Capability Analysis (Variable Data)

Transcription

1 Capability Aalysis (Variable Data) Revised: 0/0/07 Summary... Data Iput... 3 Capability Plot... 5 Aalysis Summary... 6 Aalysis Optios... 8 Capability Idices... Prefereces... 6 Tests for Normality... 7 Probability Plot... 8 Compariso of Alterative Distributios... 9 Goodess-of-Fit Tests... No-Normal Capability Idices... 3 Tolerace Chart... 5 Normal Tolerace Limits... 6 Distributio-Free Limits... 7 X or X-bar Chart... 8 MR() or R Chart... 9 Calculatios... 3 Summary The Capability Aalysis procedures for variable data are desiged to compare a sample of measuremets collected from a process to established specificatio limits for that variable. A estimate is derived of the percetage of items likely to be out of spec. Also calculated are a variety of capability idices that compare the observed performace to the specificatio limits. Methods are available for hadlig data from both ormal ad o-ormal distributios. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) -

2 Sample StatFolio: capability.sgp Sample Data The file bottles.sgd cotais the measured burstig stregth of = 00 glass bottles, similar to a dataset cotaied i Motgomery (005). Each row cosists of a sample tested at 0 miute itervals. The table below shows a partial list of the data from that file: stregth time 55 0:0 3 0:0 8 0: : :50 33 :00 40 :0 55 :0 Bottles are required to have a burstig stregth betwee 00 psi ad 300 psi. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) -

3 Data Iput There are two meu selectios that perform a capability aalysis for variables data, oe for idividuals data ad oe for grouped data. Case #: Idividuals The data to be aalyzed cosist of a sigle umeric colum cotaiig observatios. The data are assumed to have bee take oe at a time. Data: umeric colum cotaiig the data to be aalyzed. Date/Time/Labels: optioal labels for each observatio. USL: upper specificatio limit, if ay. Nomial: optioal omial or target value for the variable. If ot supplied, certai capability idices will ot be calculated. LSL: lower specificatio limit, if ay. Select: subset selectio. At least oe of the specificatio limits must be etered. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 3

4 Case #: Grouped Data The data to be aalyzed cosist of oe or more umeric colums. The data are assumed to have bee take i groups, i sequetial order by rows. Data: 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 m 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. Date/Time/Labels or Size: If each set of m rows represets a group, eter the sigle value m. For example, eterig a 5 as i the example above 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. USL: upper specificatio limit, if ay. Nomial: optioal omial or target value for the variable. If ot supplied, certai capability idices will ot be calculated. LSL: lower specificatio limit, if ay. Select: subset selectio. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 4

5 frequecy At least oe of the specificatio limits must be etered. Capability Plot The best picture of the aalysis performed by this procedure is give by the Capability Plot. Process Capability for stregth LSL = 00.0, Nomial = 50.0, USL = Normal Mea=54.64 Std. Dev.= Cp =.64 Pp =.56 Cpk =.49 Ppk =.4 K = stregth The Capability Plot shows:. A frequecy histogram for the sample data.. Tall vertical lies at the specificatio limits ad the omial value. 3. A probability desity fuctio that has bee fit to the data. By default, a ormal distributio is fit, although this may be chaged usig Aalysis Optios. 4. Shorter vertical lies at specified percetiles of the fitted distributio. By default, the percetiles are positioed to cover 99.73% of the fitted distributio, which for a ormal distributio covers a 6-sigma iterval cetered at the sample mea. For a capable process, the percetiles will be iside the specificatio limits. Pae Optios 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 5

6 Number of classes: the umber of itervals ito which the data will be divided. Itervals are adacet to each other ad of equal width. The umber of itervals ito which the data is grouped by default is set by the rule specified o the EDA tab of the Prefereces dialog box o the Edit meu. Lower Limit: lower limit of the first iterval. Upper Limit: upper limit of the last iterval. Hold: maitais the selected umber of itervals ad limits eve if the source data chages. By default, the umber of classes ad the limits are recalculated wheever the data chages. This is ecessary so that all observatios are displayed eve if some of the updated data fall beyod the origial limits. Aalysis Summary The Aalysis Summary summarizes the iput data ad displays several importat results. Process Capability Aalysis - stregth Data variable: stregth Trasformatio: oe Distributio: Normal sample size = 00 mea = std. dev. = Sigma Limits +3.0 sigma = mea = sigma = by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 6

7 Observed Estimated Defects Specificatios Beyod Spec. Z-Score Beyod Spec. Per Millio USL = % % 0.87 Nomial = LSL = % % 0.6 Total % %.03 Data variable the colum or colums cotaiig the data. Trasformatio if a selectio was made i the Data Trasformatio field o the Aalysis Optios dialog box, the maer i which the data was trasformed is summarized. Distributio the assumed distributio for the data ad the estimated or user-specified parameters. By default, a ormal distributio is assumed uless chaged usig Aalysis Optios. Distributios are fit usig maximum likelihood estimatio (MLE) o the combied data from all of the subgroups. Sigma Limits for a ormal distributio, this displays the sample mea plus ad mius a multiple of sigma. Uless chaged usig Aalysis Optios, the rage of values displayed covers 6 times the estimated stadard deviatio, which correspods to 99.73% of a ormal distributio. If a distributio other tha the ormal is selected, the output shows Equivalet Sigma Limits coverig the same percet of the populatio as the ormal limits. For example, the output whe fittig a Laplace distributio is show below: Note that the iterval for the Laplace distributio is 08.5 to 30.5, which is cosiderably wider tha the ormal iterval of.6 to This is because the Laplace distributio has cosiderably loger tails. A correct aalysis depeds o selectig the proper distributio, which is discussed i the sectio o Compariso of Alterative Distributios. Specificatios this table shows the specificatios for the data ad several importat statistics: Observed Beyod Spec. the percetage of the iput data that is beyod the specificatio limits. Z-score for a ormal distributio, the distace from the specificatio limit to the sample mea, divided by the sample stadard deviatio. For o-ormal distributios, a equivalet Z-score is displayed based o the percetage of the fitted distributio that is beyod the specificatio limit. Cosequetly, a Z-score of 3, which correspods to a specificatio limit that is 3 stadard deviatios away from the mea i the case of a ormal distributio, correspods to the idetical 0.4% beyod the specificatio limit regardless of what distributio is selected. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 7

8 Estimated Beyod Spec. the percetage of the fitted distributio that is beyod the specificatio limits. This estimates the percetage of o-coformig product beig produced. Defects Per Millio the Estimated Beyod Spec. expressed i terms of the umber of o-coformig items out of every millio produced. Assumig a ormal distributio for bottle burstig stregths, it is estimated that bottles out of every millio produced will be outside the specificatio limits. Aalysis Optios Distributio the assumed distributio for the data. Select Compariso of Alterative Distributios to compare the goodess-of-fit for various distributios. Iclude the type of capability idices to be calculated ad displayed o the Capability Plot. The choices are: Log-term ad short-term calculate both log-term ad short-term idices. Log-term oly (labeled P) calculate oly log-term idices ad label them with the letter P, as i Ppk. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 8

9 Log-term oly (labeled C) calculate oly log-term idices ad label them with the letter C, as i Cpk. Short-term oly calculate oly short-term idices. These are always labeled with the letter C, as i Cpk. Data Trasformatio the type of trasformatio applied to the data before fittig the selected distributio. Ofte, if a ormal distributio does ot fit the origial data well, it will fit some trasformatio of the data. The choices are: Noe fits the origial data. Logarithm fits the atural logarithms of the data. Power fits the data after raisig each value to the specified power. Box-Cox (optimized) fits the data after raisig each value to a power determied usig the Box-Cox procedure. Note that eve whe a trasformatio is selected for aalysis, most graphs ad tables display results i the origial (ot the trasformed) metric. Lower Threshold for distributios defied by a lower threshold, the value of that lower limit. This icludes all distributios that idicate the umber of parameters after their ame. Note that the "Lower Threshold" correspods to - times the "Added" determied by the Box-Cox trasformatio i several other procedures such as the Process Capability Aalysis (Variables) Statlet. Sigma Limits the sigma spread used to plot the limits o the Capability Plot. This value is usually set to 6. Short-term Sigma the method used to estimate the process stadard deviatio. For grouped data, the choices are: o From average rage estimate sigma from a weighted average of the subgroup rages. o From average s estimate sigma from a weighted average of the subgroup stadard deviatios. o From pooled s estimate sigma from the withi-group mea squared error as i a oeway aalysis of variace. For idividuals data, the choices are: o From average MR estimate sigma from the average movig rage of cosecutive observatios. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 9

10 o From media MR estimate sigma from the media movig rage of cosecutive observatios. o From mea SSD estimate sigma from the squared successive differeces (the squared differeces betwee cosecutive observatios). Apply s bias correctio - if checked, correctios are applied to the estimates of to remove ay bias. The formulas affected iclude the grouped estimates based o the average ad pooled s, ad the idividuals estimate based o the mea SSD. This settig also affects the log-term estimate of sigma. Parameters Push this butto to specify values for the distributio parameters. Normally, the parameters will be estimated from the data. However, this optio allows you to fix the values of those parameters. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 0

11 Capability Idices Capability idices summarize the performace of a process i compariso to the specificatio limits. Depedig o the selectios o the Pae Optios dialog box, STATGRAPHICS will calculate ad display a large umber of idices. Capability Idices for stregth Specificatios USL = Nom = 50.0 LSL = 00.0 Short-Term Log-Term Capability Performace Sigma Cp/Pp CR/PR CM/PM Zusl Zlsl Zmi Cpk/Ppk Cpk/Ppk (upper) Cpk/Ppk (lower) CCpk.6398 Cpm.499 K % beyod spec DPM Sigma Quality Level Based o 6 sigma limits. Short-term sigma estimated from average movig rage. The Sigma Quality Level icludes a.5 sigma drift i the mea. The above table shows two colums of idices, oe labeled short-term ad the other labeled log-term. Short-term idices are calculated by lookig at the variatio withi subgroups (if the data are grouped) or betwee cosecutive observatios (if the data are collected as idividuals). Log-term idices are calculated by lookig at the variatio over the etire samplig period. Some aalysts prefer to call the short-term variability the withi variability ad the log-term variability the total variability. Note: short-term capability is oly estimated for data assumed to come from a ormal distributio. The values i the table are described below. Sigma The key distictio betwee short-term ad log-term idices is the method used to estimate, the stadard deviatio of the process. The log-term sigma is estimated from the sample stadard deviatio of the etire data set. This could potetially iclude variatio caused by drifts i the process durig the period i which the data was collected, so that a large estimate could be a sig of either large iheret variatio or poor process cotrol. The short-term estimate is obtaied from either the movig rage, the mea squared successive differeces, or the subgroup stadard deviatios, depedig o the settigs o the Capability tab of the Prefereces dialog box, accessible from the Edit meu. Sice this estimate is take from observatios close together i time, it is much less iflueced by lack of cotrol over 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) -

12 the process. However, it does ot ecessarily represet the curret performace of the process. Cp/Pp This two-sided capability idex compares the distace betwee the specificatio limits to k-sigma: C P USL LSL () k Note: k is ormally set to 6, although it may be chaged usig Pae Optios. May compaies require that Cp be at least.33. CR/PR The capability ratio, defied by: C R k 00 % USL LSL () This ratio is the reciprocal of Cp. CM/PM The machie capability idex, defied by: C M USL LSL (3) 8 The deomiator of this idex is fixed at 8-sigma. Zusl A Z-score for the upper specificatio limit: Z USL USL (4) Zlsl A Z-score for the lower specificatio limit: Z LSL LSL (5) Zmi The smaller of the calculated Z-scores: MIN LSL USL Z mi Z, Z (6) A Z-score of 4 correspods to a Cpk of.33 if the data come from a ormal distributio. Cpk(upper) a oe-sided capability idex based o the upper specificatio limit: C PK ( upper ) USL (7) ( k / ) 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) -

13 Cpk(lower) a oe-sided capability idex based o the lower specificatio limit: C PK ( lower ) LSL (8) ( k / ) Cpk The smaller of the two oe-sided idices: PK( upper), PK( lower) C mi C C (9) PK Note: k/ is ormally set to 3, although it may be chaged usig Pae Optios. May compaies require that Cpk be at least.33. CCpk A modified versio of Cpk based o the target or omial value T, rather tha the estimated process mea: USL T T LSL CC PK mi, (0) ( k / ) ( k / ) If the omial value is ot specified, the T is set halfway betwee the upper ad lower specificatio limits. Cpm A modified versio of Cp measurig variatio aroud the target or omial value T rather tha the estimated process mea: C PM CP () T Cpm may be substatially less tha Cp if the process is sigificatly off-ceter. K a measure of the distace from the target to the estimated process mea, scaled by the distace betwee the specificatio limits: If T : T K, else: USL T T K () T LSL % beyod spec. the estimated percet of the populatio beyod the specificatio limits, based o the fitted distributio. DPM the estimated defects per millio, based o the estimated % beyod spec. Sigma Quality Level a idex of the level of quality for the process developed as part of the Six Sigma process. If there is oly oe specificatio limit, the SQL equals either Zmi or (Zmi+.5), depedig o the.5 Sigma Shift settig o the Pae Optios dialog box. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 3

14 For the bottle burstig stregth, ote that both Cpk ad Ppk are greater tha.33, which would ormally be cosidered acceptable performace. The value of K = 0.09 idicates that the estimated process mea exceeds the target value T by approximately 9% of the distace from the target value to the upper specificatio limit. STATGRAPHICS also calculates cofidece itervals or bouds for several of the capability idices: 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 4

15 95.0% Cofidece Itervals Idex Lower Limit Upper Limit Cp Pp Cpk Ppk Cpm Sice the idices are statistics calculated from data, there is a margi of error i their ability to estimate the true process capability. For small samples, that margi of error ca be substatial. Based o the above table, we may state with 95% cofidece that the true Cpk for the bottle burstig stregths is somewhere betwee.7 ad.70. Pae Optios Display select the idices to display. The selectio o this dialog box affects both the tabulated idices ad the Capability Plot. To chage the default selectios, use the Capability tab o the Prefereces dialog box accessible through the Edit meu. Cofidece Limits the type of limits to be displayed for the capability idices. Either a two-sided cofidece iterval or a oe-sided lower cofidece boud may be calculated. Based o k, the multiple of sigma used to calculate idices such as Cp ad Cpk. This is usually set to 6. Cofidece Level the level of cofidece for the cofidece limits. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 5

16 Prefereces The Capability tab of the Prefereces dialog box, accessible from the Edit meu, sets various defaults that affect the maer i which the capability aalysis is performed. Display the idices displayed by default i the capability aalysis procedures. Iclude the type of capability idices to be calculated ad how they will be labeled. The choices are: Log-term ad short-term calculate both log-term ad short-term idices. Log-term oly (labeled P) calculate oly log-term idices ad label them with the letter P, as i Ppk. Log-term oly (labeled C) calculate oly log-term idices ad label them with the letter C, as i Cpk. Short-term oly calculate oly short-term idices. These are always labeled with the letter C, as i Cpk. Cofidece Limits the type of limits to be displayed for the capability idices. Short-term sigma grouped data the method used to estimate the process stadard deviatio for grouped data. The choices are: 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 6

17 From average rage estimate sigma from a weighted average of the subgroup rages. From average s estimate sigma from a weighted average of the subgroup stadard deviatios. From pooled s estimate sigma from the withi-group mea squared error as i a oeway aalysis of variace. Short-term sigma idividuals the method used to estimate the process stadard deviatio whe the data are ot grouped. The choices are: From average MR estimate sigma from the average movig rage of cosecutive observatios. From media MR estimate sigma from the media movig rage of cosecutive observatios. From mea SSD estimate sigma from the squared successive differeces (the squared differeces betwee cosecutive observatios). Apply bias correctio for s if checked, correctios are applied to the estimates of to remove ay bias. The formulas affected iclude the grouped estimates based o the average ad pooled s, ad the idividuals estimate based o the mea SSD. This settig also affects the log-term estimate of sigma. Idices for o-ormal data whe costructig capability idices for o-ormal distributios, cotrols whether the idex is based o correspodig Z-scores or the distace betwee percetiles. If Correspodig Z-Scores is selected, the the relatioship betwee the capability idices ad the percetage of the populatio beyod the specificatio limits is the same for all distributios. If Use Distace betwee Percetiles is selected, the the defiitio of the capability idices as ratios of distaces is retaied, but a Z-score of 4 equates to differet percetages of ocoformig items for differet distributios. Sigma limits for plottig defies the distace betwee the tolerace limits show o the capability plots. Sigma limits for idices defies the umber of stadard deviatios used i the deomiator of Cp ad related capability idices. This value is usually set to 6. Tests for Normality The estimated process capability show above is highly depedet o the assumed distributio of the observatios. By default, it is usually assumed that the data follow a ormal distributio. The Tests for Normality pae performs oe or more tests to determie whether or ot this is a reasoable assumptio. For each test, the hypotheses of iterest are: Null hypothesis: data are idepedet samples from a ormal distributio Alt. hypothesis: data are ot idepedet samples from a ormal distributio 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 7

18 stregth Tests for Normality for stregth Test Statistic P-Value Shapiro-Wilks W Skewess Z-score Kurtosis Z-score The tests to be ru are selected usig Pae Optios. Each test is displayed with its associated test statistic ad P-Value. Small P-values (below 0.05 if operatig at the 5% sigificace level) lead to a reectio of the ull hypothesis ad thus to a reectio of the ormal distributio. I the above table, the P-Values are all well above 0.05, so there is ot statistically sigificat oormality i the data. For a detailed descriptio of the tests, see the documetatio for Distributio Fittig (Ucesored Data). Pae Optios Iclude select the tests to iclude i the output. The default tests are defied o the Dist. Fit tab of the Prefereces dialog box accessed from the Edit meu. Probability Plot The Probability Plot is aother method by which oe ca udge whether or ot the curretly selected distributio adequately describes the data. Probability Plot Mea=54.64 Std. Dev.= Normal Distributio 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 8

19 The plot shows the data values, sorted from smallest to largest, plotted agaist equivalet percetiles of the fitted distributio. If the fitted distributio is a reasoable model for the data, they will fall approximately alog a straight lie. For the bottle stregths, the ormal distributio does a reasoable ob. Pae Optios Directio: the axis alog which the percetiles of the fitted distributio will be plotted. Fitted lie: check this box to iclude a diagoal lie o the plot. Compariso of Alterative Distributios If there were reaso to doubt the adequacy of the ormal distributio, a differet distributio could be selected usig Aalysis Optios. To help determie a reasoable alterative, the Compariso of Alterative Distributios pae fits a variety of differet distributios ad sorts them by how well they fit the data. Compariso of Alterative Distributios Distributio Est. Parameters Log Likelihood KS D A^ Laplace Expoetial Power Logistic Loglogistic Normal Gamma Birbaum-Sauders Iverse Gaussia Logormal Weibull Smallest Extreme Value Largest Extreme Value Expoetial Pareto The table shows: Distributio the ame of the distributio fit. You may select additioal distributios usig Pae Optios. Est. Parameters the umber of estimated parameters for that distributio. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 9

20 Log Likelihood the atural logarithm of the likelihood fuctio. Larger values ted to idicate better fittig distributios. KS D, A^, ad other statistics values of various goodess-of-fit statistics, selected usig the Tests butto o the Pae Optios dialog box. Smaller values ted to idicate better fittig distributios. The distributios are sorted from best to worst accordig to oe of the goodess-of-fit colums. That colum is selected usig the Tests butto o the Pae Optios dialog box. The above table shows the distributios sorted accordig to the value of the Aderso-Darlig A statistic. Accordig to that statistic, the Laplace distributio fits best. Pae Optios Alteratives: select the distributios to be fit to the data. Most Commo push this butto to select the most commoly used distributios. Locatio-Scale push this butto to select all distributios that are parameterized by a locatio parameter (such as a mea) ad a scale parameter (such as a stadard deviatio). Threshold push this butto to select all distributios that cotai a lower threshold parameter. All push this butto to select all distributios. Clear push this butto to deselect all distributios. Tests push this butto to display the dialog box used to specify the desired goodess-of-fit statistics: 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 0

21 frequecy Iclude select the goodess-of-fit statistics to be icluded i the table. The list icludes the likelihood fuctio ad various statistics displayed o the Goodess-of-Fit pae. Sort By select oe of the icluded statistics to use to sort the distributios from best to worst. Example Fittig a Laplace Distributio It is iterestig to examie the differeces if a Laplace distributio is fit to the data rather tha a ormal distributio. A Laplace distributio is very peaked i the ceter ad has relatively log tails. 5 0 Process Capability for stregth LSL = 00.0, Nomial = 50.0, USL = Laplace Mea=55.0 Scale= stregth The shorter vertical lies are ow placed at positios that cover the ceter 99.73% of the Laplace distributio, rather tha plus ad mius 3 sigma. This turs out to be a substatially wider rage tha before. Some other iterestig comparisos are show i the followig table: Normal Laplace Distributio Distributio DPM log-term by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) -

22 Ppk Sigma Quality Level Clearly, the estimated performace of the process is heavily depedet o which distributio is used. Goodess-of-Fit Tests If a distributio other tha the ormal is fit to the data, its adequacy as a model for the data ca be tested usig the Goodess-of-Fit Tests. Up to 7 differet tests may be performed. For all tests, the hypotheses of iterest are: Null hypothesis: data are idepedet samples from the specified distributio Alt. hypothesis: data are ot idepedet samples from the specified distributio Goodess-of-Fit Tests for stregth Kolmogorov-Smirov Test Laplace Distributio DPLUS DMINUS 0.05 DN 0.05 P-Value EDF Statistic Value Modified Form P-Value Aderso-Darlig A^ >=0.0 *Idicates that the P-Value has bee compared to tables of critical values specially costructed for fittig the curretly selected distributio. Other P-values are based o geeral tables ad may be very coservative. The tests to be ru are selected usig Pae Optios. Two commo tests are show above. I each case, if the P-Value is large (greater or equal to 0.05 if operatig at the 5% sigificace level), the the distributio is a reasoable model for the data. For a detailed descriptio of the tests, see the documetatio for Distributio Fittig (Ucesored Data). Pae Optios 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) -

23 Iclude: select oe or more tests to perform. For the chi-squared test, select use equiprobable classes to group data ito classes with equal expected frequecies. If this optio is ot checked, classes will be created that match the Capability Plot. Calculate distributio-specific P-Values if checked, the P-Values will be based o tables or formulas specifically developed for the distributio beig tested. Otherwise, the P-Values will be based o a geeral table or formula that applies to all distributios. The geeral approach is more coservative (will ot reect a distributio as easily) but may be preferred whe comparig P-Values amogst differet distributios. No-Normal Capability Idices Rather tha fidig a alterative distributio if the ormal distributio does ot fit the data, oe ca istead calculate capability idices specifically desiged to describe o-ormal data. STATGRAPHICS calculates two types of o-ormal idices:. a idex referred to as Cp(q) based o the quatiles of a Pearso curve selected to match the skewess ad kurtosis of the data.. a idex called Cpc that is based o the average absolute distace of the data values from their target. Noormal Capability Idices for stregth Selected Pearso Curve Percetage Percetile Specificatio Idex Estimate Pp(q) by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 3

24 Ppk(q).3690 Ppk (upper)(q).3690 Ppk (lower)(q).4655 K(q) Idex Estimate Lower 95% C.L. Upper 95% C.L. Cpc NOTE: these idices are based o the distace betwee equivalet 6-sigma limits ad correspod to log-term performace. The idices are calculated as follows: Cp(q) To calculate this idex:. Calculate the skewess ad kurtosis of the data.. If the skewess is less tha -4 or greater tha +4, or if the kurtosis is less tha -.4 or greater tha 36.6, o idices are computed. Otherwise, a Pearso curve is selected to match the sample skewess ad kurtosis. 3. Estimates of the 0.35 percetile, the media, ad the percetile are obtaied from the selected Pearso curve. 4. Capability idices are the calculated i the usual maer, except that the distace betwee the percetiles replaces the usual 6 spread of the ormal distributio. Note the effect o several commo idices: Assumig Usig Cp(q) approach ormality Pp.56.4 Ppk.4.37 K The estimated capability is a little worse tha with the stadard idices, which is i lie with the slightly log tails. However, the effect is ot early as dramatic as if a Laplace distributio had bee selected. Cpc To calculate this idex, let: C PC k USL LSL xi T i (3) where k is commoly set to 6. The iterestig feature of this statistic is that it is based o the average absolute distace of the observatios from the target. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 4

25 stregth Tolerace Chart The Tolerace Chart plots the data i row order with horizotal lies idicatig the target ad specificatio limits. Tolerace Chart Observatio Pae Optios Decimal Places for Limits umber of decimal places for displayig the values to the right of the graph. Color Zoes: check this box to display gree, yellow ad red zoes. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 5

26 Normal Tolerace Limits If the data idicate that the curret specificatio limits caot be met, it is iterestig to ask what limits could be met. Oe approach to establishig ew limits is to calculate Normal Tolerace Limits. Normal tolerace limits give a rage of values for X such that oe may be 00(-)% cofidet that P percet of the populatio from which a data sample comes falls withi that rage. Assumig that the data come from a ormal distributio, a two-sided tolerace limit may be calculated by takig the sample mea plus ad mius a multiple of the stadard deviatio, accordig to x Ks (4) The factor K depeds upo the sample size, the level of cofidece (-), ad the specified percetage P. Normal Tolerace Limits for stregth Normal distributio Sample size = 00 Mea = Sigma = Specificatios USL = Nom = 50.0 LSL = % tolerace iterval for 99.99% of the populatio Xbar +/ sigma Upper: Lower: 07.7 For example, the above table states that oe may be 95% cofidet that 99.99% of all bottles produced will have burstig stregths betwee 07 ad 30. Cosequetly, establishig a spec based o these limits should give a defect rate of o more tha bottle out of every 00,000. It is importat to ote that the above iterval is ot simply the iterval uder the fitted ormal curve cotaiig a area of 99.99%. It is wider tha that sice it allows for samplig variability i both the mea ad stadard deviatio. Pae Optios 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 6

27 Cofidece Level specify the level of cofidece for the tolerace limits, i.e., 00(-)%. Populatio Proportio specify the percetage of the populatio P that the tolerace limits boud. Distributio-Free Limits The k-th smallest ad k-th largest values i a data sample may be used to costruct tolerace limits for the populatio from which the data come without assumig ay specific distributio. The resultig tolerace limits give a rage of values for X such that oe may be 00(-)% cofidet that at least P percet of the populatio from which a data sample comes falls withi that rage. The iterval ca be quite coservative, with the actual percetage beig much larger tha that stated. Distributio-Free Tolerace Limits for stregth Data summary Cout = 00 Maximum = 8.0 Media = 55.0 Miimum = 5.0 Specificatios USL = Nom = 50.0 LSL = % tolerace iterval for % of the populatio Upper: 8.0 Lower: 5.0 (Based o a iterval depth = ) For example, the above table takes the most extreme values of stregth ad states that oe ca be 95% cofidet that at least % of all bottles would have burstig stregths betwee 5 psi ad 8 psi. I this procedure, you ca select Pae Optios to choose either the level of cofidece 00(-) or the populatio percetage P, but ot both. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 7

28 stregth Pae Optios Iput - specify either the level of cofidece for the iterval 00(-) or the populatio percetage P. Iterval Depth specify the value of k used to select the order statistics upo which the limits are based. I creatig the iterval, the procedure uses the k-th smallest ad k-th largest data values. X or X-bar Chart A stadard X chart is created if the data are idividuals, while a X-bar chart is created for grouped data. 300 X Chart Out-of-cotrol poits will be flagged, as will rus rules violatios if your Cotrol Charts prefereces are to iclude rules violatios o cotrol charts. Ay out-of-cotrol sigals should be carefully evaluated sice they ca impact the estimated process capability. Pae Optios Observatio by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 8

29 MR() Decimal Places for Limits umber of decimal places for displayig the values to the right of the graph. Color Zoes: check this box to display gree, yellow ad red zoes. MR() or R Chart A MR() chart is created if the data are idividuals, while a R chart is created for grouped data. 50 Rage Chart Observatio Pae Optios Decimal Places for Limits umber of decimal places for displayig the values to the right of the graph. Color Zoes: check this box to display gree, yellow ad red zoes. 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 9

30 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 30

31 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 3 Calculatios Log-term Sigma If o bias correctio: x x i i (5) If bias correctio: ) ( 4 c x x i i (6) Short-Term Sigma Idividuals From average MR: () d R (7) From media MR: () ~ d 4 R (8) From mea SSD with o bias correctio: x x i i i (9) From mea SSD with bias correctio: ) ( 4 c x x i i i (0)

32 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 3 Short-Term Sigma Grouped Data From average rage: k k f d R f () where i d d f 3 () From average s with o bias correctio: k k s (3) From average s with bias correctio: k k h c s 4 (4) where c c h 4 4 (5) From pooled s with o bias correctio: k k s (6) From pooled s with bias correctio:

33 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 33 m k d c s 4 ) ( (7) where k d (8) Cofidece Iterval for Cp, /, / C C C P P P (9) Cofidece Iterval for Cpk 9 9 / / C Z C C C Z C PK PK PK PK PK (30) Cofidece Iterval for Cpc c s t C C c s t C C PC PC C PC / /, /, / (3) where T x c i i, i i C c T x s (3)

34 Cofidece Iterval for Cpm C PM /, /, C PM C (33) PM v where, T (34) 07 by Statgraphics Techologies, Ic. Capability Aalysis (Variable Data) - 34