Copyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Dr. Wayne A. Taylor

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1 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart Copyrght 07 by Taylor Enterprses, Inc., All Rghts Reserved. ormalzed Indvduals (I) Control Chart Dr. Wayne A. Taylor Abstract: The only commonly used control chart that cannot be normalzed s the Indvduals (I) chart. A procedure, called a ormalzed Indvduals (I) chart s provded for normalzng data assocated wth an I chart. The I chart works nearly dentcal to the Laney U and P charts for count data. The I chart has certan theoretcal advantages as the estmates of the standard devaton reman unbased n all stuatons where the process s stable. The I chart also has the advantage that t can be used for other applcatons not nvolvng counts..0 Introducton X -, U, P, Laney U and Laney P control charts all allow the charts to be normalzed based on the sample sze or number of opportuntes. The only commonly used control chart that cannot be normalzed s the Indvduals (I) chart. A procedure, called a ormalzed Indvduals (I) chart s provded for normalzng data assocated wth an I chart. Ths chart has been mplemented n Taylor (07c), ncludng an Excel spreadsheet. Donald Wheeler (0) recommends an I chart for handlng count data, whch he refers to as an XmR chart: In contrast to ths use of theoretcal models whch may or may not be correct, the XmR chart provdes us wth emprcal lmts that are actually based upon the varaton present n the data. Ths means that you can use an XmR chart wth count based data anytme you wsh. nce the p-chart, the np-chart, the c-chart, and the u-chart are all specal cases of the chart for ndvdual values, the XmR chart wll mmc these specalty charts when they are approprate and wll dffer from them when they are wrong. Rchard Laney (00) ponts out that the I chart cannot be normalzed to account for dfferences n sample sze or opportuntes, resultng n constant control lmts. He provdes the Laney U and P charts that address ths ssue for count data. The I chart works nearly dentcal to the Laney U and P charts for count data, so s equally effectve at addressng the constant lmts concern. The I chart has certan theoretcal advantages as the estmates of the standard devaton reman unbased n all stuatons where the process s stable. The I chart also has the advantage that t can be used for other applcatons not nvolvng counts. Ths ncludes control charts of lots wth between lot varaton and unequal sample szes. It also ncludes control charts of stablty data for out of trend values wth unequal tme perods. The X and I control charts handle most needs, smplfyng the selecton of a control chart. Revson : eptember 4, 07

2 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart.0 Indvduals (I) Chart for the ormal Dstrbuton The I chart s the bass for the other procedures provded. Assume the values are represented by X, X,, X n, where the X are ndependent normal wth common standard devaton σ. They may have dfferent means. ( µ σ ) X ~, The averages µ are subject to or more shfts. Ths means µµ + n most cases except possbly for a small number of nstances where a mean shft occurs. ETIMATIG THE TADARD DEVIATI Because the average may shft or more tmes, the total standard devaton of the X may overestmate σ. A more robust estmator of σ s based on X X, whch has dstrbuton X X ~ ( µ µ, σ ). In most cases µµ, so: X X ~ ( 0, ) σ As a result, Mean: X X σ has the standard half-normal dstrbuton wth: Medan: Φ ( 0.75) tandard Devaton: When µµ, ths results n the followng unbased estmates of the standard devaton σ: X X R X X d where R X X and the constant d Ths results n the followng estmate of the standard devaton: R Average, 3,, n d ( ) where R Average ( R, R,, R ) 3 n Revson : eptember 4, 07

3 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart s an unbased estmate of σ so long as no shfts occur. If shfts occur some of the are based. A more robust, but slghtly less powerful, estmate of σ s: Φ Medan,,, ( 0.75) ( ) 3 n Medan R, R,, R Medan R, R,, R Φ 0.75 d 0.75 ( 3 n) ( 3 n) Φ CTRL LIMIT FR THE IDIVIDUAL (I) CHART Control lmts are the average plus and mnus 3 standard devatons of the values beng plotted. For an I chart the values X are plotted. The estmated average of the X s: X + X X X n n Usng the estmates of σ from the prevous secton: CL X ± 3 R X± 3 X± R d CL X ± 3 ( 3, Rn) Φ ( 75) Medan R, R, X ± 3 X ± Medan R, R,, R 0. CTRL LIMIT FR THE MVIG CHART For the Movng chart are plotted. E { } σ { } has: D σ σ UCL{ } can be substtuted for n the above equaton. ( ) 3 n Revson : eptember 4,

4 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart Table : Control Lmts for Indvduals (I) Chart Type of Chart Estmator Formulas Indvduals Chart: Plot of values X Uses: X + X X X n n Movng Chart: Plot of X X Average,,, CL X ± 3 ( 0.75) R R 3 n Medan,,, Φ CL X ± 3 R Average R, R,, R 3 n 3 n 3 C L X± R X± R R Medan R, R,, R 3 n R CL X ± 3 Φ 0.7 ( 5) X ± R UCL UCL Movng Range Chart: Plot of R X X R UCL + 3 R R R UCL 3 + R R Φ ( 0.75) ote that the Movng and Movng R charts dffer by a factor of d and provded essentally the same nformaton. However, the estmates and are hander for other applcatons ncludng estmatng process capablty, so the Movng chart s preferred. Revson : eptember 4,

5 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart CTRL LIMIT FR THE MVIG R CHART For the Movng R chart R d are plotted. The control lmts are smlarly scaled by d : UCL{ R} ducl{ } + 3 R R UCL{ R } d UCL{ } + 3 Medan R, R,, R Φ Medan R, R,,R ( ) ( ) 3 n 3 n The lower control lmts of the movng and R charts are negatve. 3.0 ormalzed Indvduals (I) Chart for the ormal Dstrbuton The ormalzed Indvduals (I) chart assumes nstead: ( µ σ ) X ~, The I chart s a specal case of the I chart wth n. The represent the sample sze or number of opportuntes that the X are based on. The relatonshp between the average and standard devaton above s based on the effect of addton. Assume X Y+ Y + Y3 + + Y. Assumng the Y s are ndependent, regardless of the dstrbuton of the Y s, µ µ and X Y σ X σ Y. µµ n most cases, except possbly for a small number of nstances where a mean shft occurs. The normalzed values are then: X ~ µ, The normalzed values σ are plotted on the I chart. ETIMATIG THE TADARD DEVIATI Because the average may shft or more tmes, the total standard devaton of the wll overestmate σ f there are shfts n the average. A more robust estmate of σ s based on: ~ µ µ, σ + Revson : eptember 4,

6 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart In most cases µµ, so: I σ + I σ + ~ 0, has the standard half-normal dstrbuton When µµ, ths results n the followng unbased estmates of the standard devaton σ: I + Ths results n the followng estmate of the standard devaton: Average,,, 3 n s an unbased estmate of σ so long as no shfts occur. If shfts occur some of the are based. A more robust, but slghtly less powerful, estmate of σ s: ( 3, n) ( 0.75) Medan,, Φ CTRL LIMIT FR THE RMALIZED IDIVIDUAL (I) CHART Control lmts are the average plus and mnus 3 standard devatons of the statstc beng plotted. For the I chart the values are plotted. The estmated average of the s: X + X X n n Usng the estmates of σ from the prevous secton the control lmts for the th pont are: ± or CL ± 3 CL 3 CTRL LIMIT FR THE RMALIZED MVIG CHART For the ormalzed Movng chart E { } σ { } are plotted. D σ Revson : eptember 4, has:

7 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart UCL{ } can be substtuted for n the above equaton. The lower control lmts of the normalzed movng chart are negatve. Based on: σ ~ Half-normal(0,) wth dstrbuton functon F( x) Φ( x) σ ~ χ Exact control lmts usng Φ( 3) and ( 3) percentles are: Φ 3 LCL{ } Φ Φ χ ( Φ( 3) ) UCL{ } Φ Φ χ ( Φ ( 3) ) can be substtuted for n the above equaton. 4.0 Comparson to Laney U Chart The Laney U chart has control lmts: CL ± 3 σ Z The resultng estmate of the standard devaton s: n n Ŝ σ Z d n n ( ) ( ) Compare ths to the estmate of the standard devaton for the I chart: n + ( n ) ( n ) n I Revson : eptember 4,

8 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart Table : Control Lmts for ormalzed Indvduals (I) Chart Type of Chart Estmator Formulas ormalzed Indvduals Chart: Plot of Average,,, ± CL 3 3 n X Uses: X + X X n n ormalzed Movng Chart: Plot of Medan,,, Φ ( 0.75) CL ± 3 3 n UCL I + UCL When the are all equal, the two estmates are equvalent. n ( ) ˆ n - The two estmates dffer wth how they handle a changng number of opportuntes. s an unbased estmate, as prevously shown. Ŝ can be based. Ths s due to the fact that the correcton factor d assumes the two tems subtracted are ndependent of each other. They are not ndependent because they nclude the common term. For larger sets of data, the estmate s more precse and the two parts are nearly ndependent. Table 3 shows the performance of the followng two estmators for the case where the standard devaton s. Dfference combnatons of, and 3 are shown. 3 s number of opportuntes for all the other data ponts combned. Revson : eptember 4,

9 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart - - Laney: Ŝ Taylor: + Table 3: Comparson of Laney and Taylor Estmators of the Movng tandard Devaton ˆ 3 Average D Average D Table was generated usng smulatons of 00,000,000 trals each, gvng 4 dgts of precson. nly 3 dgts are shown. Table confrms: s unbased Ŝ, and thus unbased when. Ŝ s based when. However, ths bas s % or less when << 3. Whle s the theoretcally better estmator, for all practcal purposes the two estmators perform the same. Ether can be used. ne nce feature of the Laney U chart s that σz s a useful measure of over dsperson. A value close to suggests a U chart could be used. For an I chart, σ Z could be defned as below for applcatons nvolvng normalzed count data: σ Z It s common practce to use a par of charts to show the average and varaton ( X -, I-MD). mlarly, a Laney U or P chart can be pared wth a movng σ chart. Z Revson : eptember 4,

10 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart 5.0 Examples of Applcatons CMPLAIT DATA The frst set of data s the complant data shown n Table 4. There are 0 values. The sales volume, representng the number of opportuntes, steadly ncreases. The data s over dspersed relatve to the Posson dstrbuton wth about half the ponts fallng outsde the control lmts on a U-chart. Month Table 4: Example Complant Data Complants (X) ales Volume () ormalzed Complants Rates () Fgure shows the Laney U Chart and Fgure shows the I chart of ths data. They are nearly dentcal and result n the same concluson that the complant rate s unchanged. gmaz values are also shown, whch are smlar. The two charts use dfferent estmators of the standard devaton, so there wll be slght dfferences between the charts for each ndvdual set of data. The I chart can be used anytme a Laney U or P chart can be used. A ormalzed Movng chart s shown n Fgure 3. A Movng σ Z chart can be added to a Laney U chart. Fgure 4 shows a movng σ Z chart for the complant data. It looks nearly dentcal to the ormalzed Movng chart n Fgure 3, except for the scale. The Revson : eptember 4,

11 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart I chart also has an opton of usng the medan rather than average to estmate the standard devaton. Ths opton can also be extended to the Laney U chart. 0.0 Laney U' Chart - gmaz Rate Complants UCL (3 D) Center LCL (3 D) Month Fgure : Laney U Chart of Complant Data 0.0 ormalzed Indvduals Chart - gmaz ormalzed Value ormalzed Complants UCL (3 D) Center LCL (3 D) Month Fgure : I Chart of Complant Data ormalzed Movng Chart UCL (3 D) Center Month Fgure 3: ormalzed Movng Chart of Complant Data Revson : eptember 4, 07

12 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart Movng gmaz Chart gmaz gmaz UCL (3 D) Center 4 0 Month Fgure 4: Movng gmaz Chart of Complant Data BETWEE/WITHI LT VARIATI Whle for count data the Laney U and I charts are nterchangeable, there are many other applcatons of the I chart to non-count data where only the I chart s applcable. For example, the X chart assumes there s a sngle source of varaton. An alternatve model better fttng some processes s the between/wthn lot varaton model. Ths model assumes there are two sources of varaton, one for the lot averages and one for ndvdual unts wthn a lot around the lot average. An I chart of the lot averages s recommended n ths case. However, f the sample sze vares from lot-to-lot, an I chart s more approprate. Table 5 shows an example set of data. Fgure 5 shows an I chart of lot averages where some averages are based on 5 samples and other 3. 0 ormalzed Indvduals Chart ormalzed Value ormalzed um UCL (3 D) Center LCL (3 D) Lot Fgure 5: I Chart of Lot Averages wth Unequal ample zes Revson : eptember 4, 07

13 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart Table 5: Example Between Lot Varaton Data Lot Value (um) ormalzed Value (Average) Table 6: Example tablty Data Month Value LIEAR TRED WITH UEQUAL ITERVAL Another applcaton s when there s a lnear trend over tme but values are collected at unequal ntervals. An example s the detecton of out of trend (T) values durng a Revson : eptember 4,

14 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart stablty study where data s collected at tmes 0, 3, 6, 9,, 8, 4, 36 and 48 months. Table 6 shows an example set of data. Fgure 6 shows a lnear regresson of the data. The twelve-month data pont appears to be hgher than expected, however, falls wthn the 95% predcton nterval. Flaggng a pont outsde the 95% predcton nterval s a poor approach to detectng T values. It would result n false sgnal for around 5% of the stablty ponts. Ths translates to close to 50% of stablty studes sgnalng an T pont. Further the T value has wdened the predcton nterval so t does not fall outsde them. Robust regresson estmators could solve the second ssue but not the frst. Ftted Lne Plot Value Months Value Regresson 95% CI 95% PI R-q 99.4% R-q(adj) 99.3% Months Fgure 6: Lnear Regresson of tablty Data Before trendng the data on an I chart, the dfferences between consecutve values must be calculated as shown n Table 7. When ths s done, the normalzed values are the slopes. Table 7: Changes and lopes of Example tablty Data Month Change Length Interval ormalzed Change (lope) Revson : eptember 4,

15 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart Fgure 7 shows the resultng I chart. The medan standard devaton estmator was used to avod any T pont nflatng the estmated varaton and wdenng the control lmts. It shows the -month pont s T. 0. ormalzed I Chart 0. ormalzed Value ormalzed Change UCL (3 D) Center LCL (3 D) Month Fgure 7: I Chart of Changes to Detect ut-f-trend Pont It s clear that the assumpton X ~(, ) µ σ apples to the complant and between lot data. Both are addtve sets of data for whch ths assumpton s assured to be met. It s not as clear t s met for the stablty data. There are numerous sources of varaton ncludng measurement error, varaton n the startng values for each unt and varaton n the slopes for each unt. ome of these are constant and some grow lnearly wth tme. Combnng all these sources of varaton results n somethng somewhere n between, makng the assumpton reasonable. 6.0 Conclusons Based on the methods and comparsons presented, the followng recommendatons are made relatve to control chartng practce: The I chart handles count and pass/fal data where a Laney U or P chart mght be used. It also handles many other stuatons nvolvng non-count data where a Laney U or P chart do not apply. The X and I charts handle most needs, smplfyng the selecton of a chart. These are the only charts needed n most cases. The decson between them s based on whether there are multple values per tme perod or not. The excepton to ths rule s nonnormal data. ne such case s when counts are low and follow the bnomal or Posson dstrbutons. In ths case U and P charts wth adjusted control lmts should be used as descrbed n Taylor (07a, b). If the Laney U or P chart s used, consder accompanyng t wth a movng Z σ chart. It s nconsstent to show X -, and I-Movng charts, but not to do the Revson : eptember 4,

16 Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart same for a Laney U or P chart, as they are all based on a tme ordered seres of estmates of the standard devaton. and movng charts are preferable to R and Movng R charts because the estmates and are useful for other applcatons ncludng estmatng process capablty. 7.0 References Laney, Davd (00), Improved Control Charts for Attrbutes, Qualty Engneerng, 4(4), Wheeler, Donald (0), What About p-charts?, Qualty Dgest, Taylor, Wayne (07a), Adjusted Control Lmts for U Charts, Taylor Enterprses, Inc., Taylor, Wayne (07b), Adjusted Control Lmts for P Charts, Taylor Enterprses, Inc., Taylor, Wayne (07c), tatstcal Procedures for the Medcal Devce, Taylor Enterprses, Inc., Revson : eptember 4,