X-Bar and S-Squared Charts

Transcription

1 STATGRAPHICS Rev. 7/4/009 X-Bar ad S-Squared Charts Summary The X-Bar ad S-Squared Charts procedure creates cotrol charts for a sigle umeric variable where the data have bee collected i subgroups. It creates both a X-bar chart to moitor the subgroup meas ad a S chart to moitor the subgroup variaces. Out-of-cotrol sigals are highlighted, icludig both poits beyod the cotrol limits ad ay uusual rus i the data. The charts may be costructed i either Iitial Study (Phase ) mode, where the curret data determie the cotrol limits, or i Cotrol to Stadard (Phase ) mode, where the limits come from either a ow stadard or from prior data. The output ad optios available i the X-Bar ad S-Squared Charts procedure are similar to those i the X-Bar ad R Charts procedure. This documet will highlight oly the differeces betwee the two procedures. For a detailed discussio of the commo optios, refer to the documetatio for X-Bar ad R Charts. Sample StatFolio: xbarsschart.sgp Sample Data: The file pistorigs.sgd cotais measuremets made o the diameter of automobile egie pisto rigs, tae from Motgomery (005). The data cosist of m = 5 samples of 5 rigs each. The table below shows a partial list of the data i that file: Sample X X X3 X4 X STATGRAPHICS refers to each row of the file as a subgroup. The subgroup size = by StatPoit Techologies, Ic. X-Bar ad S-Squared Charts -

2 STATGRAPHICS Rev. 7/4/009 Data Iput I eterig data for this procedure, you may eter either:. the origial measuremets.. the subgroup meas ad variaces. Case #: Eterig Origial Measuremets I this case, the data to be aalyzed cosist of the origial measuremets tae o a sigle variable. Observatios: oe or more umeric colums. If more tha oe colum is etered, each row of the file is assumed to represet a subgroup with subgroup size equal to the umber of colums etered. If oly oe colum is etered, the the Date/Time/Labels or Size field is used to form the groups. 009 by StatPoit Techologies, Ic. X-Bar ad S-Squared Charts -

3 STATGRAPHICS Rev. 7/4/009 Date/Time/Labels or Size: If each set of rows represets a group, eter the sigle value. For example, eterig a 5 implies that the data i rows -5 form the first group, rows 6-0 form the secod group, ad so o. If the subgroup sizes are ot equal, eter the ame of a additioal umeric or o-umeric colum cotaiig group idetifiers. The program will sca this colum ad place sequetial rows with idetical codes ito the same group. LSL, Nomial, USL: optioal lower specificatio limit, omial (target) value, ad upper specificatio limit. Select: subset selectio. Case #: Eterig Subgroup Statistics I this case, the statistics for each subgroup have bee computed elsewhere ad etered ito the datasheet, as i the table below: Sample Meas Variaces Sizes by StatPoit Techologies, Ic. X-Bar ad S-Squared Charts - 3

4 STATGRAPHICS Rev. 7/4/009 Subgroup Statistics: the ames of the colum cotaiig the subgroup meas, subgroup variaces, ad subgroup sizes. Dare/Time/Labels: optioal labels for each subgroup. LSL, Nomial, USL: optioal lower specificatio limit, omial (target) value, ad upper specificatio limit. Select: subset selectio. 009 by StatPoit Techologies, Ic. X-Bar ad S-Squared Charts - 4

5 STATGRAPHICS Rev. 7/4/009 X-Bar Chart This chart plots the subgroup meas x. X-bar Chart for X-X UCL = CTR = LCL = X-bar Subgroup I Phase (Iitial Studies) mode, the ceterlie ad cotrol limits are determied from the data. The ceterlie is located at the weighted average of the subgroup meas: x m m x () The cotrol limits are placed above ad below the ceterlie at: ˆ x () where is the sigma multiple specified o the Cotrol Charts tab of the Prefereces dialog box ( = 3 except i rare cases), ˆ is the estimate of the process sigma, ad is the subgroup size. If the subgroup sizes are ot equal, the depedig o Aalysis Optios, is replaced by either: (), the average subgroup size. I this case, the cotrol limits are the same for all subgroups. (), the idividual subgroup sizes. I this case, the cotrol limits are step fuctios. The method for estimatig the process sigma also depeds o the settigs o the Cotrol Charts tab of the Prefereces dialog box, as discussed i the Aalysis Summary sectio below. Ay poits beyod the cotrol limits will be flagged usig a special poit symbol. Ay poit excluded from the aalysis, usually by clicig o a poit o the chart ad pressig the Exclude/Iclude butto, will be idicated by a X. If so idicated o the Pae Optios dialog 009 by StatPoit Techologies, Ic. X-Bar ad S-Squared Charts - 5

6 STATGRAPHICS Rev. 7/4/009 box, uusual sequeces of poits may also be flagged. I the curret chart, o uusual poits or out-of-cotrol sigals are idicated. Pae Optios Outer Warig Limits: chec this box to add warig limits at the specified multiple of sigma, usually at sigma. Ier Warig Limits: chec this box to add warig limits at the specified multiple of sigma, usually at sigma. Movig Average: chec this box to add a movig average smoother to the chart. I additio to the subgroup meas, the average of the most recet q poits will also be displayed, where q is the order of the movig average. The default value q = 9 sice the -sigma ier warig limits for the origial subgroup meas are equivalet to the 3-sigma cotrol limits for that order movig average. Expoetially Weighted Movig Average: chec this box to add a EWMA smoother to the chart. I additio to the subgroup meas, a expoetially weighted movig average of the subgroup meas will also be displayed, where is the smoothig parameter of the EWMA. The default value = 0. sice the -sigma ier warig limits for the origial subgroup meas are equivalet to the 3-sigma cotrol limits for that EWMA. Decimal Places for Limits: the umber of decimal places used to display the cotrol limits. Mar Rus Rules Violatios: flags with a special poit symbol ay uusual sequeces or rus. The rus rules applied by default are specified o the Rus Tests tab of the Prefereces dialog box. 009 by StatPoit Techologies, Ic. X-Bar ad S-Squared Charts - 6

7 STATGRAPHICS Rev. 7/4/009 Color Zoes: chec this box to display gree, yellow ad red zoes. Display Specificatio Limits: whether to add horizotal lies to the chart displayig the locatio of the specificatio limits (if ay). S-Squared Chart This chart plots the subgroup variaces s. (X 0.000) 5 S-squared S-squared Chart for X-X Subgroup UCL = CTR = LCL = I Phase (Iitial Studies) mode, the ceterlie ad cotrol limits are determied from the data. The ceterlie is located at: CL ˆ (3) The cotrol limits are placed at locatios determied from critical values of a chi-squared distributio: UCL ˆ /, (4) LCL ˆ /, (5) where is the false alarm probability of the chart,ˆ is the estimate of the process sigma, ad is the subgroup size. is set to give the same false alarm probability as the sigma multiple specified o the Cotrol Charts tab of the Prefereces dialog box ( = except i rare cases). If the subgroup sizes are ot equal, Aalysis Optios specifies whether to use the average subgroup size or the idividual subgroup sizes. The S chart for the sample data shows o uusual sigals. 009 by StatPoit Techologies, Ic. X-Bar ad S-Squared Charts - 7

8 Pae Optios The same optios exist as for the X-Bar chart. STATGRAPHICS Rev. 7/4/009 Subgroup Reports This pae tabulates the values plotted o the cotrol charts: Subgroup Reports All Subgroups X = Excluded * = Beyod Limits Subgroup Size X-bar S-squared Out-of-cotrol poits are idicated by a asteris. Poits excluded from the calculatios are idicated by a X. Pae Optios 009 by StatPoit Techologies, Ic. X-Bar ad S-Squared Charts - 8

9 Display: specify the subgroups to display i the report. STATGRAPHICS Rev. 7/4/009 Aalysis Summary The Aalysis Summary summarizes the data ad the cotrol charts. X-bar ad S-squared Charts - X-X5 Number of subgroups = 5 Subgroup size = subgroups excluded Distributio: Normal Trasformatio: oe X-bar Chart Period #-5 UCL: +3.0 sigma Ceterlie LCL: -3.0 sigma beyod limits S-squared Chart Period #-5 UCL: +3.0 sigma Ceterlie LCL: -3.0 sigma beyod limits Estimates Period #-5 Process mea Process sigma Pooled s Sigma estimated from pooled s with bias correctio Icluded i the table are: Subgroup Iformatio: the umber of subgroups m ad the average subgroup size m m (6) If ay subgroups have bee excluded from the calculatios, that umber is also displayed. Distributio: the assumed distributio for the data. By default, the data are assumed to follow a ormal distributio. However, oe of 6 other distributios may be selected usig Aalysis Optios. Trasformatio: ay trasformatio that has bee applied to the data. Usig Aalysis Optios, you may elect to trasform the data usig either a commo trasformatio such as a square root or optimize the trasformatio usig the Box-Cox method. 009 by StatPoit Techologies, Ic. X-Bar ad S-Squared Charts - 9

10 STATGRAPHICS Rev. 7/4/009 X-bar Chart: a summary of the ceterlie ad cotrol limits for the x-bar chart, which plots the subgroup meas x. Usig Aalysis Optios, separate limits may be calculated for differet periods (sets of subgroups). S-Squared Chart: a summary of the ceterlie ad cotrol limits for the S chart, which plots the subgroup stadard variaces s. Estimates: estimates of the process mea ad the process stadard deviatio. The process mea is estimated from the weighted average of the subgroup meas: ˆ x (7) The process sigma may be estimated i ay of 4 ways, depedig upo the settigs o the Cotrol Charts tab of the Prefereces dialog box, accessible through the Edit meu. There are 4 optios: () From average s with o bias correctio: the process sigma is estimated from a weighted average of the subgroup stadard deviatios. () From average s with bias correctio: the process sigma is estimated from a weighted average of the subgroup stadard deviatios ad the multiplied by a factor that maes the result a ubiased estimate of. (3) From pooled s with o bias correctio: the process sigma is estimated from the pooled withi group variace, as i a oeway ANOVA. I this case, s is a ubiased estimate of but s is a biased estimator for. (4) From pooled s with bias correctio: the process sigma is estimated from the pooled withi group variace ad the multiplied by a factor that maes the result a ubiased estimate of. Pooled s: the pooled withi subgroup stadard deviatio: s p s (8) 009 by StatPoit Techologies, Ic. X-Bar ad S-Squared Charts - 0

11 STATGRAPHICS Rev. 7/4/009 Aalysis Optios Type of Study: determies how the cotrol limits are set. For a Iitial Study (Phase ) chart, the limits are estimated from the curret data. For a Cotrol to Stadard (Phase ) chart, the cotrol limits are determied from the iformatio i the Cotrol to Stadard sectio of the dialog box. Normalize: if selected, all statistics will be ormalized by calculatig Z-scores ad the Z- scores plotted o the charts. The ceterlie o such a chart is always located at 0, ad the cotrol limits are always located at. Avg. Subgroup Size: if checed, the cotrol limits will be horizotal lies based o the average subgroup size. If ot checed, the idividual subgroup sizes will be used, resultig i step fuctio cotrol limits if the subgroups sizes are ot all equal. Use Zoe Format: if checed, the statistics will be plotted usig a zoe chart rather tha the usual format. See the example below. Recalculate at: the cotrol limits of the chart may be recalculated at up to 4 locatios alog the X axis by specifyig the subgroup umbers at which ew estimates are to be iitiated. Separate estimates of the process mea ad sigma will be obtaied usig the data i the differet sectios. I such cases, the cotrol limits will be adusted at the start of each ew period. X-bar Cotrol Limits: specify the multiple to use i determiig the upper ad lower cotrol limits o the X-bar chart. To suppress a limit completely, eter 0. S-squared Cotrol Limits: specify the multiple to use i determiig the upper ad lower cotrol limits o the S chart. To suppress a limit completely, eter 0. The value will be coverted to a equivalet ris for creatio of the cotrol limits. 009 by StatPoit Techologies, Ic. X-Bar ad S-Squared Charts -

12 STATGRAPHICS Rev. 7/4/009 Cotrol to Stadard: to perform a Phase aalysis, select Cotrol to Stadard for the Type of Study ad the eter either of the followig: Specify Parameters: specify the established stadard process mea ad sigma (or other parameters if ot assumig a ormal distributio). These values will the be used to positio the ceterlies ad cotrol limits o the charts. Specify Cotrol Limits: specify the locatio of the ceterlies ad cotrol limits exactly where you wish them to be placed. Exclude butto: Use this butto to exclude specific subgroups from the calculatios. Trasform Butto: Use this butto to specify a trasformatio or o-ormal distributio. Example: Defiig Multiple Sectios The time axis of the cotrol chart ca be divided ito sectios by requestig that the cotrol limits be recalculated at differet poits i time. For example, the Recalculate at field o the followig Aalysis Optios dialog box ass that ew sectios begi at subgroups 6 ad 5: The cotrol charts the display differet limits for each sectio: 009 by StatPoit Techologies, Ic. X-Bar ad S-Squared Charts -

13 STATGRAPHICS Rev. 7/4/009 X-bar Chart for X-X5 X-bar UCL = CTR = LCL = Subgroup The labels alog the right axis show the most recet limits. Save Results The followig results ca be saved to the datasheet:. Meas the subgroup meas.. Variaces the subgroup variaces. 3. Sizes the subgroup sizes. 4. Labels the subgroup labels. 5. Process Mea the estimated process mea. 6. Process Sigma the estimated process stadard deviatio. 009 by StatPoit Techologies, Ic. X-Bar ad S-Squared Charts - 3

14 STATGRAPHICS Rev. 7/4/ by StatPoit Techologies, Ic. X-Bar ad S-Squared Charts - 4 Calculatios Estimate of Process Sigma () From average s with bias o correctio: s ˆ (9) () From average s with bias correctio: h c s f 4 ˆ (0) where c c h 4 4 () (3) From pooled s with o bias correctio: s ˆ () (4) From pooled s with bias correctio: m s d c 4 ) ( ˆ (3) where (4) d