 5 4 7 5 9 6 8 7 9 8 4 9 9 10 12 80 തc Prop Def LCL UCL Not Reqd 80/10 =8.0 per day 16.5-0.5 16.5-0.5 16.5-0.5 16.5-0.5 16.5-0.5 16.5-0.5 16.5-0.5 16.5-0.5 16.5-0.5 16.5-0.5 41 18 16.5 16 14 12 10 8 8.0 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 11 Day Cot Limits = തc ± z തc Here, the SD തc is costat LCL &UCL for all observatios are also costat, givig straight lies തc = 80 = 8.0, z=3 10 UCL, LCL = 8.0 + 3 8.0 = 16.5 17, -0.5 0 u-chart 1/2 Example 9: A costructio compay trasports its workers to project site i buses. The umber of workers o the compay s register over the last 10 days, ad those who missed the buses are as tabulated. Develop a 3-Sigma u- Chart for the defaultig workers Day Number of Workers i.e. Size () Those who missed the buses i.e. Defaults/ Defects (d) 1 98 9 2 100 8 3 100 5 4 102 7 5 100 9 6 99 8 7 99 9 8 100 4 9 100 9 10 102 12 Cot Limits = ഥu ± z where ഥu = CL & mea proportioal umber of defects z = Quality stadard, eg 3 sigma or more, or degree of cofidece ഥu 42 21
Day Size () Defaults/ Defects (d) Prop Defaults (d/) Proportioal Defectives u-chart 2/2 Prop Defaul ters (d/) LCL UCL 1 98 9 0.092-0.006 0.166 2 100 8 0.080-0.005 0.165 3 100 5 0.050-0.005 0.165 4 102 7 0.069-0.004 0.164 5 100 9 0.090-0.005 0.165 6 99 8 0.081-0.005 0.165 7 99 9 0.091-0.005 0.165 8 100 4 0.040-0.005 0.165 9 100 9 0.090-0.005 0.165 10 102 12 0.118-0.004 0.164 1,000 80 ഥu 80/1,000 =0.08 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.080 0.166 0.165 0.165 0.164 0.165 0.165 0.165 0.165 0.165 0.164 0.00 0 1 2 3 4 5 6 7 8 9 10 11 (Box No) Cot Limits = ഥu ± z ഥu 43 I each case (for each day), the sample size varies. the stadard deviatio ad hece LCL &UCL for each case vary, ad form wiggly lies ഥu, p-chart 1/2 Example 10: O a large costructio project, 10 boxes of electrical switches have arrived. Each box has 1,000 switches. Radomly, the procuremet maager picks up 20 switches each from the 10 boxes. He fids 3, 3, 4, 2, 1, 3, 2, 3, 2 & 1 switches defective, i the te boxes. Develop a 3- sigma p-chart for the samplig doe. where Cot Limits = ഥp ± zσ p = ഥp ± z ഥp(1 ഥp) ഥp = CL & mea of sample proportio defectives = total defectives total observatios z = stadard deviatio of sample meas, eg 3 sigma or more, or degree of cofidece σ p = Std Dev of the Average Proportio Defective = = mea sample size ഥp(1 ഥp) 44 22
Actual Number of Defects p-chart 2/2 Def Box No Size () Defects (d) 1 20 3 2 20 3 3 20 4 4 20 2 5 20 1 6 20 3 7 20 2 8 20 3 9 20 2 10 20 1 200 24 ഥp 24/200 = 0.12 (Prop defects (d/) LCL UCL Not Reqd -1.96 6.76-1.96 6.76-1.96 6.76-1.96 6.76-1.96 6.76-1.96 6.76-1.96 6.76-1.96 6.76-1.96 6.76-1.96 6.76 8 7 6 5 4 3 2 1 6.76 2.40 0 0 1 2 3 4 5 6 7 8 9 10 11 Defective (Box) No Cot Limits =ഥp ± zσ p =ഥp ± z ഥp(1 ഥp) 45 I each case (for each box), the sample size is the same. the SD ഥp 1 ഥp, ad hece LCL &UCL for each case are also costat, givig straight lies UCL, LCL = 20x0.12+3 20x0.12(1 0.12) = 6.76 7, -1.96 0 p-chart 1/2 Example 11: O a large costructio project, 10 boxes of electrical switches have arrived. Each box has 1,000 switches. Radomly, the procuremet maager picks up 22, 20, 18, 18, 18, 20, 15, 18, 18 & 20 switches from the 10 boxes. He fids 3, 2, 1, 2, 1, 3, 2, 1, 2 & 3 switches defective, respectively, i the te boxes. Develop a 3-sigma p-chart for the samplig doe. ഥp(1 ഥp) where ഥp = CL & mea of sample proportio defectives = total defectives total observatios z = stadard deviatio of sample meas, eg 3 sigma or more, or degree of cofidece σ p = Std Dev of the Average Proportio Defective = = mea sample size Cot Limits = ഥp ± zσ p = ഥp ± z ഥp(1 ഥp) 46 23
Proportioal Defects p-chart 2/2 Def Box No Defects Size () (d) 1 21 3 2 20 2 3 18 1 4 20 2 5 17 1 6 20 3 7 18 3 8 18 2 9 18 1 10 20 1 190 19 ഥp 19/190=0.100 Prop Defects (d/) LCL UCL 0.14-0.10 0.30 0.10-0.10 0.30 0.06-0.11 0.31 0.10-0.10 0.30 0.06-0.12 0.32 0.15-0.10 0.30 0.17-0.11 0.31 0.11-0.11 0.31 0.06-0.11 0.31 0.05-0.10 0.30 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.10 0.30 0.30 0.31 0.30 0.32 0.30 0.31 0.31 0.31 0.30 0.00 0 1 2 3 4 5 6 7 8 9 10 11 Defective (Box) No Cot Limits = ഥp ± zσ p = ഥp ± z ഥp(1 ഥp) 47 I each case (for each box), the sample size varies. the SD hece LCL & UCL for each case vary, ad form wiggly lies ഥp(1 ഥp), ad 24