Comparison of statistical methods for outlier detection in proficiency testing data on analysis of lead in aqueous solution

Transcription

1 Amercan Journal of Theoretcal and Appled Statstcs 2013; 2(6): Publshed onlne December 10, 2013 ( do: /j.ajtas Comparson of statstcal methods for outler detecton n profcency testng data on analyss of lead n aqueous soluton Sushree SwarupaTrpathy *, Rajv Kumar Saxena, Prabhat Kumar Gupta Analytcal Chemstry, Sophstcated Analytcal Support Dvson, CSIR- Natonal Physcal Laboratory, Dr K. S. Krshnan Marg, New Delh , INDIA Emal address: trpathyss@mal.nplnda.org (S. S. Trpathy) To cte ths artcle: Sushree SwarupaTrpathy, Rajv Kumar Saxena, Prabhat Kumar Gupta. Comparson of Statstcal Methods for Outler Detecton n Profcency Testng Data on Analyss of Lead n Aqueous Soluton. Amercan Journal of Theoretcal and Appled Statstcs. Vol. 2, No. 6, 2013, pp do: /j.ajtas Abstract: Profcency testng s the regular testng of the performance of ndvdual laboratores by an external agency. Stable and effectvely homogeneous elemental solutons of dfferent concentratons rangng were prepared at NPL and were certfed by round robn test. These certfed reference materals (CRMs) along wth reports contanng all nformaton, where the laboratores were dentfed only by a reference number were beng dstrbuted to sxty seven partcpant laboratores. In ths paper the data of lead (Pb) elemental soluton of 1-5 mg/l s only presented for outler detecton. The results receved from the sxty four numbers of laboratores for Pb elemental soluton statstcally evaluated wth dfferent approaches vz. Cochran s test, Grubb s test, Hampel s test, classcal z-score, medan and NIQR method, robust statstcal analyss : Algorthm A (ISO 13528) and NATA method. The robust estmate of average and uncertanty values derved from ISO method are very close to the reference value for the 1 and 2 mg/l of Pb elemental solutons. The performance of the laboratores was expressed by z-score and the laboratores havng z < 2 are classfed as satsfactory, 2 < z < 3 are classfed as questonable and z > 3 are consdered as unsatsfactory. Among all the methods, the hghest number of outler.e. about 30 % obtaned by NATA statstcal analyss. As NATA method consders varance n both wthn and between laboratory results so t seems to be the most sutable method for outler detecton for the present data set evaluated n ths study. Keyword: Profcency Testng, Statstcal Analyss, Z-Score, Outler 1. Introducton Growth n world trade has shown the ways to remove the techncal barrer so that the goods can cross the fronters wthout delay and added cost. The mportng economy needs to ensure the qualty of the product beng mported as per ther requrement. It should have full confdence n the techncal competence of the exporter or ther testng laboratory. The qualty assurance and qualty control of the product could be acheved by mplementaton of ISO/IEC [1] and use of certfed reference materals n calbraton of analytcal equpment and valdaton of test methods s mandatory. Thrd party accredtaton and partcpaton n profcency testng (PT) program enhance global acceptance of test reports. Accordng to ISO/IEC 17025, partcpaton n profcency testng programs or other nter laboratory comparsons s mandatory part of a laboratory s qualty control procedures. PT program s an ndependent and unbased technque to assess the capablty of the laboratory to perform specfc test or measurement wthn desred uncertanty lmts [2-6]. Generally PT organzer gves a wde publcty about the detals of the program ncludng characterstcs of the test to be undertaken to attract the attenton of testng laboratory. PT organzer prepared or got prepares the materal under ther control or some tme certfed reference materals could also be used n PT program. After ensurng the stablty and homogenety of the test materal, t was sent to each partcpatng laboratores wth a code number along wth protocol of

2 234 Sushree SwarupaTrpathy et al.: Comparson of Statstcal Methods for Outler Detecton n Profcency Testng Data on Analyss of Lead n Aqueous Soluton analyss, nstructons and format of the test report. Each partcpatng laboratory carred out analyss of the characterstcs of the test materal and reports the results on gven format wthn fxed deadlne. The data of the partcpants has to be compled and statstcally computed to calculate the values to assess the qualty of the performance of the laboratores and to provde a score to the partcpant laboratores. Certfed reference materals of Pb, Cr, Zn mono elemental solutons of dfferent concentraton have been dstrbuted to 67 NABL accredtated laboratores for measurement of metal n water under the PT program carred out by the NPL-I. In ths paper detaled of statstcal analyss and descrpton of the PT programme has been presented for Pb elemental soluton of dfferent concentratons rangng from 1-5 mg/l. Results obtaned were evaluated by dfferent statstcal approaches vz. Cochran s test, Grubb s test, Hampel s test, classcal z- score, Medan and NIQR method, robust statstcal method and NATA method to establsh the best method to present the performance of the partcpant along wth the outler detecton. When the number of data s more and there s a great dfference between the laboratores results, n that case the detecton of outler s a tough job. Thus, n the present paper emphass has been gven to establsh a sutable method to fnd out the outler n such a way that the extreme values provded by some PT partcpants would not nfluence the results of the partcpants close to the reference value. Z-score s the performance score of one laboratory wth the other laboratores. The qualty of the measurement ncreases wth the decrease n the z-score value. In ths study the laboratores havng z-score value between ± 2 to ± 3 are consdered as questonable are advsed to take close look at ther results to rectfy the fault. The cut off value of z-score evaluated by any method s selected as outsde the range of ± 3 whch ndcates there s a problem wth the measurement. 2. Statstcal Technques The statstcal evaluaton of the data obtaned from the partcpants has been llustrated n ths paper usng both numercal technque such as Cochran s, Grubb s, Hampel s statstc and graphcal technques such as classcal z-score, Medan & NIQR, ISO 13528: Algorthm A, NATA statstcal method [4-6] Cochran s Test Accordng to ISO , Cochran s test s recommended for the detecton of outlers n a gven set of ntra laboratory varablty test. It s a one sded outler test as the crteron of the test examnes only the greatest standard devaton and allows to elmnate the problematc result wth the wthn laboratores reproducblty / repeatablty. Cochran s statstc C s calculated usng the followng formula C = p SD j= 1 2 max 2 SD j where, SDmax s the maxmum standard devaton among the nvestgated laboratores SDj s the standard devaton of data from the laboratory p s the number of partcpated laboratores The calculated C value can be compared wth the crtcal value for a gven n value.e. the number of results gven by each laboratory Grubb s Test Grubb s test detects the outler n the nter laboratory varablty test n a gven set of result dfferng sgnfcantly from others. In the Grubb s test the hghest and smallest values n the set of results ( X j where j=1,2,...p ) are to be tested for outler. The value of parameter G p can be calculated for the largest value n the set s gven as follows G p X p X SD (1) = (2) where, Gp s the Grubb s statstcs for hghest value n the set of result X p s the hghest value X s the mean value SD s the standard devaton G p value can compare wth the crtcal value for the p number of laboratores X and SD can be calculated as X 1 p X j p j = 1 = (3) p 1 SD = ( X ) 2 j X (4) 1 p j = 1 Grubb s statstcs for the lowest value n the set of result can be calculated as follows G X l s the lowest value X X l l = (5) SD

3 Amercan Journal of Theoretcal and Appled Statstcs 2013; 2(6): Hampel s Test The Hampel s test [6] can be performed as r ( x M ) = (6) e descrbed n ISO and s not affected by the results far from the reference value. The flow chart of the algorthm s shown n schematc dagram 1. There are p data from partcpants: x 1, x 2,, x,, x p Me s the medan of the data set r s the devaton from the medan value x s the th data of the data set where = 1 to p, number of all results The outler value from the data set can be obtaned by comparng the r value from the condton gven below r 5M 4. er (7) M e r s the medan for devaton 2.4. Z-Score Analyss The z-score s the score gven to the partcpant as per ther performance. The classcal z-score can be calculated as follows z = ( X X ) SD (8) lab mean / where, X lab s the result of the ndvdual laboratory X mean s the mean value of the analyte obtaned by the partcpants result SD s the standard devaton of the data To smplfy the nterpretaton of the z-scores, the followng agreements were made as z < 2 mply the result s satsfactory 2 < z < 3 mply the result s questonable z > 3 mply the result s unsatsfactory The robust z-score obtaned by medan & normalzed nter quartle range (NIQR) method can be calculated as follows z / = ( X lab X medan) SDNIQR (9) where, Xlab s the result of the ndvdual laboratory Xmedan s the medan value of the analyte SDNIQR s the normalzed nter quartle range of the data set The z-score s nterpreted n the smlar way as descrbed above 2.5. ISO 13528: Robust Analyss Algorthm A The robust statstcs can be calculated accordng to ISO 13528: Robust analyss Algorthm A. The robust average (x*) and the standard devaton value (s*) of the results of the partcpants could acheve by an teratve calculaton as Schematc dagram 1. Flow chart of robust analyss algorthm A. The assgned value of the analyte s the robust average value and the standard uncertanty (ux) of the assgned value of the analyte can be estmated usng the robust standard devaton as follows * ux = 1.25 s / p (10) Usng the assgned value and the robust standard devaton the outler detecton may be assessed accordng to the z-score value NATA Method The process of the performance evaluaton of the partcpatng laboratores s llustrated by the z-score values of the results obtaned n the PT program n accordance to the Natonal Assocaton of Testng Authortes, Australa (NATA) gudelnes [4]. The values of standardzed sum ( S ) and standardzed dfference ( D ) between two results of laboratory have been calculated usng equatons (11) and (12). ( A B )/ 2 S + = (11) D = ( A B )/ 2 f medan ( A) > medan ( B ) (12)

4 236 Sushree SwarupaTrpathy et al.: Comparson of Statstcal Methods for Outler Detecton n Profcency Testng Data on Analyss of Lead n Aqueous Soluton Whereas A and B are the two measurement values of the laboratory Usng the values of S and D the values of both the Z- scores.e wthn the laboratory (Zw) and Z-score between the laboratores (Zb) can be calculated usng equatons (13) and (14). S Zb = IQR S D Zw = IQR D Medan ( S ) ( ) Medan ( D ) ( ) (13) (14) The nterquartle range (IQR) s the dfference between the lower and upper quartles of data. The lower quartle (Q1) s the value below, whch a quarter of results lad and the upper quartle (Q3) s the value above, whch a quarter of results lad. The quartles are calculated analogously to the medan and IQR = Q3 - Q1. The term Normalzed IQR s comparable to a standard devaton and equals to IQR x The factor comes from the standard normal dstrbuton, whch has a mean of zero and a standard devaton equal to one. The wdth of the nterquartle range of such dstrbuton s and 1/ = z-score values Zb and Zw of laboratory are the robust z-score of ts S and D values. Fnally each partcpatng laboratory wll be assgned wth two z-score values on the bass of ther results. Generally laboratores havng the value of any z- score value outsde the range ± 3 are consdered to be the outler laboratores by NATA. 3. Sample Detals of PT The solutons provded to the laboratores are the certfed reference materals [named as Bharatya Nrdeshak Dravyas (BNDs)] whch had been certfed wth a reference value by round robn testng. Instruments lke atomc absorpton spectrometer (AAS), Inductvely Coupled Plasma - Atomc emsson spectrometer (ICP-AES) and Inductvely Coupled Plasma -Mass spectrometer (ICP- MS) had been used for the analyss of these BNDs. Three numbers of elemental solutons (vz. Pb, Cr and Zn) of dfferent concentraton had been sent to 67 nos. of laboratores. In ths paper only one metal.e. Pb elemental solutons of 1-5 mg/l concentraton s consdered and evaluated the results of the partcpants wth dfferent statstcal methods. Pb elemental solutons of 1, 2 and 5.01 mg/l were sent to 11, 34 and 22 numbers of laboratores respectvely. Out of thrty four numbers of laboratores n case of 2 mg/l Pb soluton, three numbers of laboratores dd not acknowledge the recept of the soluton whle other partcpants have been provded the results n trplcates. Detals of the sample sent to varous partcpatng laboratores and results receved from them are gven n table 1. The partcpant laboratores have used dfferent analytcal technques such as AAS-Flame, AAS-Graphte Furnace, ICP-AES, ICP-MS, APHA-3111B method, APHA-3500D method and IS-3025 method to determne the concentraton of Pb elemental solutons. Table 1. Detals of the sample recpent laboratores and ther results Sl. No. Metal Soluton Sent (Marked as C) Lab Code Number Mean Value Standard Devaton CRM: BND [Pb] = 1 mg/l CRM: BND [Pb] = 2 mg/l CRM: BND [Pb] = 5.01 mg/l

5 Amercan Journal of Theoretcal and Appled Statstcs 2013; 2(6): Sl. No. Metal Soluton Sent (Marked as C) Lab Code Number Mean Value Standard Devaton Result and Dscusson of Statstcal Analyss of PB Elemental Soluton 4.1. Numercal Statstcal Analyss As n ths PT program the CRMs of elemental solutons have been dstrbuted to the partcpatng laboratores, so the certfed value of the CRMs are used as the reference value. To see the scatterng of the data, the results of the partcpants wth standard uncertanty are presented n Fgure 1(a)-(c) wth the CRM reference value. The central lne represents the reference value and the dotted lnes above and below the central lne s the dsperson attrbuted to the reference value (gven n Table 1) of the Pb elemental soluton. The bar lne shows the ndvdual result of the partcpant wth standard uncertanty. It can be seen from the fgures that laboratory code 6; 18, 33, 44 and 46, 51 n case of 1 mg/l, 2 mg/l and 5.01 mg/l of Pb solutons respectvely are far from the reference value whch gve the mpresson as outlers. In case of 1 mg/l result, the bar lne for lab code 5 & 6 are sgnfcantly hgh mples much varaton n the replcates. In ths paper emphass has been gven to fnd out the outlers so as to assess the partcpant performance but not to ascertan the assgned value. Hence the data receved from the partcpatng laboratores have been evaluated by both numercal and graphcal method mentoned above and the assessment of the outcome of dfferent methods appled to the results s presented n ths secton. The results obtaned from the Cochran s test were compared wth the crtcal values of the Cochran s table and found that the laboratory codes 5 & 8 n case of 1mg/l; 15, 16, 17, 23 & 30 n case of 2 mg/l and 52, 59 & 62 n case of 5.01 mg/l of Pb elemental soluton measurement are classfed as outlers. Cochran s test consders only the wthn laboratores reproducblty/repeatablty result so even though the laboratory results are qute far.e. devates 75 % from the reference value s not consdered as outler. Rather the result whch devates only 1 % from the reference value s found as outler that can be reveals from the bar of the lab code 8 n fgure 1(a). As Cochran s test s the ntra laboratory varablty test, so ths method s not relable for the outler detecton. Concentraton of Pb Fgure 1(a). Reported result of the partcpant wth lab code for Pb elemental soluton 1 mg/l Concentraton of Pb Fgure 1(b). Reported result of the partcpant wth lab code for Pb elemental soluton 2 mg/l

6 238 Sushree SwarupaTrpathy et al.: Comparson of Statstcal Methods for Outler Detecton n Profcency Testng Data on Analyss of Lead n Aqueous Soluton Fgure 1(c). Reported result of the partcpant wth lab code for Pb elemental soluton 5.01 mg/l Test carred out Table 2. Outlers detected by varous statstcal analyss of detected outlers for measurement of Pb elemental solutons 1 mg/l 2 mg/l 5 mg/l Cochran s Test 5, 8 15, 16, 17, 23, 30 52, 59, 62 Grubb s Test 6 18, 33, 44 46, 51 Hampel Test 5, 6, 10 14, 18, 33, 44 46, 51, 53, 62 Classcal approaches Nl Medan & NIQR method (Robust z-score) 5, 6, 10 14, 17, 18, 33, 44 46, 51, 53, 62 ISO 13528: Alogorthm A method 5, 6, 10 14, 18, 33, 44 46, 51, 53, 62 NATA Method Zw 5, 8 15, 16, 23, 30 52, 59, 62 Zb 5, 6, 10 14, 17, 18, 31, 33 51, 53, 62 Smlarly, Grubb s test s also not sutable method to fnd out outlers. Ths test was evaluated wth consderng two largest and two smallest values of the results and compared wth the crtcal values of the Grubb s table. There s one number, three numbers and two numbers of results n case of 1, 2 and 5.01 mg/l Pb elemental soluton respectvely were evaluated as outler n Grubb s test. In ths test the hghest devaton of the result of the laboratory code 6 s found to be 75 % above from the reference value for 1 mg/l Pb elemental soluton where as the laboratores code 18, 33, 44, 46, 51 for other two concentratons are reported strange values whch gve an mpresson that these laboratores have been based wth the other laboratory. There s less number of laboratores found as outler n Grubb s test as compared to Cochran s test. The laboratores detected as outlers by Cochran s and Grubb s test for all the concentratons of Pb elemental solutons are shown n the table 2. Hampel s test whch s known as Huber test [7] was carred out followng the condton.e. the devaton value from the medan wth the medan of devaton of the data set to fnd out outler. Hampel s test s more or less a robust test towards outler detecton as the medan value of all the measurement s consdered to fnd out outlers. The number of outler detected by Hampel s test s more as compared to Cochran and Grubb s test n analyss of all the concentratons of Pb elemental soluton. It was found that the result whch s far from the reference value n the data set.e. devate at least 20 % were detected as outler and ths method seems to be a good method for outler detecton. The outler lab codes for all the concentratons of Pb are shown n table Graphcal Statstcal Analyss In PT, most often the results obtaned from the partcpatng laboratores are expressed n terms of z-score to present ther performance. The z-score of ndvdual laboratory ndcates how much ts reported result dffers from the reference/assgned value and dentfy as satsfactory, questonable and unsatsfactory results. In ths paper to assess the laboratory performance, data receved from the partcpatng laboratores have been converted nto the z-score by applyng the statstcal methods. Accordng to the classcal z-score, the outlers result nfluence the mean value and standard devaton of the data set. By classcal approach, maxmum numbers of laboratores show z-score wthn acceptable range.e. les wthn z-score ± 2. In case of robust z-score calculaton the medan and the normalzed nter quartle range (NIQR) of the data set have been consdered. In ths test laboratores 5, 6 and 10 for 1 mg/l, 14, 17, 18, 33 and 44 for 2 mg/l and 46, 51, 52 and 62 for 5.01 mg/l are showng unsatsfactory results. Ths s a very good method to fnd out the outlers as the medan of the data set used n the calculaton whch does not nfluenced much by the result far from the reference value. The z-score obtaned from the classcal and robust method s tabulated n table 3. In ISO 13528: robust analyss of algorthm A method to get the robust estmates, ntally the

7 Amercan Journal of Theoretcal and Appled Statstcs 2013; 2(6): PT results were arranged n an ncreasng order and the absolute devaton from the medan was calculated. Then by teraton.e. updatng the robust average (x*) and robust standard devaton (s*) several tmes the process converges so that the value does not change from one teraton to the next teraton tll the thrd dgt. The robust average s consdered as the assgned value and the uncertanty value was derved by puttng the robust standard devaton n Eq. 10. The consensus/assgned value wth the standard devaton/uncertanty value derved by dfferent methods Table 3. z-score obtaned by classcal and robust method along wth the reference value are tabulated n Table 4. The z-score was calculated usng the robust estmates. The laboratory result whch devates up to 30 % lower and hgher from the reference value s not consdered as outler. The laboratores found as outler by ths method are smlar wth the laboratores found n case of Hampel s test. It s observed from the evaluaton of the data; ths s the best method to derve assgned value when the reference value of the analyte dstrbuted for PT s not known. Sl. No. Values mg/l z-score Classcal z-score Robust Sl. No. Values mg/l z-score Classcal z-score Robust lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab lab Table 4. Comparson of reference value wth the mean, medan, robust assgned value obtaned by varous statstcal methods Reference value of Pb elemental soluton at 95 % confdence level Mean value Mean value Standard uncertanty Medan value Medan value NIQR Consessus value (after elmnaton of outler obtaned by Hampel s test) Consensus value NIQR Assgned value (ISO method) Assgned value Standard uncertanty ± ± ±

8 240 Sushree SwarupaTrpathy et al.: Comparson of Statstcal Methods for Outler Detecton n Profcency Testng Data on Analyss of Lead n Aqueous Soluton Zw Fgure 2(a). z-score wthn laboratores (Zw) obtaned by NATA method for Pb elemental soluton 1 mg/l Zb Fgure 2(b). z-score between laboratores (Zb) obtaned by NATA method for Pb elemental soluton 1 mg/l Zb Fgure 2(d). z-score between laboratores (Zb) obtaned by NATA method for Pb elemental soluton 2 mg/l Zw Fgure 2(e). z-score wthn laboratores (Zw) obtaned by NATA method for Pb elemental soluton 5.01 mg/l Zw Fgure 2(c). z-score wthn laboratores (Zw) obtaned by NATA method for Pb elemental soluton 2 mg/l Data receved from the partcpatng laboratores have been evaluated quanttatvely for ther qualty of measurement usng NATA method also. For ths purpose the PT data was processed to calculate two types of z- scores such as z-score wthn the laboratory (Zw) and between the laboratores (Zb) by ntegraton of the seven summary statstcs - number of the laboratores, medan, normalzed nterquartle range (IQR), robust coeffcent of varaton (CV), mnmum, maxmum and range [4]. A value of the z-scores wthn the laboratory ndcates the varance n the data produced n the same laboratory and z-score between the laboratores ndcates the varance n the data produced by dfferent partcpatng laboratores. The qualty of the measurement ncreases wth the decrease n the z- Zb Fgure 2(f). z-score between laboratores (Zb) obtaned by NATA method for Pb elemental soluton 5.01 mg/l score value. The values of both the z-score wthn the laboratory and z-score between the laboratores has been calculated for all the concentratons of Pb soluton and are plotted aganst the lab code (Fgure 2(a) - (f)). From the chart of the z-score results each laboratory can compare easly ts performance relatve to ther own and other laboratores results. Fgure 2(a), (c) and (e) show that for the analyss of Pb soluton the laboratory nos. 5, 8 for 1 mg/l; 15, 16, 23, 30 for 2 mg/l and 52, 59, 62 for 5.01 mg/l concentraton are havng greater varance whch Zw s outsde the range ± 3. Smlarly, laboratory code nos. 5, 6, 10 for 1 mg/l; 14, 17, 18, 31, 33 for 2 mg/l and lab code no. 51, 53, 62 for 5.01 mg/l are exceedng Zb value ± 3 when the medan value of ndvdual laboratory s compared wth

9 Amercan Journal of Theoretcal and Appled Statstcs 2013; 2(6): the medan value of all the partcpatng laboratores [Fgure 2(b), (d) and (f)]. Hence these laboratores havng z-score outsde the range ± 3 are an ndcaton of a problem wth ther results and are consdered to be outler laboratores for the analyss of Pb soluton. The laboratory wth code no. 17 n case of Zw and 20, 42, 45, 52, 60 n case of Zb exceed the cut off value of the z-score.e. outsde the range of ± 2 and below ± 3; whch are advsed to have a look nto ther result. There are 19 numbers of laboratores found as outler by ths method consderng both the wthn and between laboratory data varablty Comparson of Statstcal Methods Appled for Laboratores Performance The overall result of the statstcal evaluaton of the analyss of all the concentratons of Pb elemental solutons s compared n fgure 3 whch gves a clear dea about the percent of outlers found by each statstcal method. Except lab code 17, the other laboratores found as outler n the Cochran s test s totally matches wth the outler result of the wthn laboratory varance of NATA method. The percent of outler n case of classcal approach for calculaton of z-score s very less.e. 0, < 4 and < 5 % for 1, 2 and 5.01 mg/l Pb soluton respectvely and t seems to be not sutable method for outler detecton as the mean of the data set s based by the extreme result of the partcpant. There s much smlartes n the outler result obtaned by Hampel s test, Medan & NIQR method, ISO method and only the between laboratory varance of NATA method n whch the medan value was consdered for the z-score calculaton. The medan value does not affected by the absurd result gven by the partcpant. More or less 30 % of the results n all the concentratons of Pb solutons are found as outler by NATA method. % outlers Cochran Grubbs Hampel z-classcal z-robust Alogorthm Methods A NATA Fgure 3. Comparson of statstcal methods used for all the concentratons of Pb elemental solutons 1 mg/l 2 mg/l 5 mg/l [Pb] Usually when the reference value of the measurand s not known n that case elmnaton of the outler s a crucal task to set the assgned value. In that case the consensus value can be set after elmnaton of the outler obtaned by Hampel s test. The measurement results have been checked for the consensus value after elmnatng the outlers obtaned from Hampel s test and found very close to the reference value for the 1 and 2 mg/l of Pb solutons. The mean, medan, assgned values of the data set are calculated as descrbed above and presented n the table 4. For 1 and 2 g/l Pb soluton; the medan, consensus and assgned values are almost same as the reference value but the mean value s qute far from the reference value for the same concentratons of Pb soluton. Ths reveals that the mean value mght be based by more number of absurd results n the data set. In case of 5.01 mg/l Pb soluton, except the mean value, the set values obtaned by other method are qute far from the reference value. The uncertanty value derved from ISO 13528: Algorthm A s very low for all the concentraton of Pb solutons. From the comparson of the dfferent statstcal methods evaluated n ths paper, t s revealed that NATA method s the most sutable method for the outler detecton for the present data set. However, t s dffcult to predct a sutable statstcal method for the determnaton of outler as several factors depends on t such as structure of the data set, type of outlers, number of outlers n the dataset etc. 5. Conclusons By comparng all the results t was observed that the common laboratores code numbers 5, 6, 8, 10, 14, 15, 16, 17, 18, 23, 30, 33, 44, 46, 51, 52, 53, 59 and 62 acqured the value of z-score out sde the range ± 3 n measurement of all the three concentraton of Pb elemental solutons. All these laboratores consdered as outler and are advsed to look nto the matter and take necessary correctve acton for over or under estmaton of the results. In case of the measurement result of 1 mg/l Pb soluton, the outler found n the hgher sde devate 75 % and the lower sde devate 21 % from the reference value. The outler laboratory codes 18, 33, 44 are reported more than 5.01 mg/l n the hgher sde and laboratory code 14 reported less than 1 mg/l n the lower sde for 2 mg/l Pb soluton measurement result. Smlarly the laboratory codes 51, 46 are found as outler reported less than 2 mg/l nstead of 5.01 mg/l Pb soluton measurement result. Among all the statstcal methods NATA method s the best approach to fnd out outler n a data set and for the performance assessment of the PT partcpant. Because t comprses both the wthn and between laboratory varance and moreover t s a robust analyss to calculate z-score whch does not nfluenced by the extreme value of the data set. There s no systematc trend observed n the capablty of measurement of the laboratores n analyss of lower and hgher concentraton of the Pb elemental soluton even though dfferent analytcal methods and sophstcated nstruments were used by the laboratores. More than 70 % of laboratores are shown satsfactory performance for analyss of Pb out of 64 partcpants. The dssatsfactory performance of the laboratores mght be due to not usng a reference materal of respectve metal soluton for calbraton of the nstrument/standardzaton of method. Therefore, t s advsable for all the PT partcpants to mplement qualty system n ther laboratores and use of certfed reference

10 242 Sushree SwarupaTrpathy et al.: Comparson of Statstcal Methods for Outler Detecton n Profcency Testng Data on Analyss of Lead n Aqueous Soluton materals wll be strongly recommended. The outcome of ths profcency testng clearly reveals that Pb elemental analyss of aqueous soluton can be performed n many laboratores wth hgh accuracy. Acknowledgement Authors are thankful to Natonal Accredtaton Board of Testng and Calbraton Laboratores (NABL), Inda for provdng fnancal support. Thanks are due to the Drector, NPL for gvng permsson to publsh ths paper. References [1] ISO/IEC 17025, General Requrements for the Competence of Testng and Calbraton Laboratores, Internatonal Standards Organzaton (ISO), Geneva, [2] ISO/IEC 17043:2010(E) Conformty assessment-general requrements for profcency testng. Internatonal Standards Organzaton, Geneva. Frst edton [3] ILAC-G13: Gudelnes for the requrements for the competence of profcency testng schemes, [4] New statstcs for NATA s Profcency Testng Programs, Natonal Assocaton of Testng Authortes (NATA), Australa, [5] ISO 13528:2005(E) Statstcal methods for use n profcency testng by nterlaboratory comparsons, Internatonal Standards Organzaton, Geneva. Frst edton [6] Nordtest project No ; 2000:35 methedology of nter-comparson Tests and statstcal analyss of test results. [7] Qualty assurance and qualty control n the analytcal chemcal laboratory: A practcal approaches. Potr Koneczka and Jacek Namesnk. Taylor & Francs Group, 2009.