(rorabacher et al. 1991 anal chem) q-test

8/6/2019 (Rorabacher Et Al. 1991 Anal Chem) Q-Test

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Anal. Chem. 1991. 63. 39-146 139t e t radecano l) a re separa ted a nd de tec ted , quan ti t a t ion i shindered by poor solubility and the formation of micelles.Applications. Since a prerequisite for PA D reactivity isadsorp tion of the analyte, t he presence of other surface-ad-sorbable substances, as well as electroactive compo unds, canact a s interferences. Therefore, general selectivity is achievedvia chroma tographic separation prior to PAD. This conclusiondoes not pre clude additional selectivity from control of de-tect ion param eters .T he assay for alcohols was applied to several matric es toillustrate the analytical utility of the procedure. Sepa rationon an ion-exclusion column with direc t detection is illustratedin Figure 8 for various aliphatic alcohols and polyalcohols intoothpa ste (A), liquid cold formula (B), brandy (C), and winecooler (D). Th e selectivity for alcohols in acidic media at aPt electrode contributes to decreased tim e for sample prep-aration and simplified chromatograms.Th e versatility of separation s on mixed-mode ion-exchangecolumns with selective detection is illustrated in Figure 9 bythe sim ultaneo us detection of ionic an d neutr al species in apharmaceutical preparat ion. This experim ent ut i lizes a UVdetector and PA D in series after a PCX-500 column. Underacidic conditions, th e cephalosporin antibacterial consists ofa cation (i.e., cefazolin) and neut ral an d anionic compounds(i.e., lB-hexanediol, 1,4-cyclohexanediol,and p-toluenesulfonicacid). The neutral and anionic compounds are separated bythe r eversed-p hase character of the column, while the cationiccompoun d is separa ted by a combination of cation exchange

and reversed-phase mechanism. Figure 9 shows th at thep-toluenesulfonic acid and the cefazolin are b oth d etecte d byUV a t 254 nm (A), and t he two diols, which do not have achromophore, a re easily detected by PAD (B). In addit ion,the cefazolin has a PAD signal, which has a higher limit ofdetection than UV, but may be utilized for added selectivity.LITERATURE CITED

Kisslnger, P. T. I n Laboratory Techniques inElectroanalytical Chemis-t ry ; Klssinger, P. T., Heineman, W. R. , Eds.; Marc el Dekk er: NewYork, 1984; pp 631-32.Adams, R. N. Electrochemistry at SolM Electrodes; Marcel Dekker:New York, 1969.Gllman, S. I n E/ectroana/ytica/ Chemistry; Bard, A. J., Ed.; MarcelDekker: New York, 1967; Vol. 2, pp 111-92.Fleet, B.; Little, C. J. J. Chromtogr. Sci . 1874, 72, 747.Van Rooljen, H. W.; Poppe. H. Anal. Chim. Acta 1881, 730, .Hughes, S.; Meschi, P. L.; Johnson, D. C. Anal. Chim. Acta 1881.732, .Johnson, D. C.; Lacourse, W. R. Anal. Chem. 1890. 62, 89A.Nachtmann, F.; Budna. K. W. J. Chromtogr. 1977, 736, 79.Jupille, T. J. Chromatogr. Sci. 1978, 77, 160.Beden, B.; Cetin, I.; ahyaoglu. D.; akky, D.; Lamy, C. J. Catal.1987, 704, 37.Ocon, P.; Alonso, C.; Celdran, R.; Gonzalez-Velasco , J. J. Electroanal.Chem. 1888, 206 , 179.Slingsby, R. W.; Rey, M. J. Li9. Chromatogr. 1880, 73(1), 107.Lacourse, W. R.; Jackson, W. .; Johnson, D. C. Anal. Chem. 1888,67 , 2486.

RECEIVEDor review July 20,1990. Accepted Octob er 8,1990.Th e financial support of Dionex Corporation is acknowledgedwith gratitude.

Statistical Treatment for Rejection of Deviant Values: CriticalValues of Dixons Q Parameter and Related Subrange Ratiosat the 9 5 % Confidence LevelDavid B. RorabacherD e p a r t m e n t of Chemis t ry , W ayne S ta te Un ivers i t y , Detro it , Michigan 48202

CrHlcal values at the 95% Confidence leve l for the two-tailed0 test, and related tests based upon subrange ratlos, for thestatlstkal rejectlon of outlying data have been Interpolated byapplying cublc regresslon analysls to the values orlglnallypublished by Dlxon. Correc tions to errors In Dixons orlglnaltables are also Included. The resultant values ar e judged tobe accurate to wlthln f0.002 and corroborate the fact thatcorrespondlng crltlcal values published In recent slatlstlcaltreatlses for analytical chemlsts are erroneous. I t Is recom-mended that the newly generated 9 5 % crltlcal values beadopted by analytical chemlsts as the general standard forthe rejection of outller values.

Analytical chemists depend upon the generat ion and in-terp reta tion of precise experime ntal da ta. As a result, theyare especially cognizant of the value of statistics in datatreatment, and a numbe r of statistical treatises have recentlybeen published th at are specifically written for the professionalanalytical chemist ( I ) . Included in ea ch of these publication sis a brief section dealing with tes ts for the rejection of grosslydeviant values (outliers). Although man y statistical tests havebeen proposed to deal with this topic [Ba rnet t and Lewis (2)discuss 47 different eq uations designed for this purpose], it

is interesting to note tha t these treatises, as well as essentiallyall analytical chemistry textbooks published in the U.S. uringthe past decade (3),have settled on the use of Dixons Q est(and variants thereof) ( 4 )as the prim ary method for test ingfor the rejection of outlying values.Each of the rece nt statistical treatises written for analyticalchemists has atte mp ted to include critical values of Q for the95% confidence level, values tha t were not included in Dixonspublications. However, not only do the 95% confidence valuesdiffer in each treatis e but a ll compilations contain significanterrors. Th e most legitimate set of 95% values is tha t presentedby Miller and M iller (4I I 0) ( l a ) ,which they at t r ib uteto King (5),but no such values are listed in Kings article, andthe values of Miller and Miller differ by amounts arying from0.002 to 0.007 f rom the 95% values presented in the c urrentmanuscript . Anderson ( I b ) describes the equations corre-sponding to the two-tailed tests for Dixons parametersdesignated as rl0 (for 3 5 n 5 o ) , r l l (for 8 5 n I o) , andrZ 1 fo r 11I I 3) and p urporte dly lists critical values forthe 9 0% ,9 5% , and 99% confidence levels for these samplesizes, but the values actually listed in his table a re Dixonsvalues for one-tailed tests. Thus, s applied to two-tailed tests,Andersons confidence levels should be labeled 80%,9070,nd98% (vide infra). Caulc utt and Boddy (IC), while describingonly the equation for th e Q (i.e., rl0)ratio, accurately list bo th

0003-2700/91/0363-0139$02.50/0 0 1991 American C hemical Society


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14 0 A N A L Y T I C A L C H E M I ST R Y , V O L . 63 , NO. 2 , JANUARY 15, 199195% an d 99% confidence level critical values for this ratiofor five samp le sizes, 3 5 n 5 7 ; he values given in their Tab leG (p 248) for 8 5 n 5 12 and for 13I 5 40 are clearly no tQ values bu t are somewhat s imilar to (though not identicalwith) the critical values for rI1and r21, espectively. However,the equations /o r these la t t er parameters are not euenmentioned in th is treat ise.Faced with this confusing and conflicting array of criticalvalues for the popular Q test and Dixons related tests basedon various subrange ratios, it seems likely that many erroneousstatistical inferences are being made daily in analyticalchemistry laboratories through out the world. To counteractthis possibility and to extend the utility of Dixons para mete rsfor two-tailed tests, the cu rren t paper provides a reliable setof critical values at the 95% confidence level for the Q pa-rameter and for all of the related subrange ratios proposedby Dixon. It also corrects several errors found in Dixonsoriginal published tables. At the sam e time, it is intendedto alert analyt ical chemists to the problems that exis t incurren t treatises an d, hopefully, may serve as a useful sourcefor future textbook and treatise authors dealing with this topic.

CRITERIA FOR REJECTION OF OUTLIERSDeterminate Errors and Outliers. In any set of ana-lytical m easuremen ts, one or more values may incorporate a

deter mina te (systematic) error. Such errors may involve eithera locator error (disp lacem ent of the popu lation me an) ora scalar error (increa se in the population variance ) or acombination of the two ( 4 ) . Values that are affected by adeterminate error belong to a different population of mea-surements . Thus, if the presence of a determinate error isknown to the individual making the measurem ents, the valuesso affected should automatically be rejected, regardless of theirmag nitude. (For a brief discussion on the detection andelimination of deter mina te errors in analytical measurements,see ref 3b.) However, in many instanc es, determ inate errorsare incurred that are not detected by the experimental is t .If an undetected determinate error affects only a fractionof the values in the total sample and if the error is sizablerelative to the experimental precision of the remaining values,the affected values may be significantly larger or smaller thanthe values th at are no t affected by such errors; ;.e., they willbe observed as outlier values. Recognizing this, experimen-talists tend to suspect the presence of determinate errorswhenever outliers are observed in a set of da ta, and there isa strong motivation to eliminate such outliers since theysignificantly affect the sample me an a nd , therefore, the finalestim ate of the population mean. In the absence of inde-pendent knowledge th at de terminate errors are responsiblefor the a ppea rance of outliers, however, the decision to deletedeviant values must be based upon reasonable statisticalevidence. Some statisticians look askance at any attem pt todiscard devia nt values by means of statistical criteria. Dem ing,a highly respected au thority in t he ind ustrial application ofstatistic s, has concluded t ha t a point is never to be excludedon statistical grounds alone (6). However, the majority ofthose who have addressed themselves to the s tat is t icaltrea tme nt of scientific measuremen ts would appea r to agreewith Parra tts statem ent tha t rejection on the basis of a hunchor of gene ral fear is not a t all satisfactory, and some sort ofobjective criterion is better than none (7 ) . In fact , the s ta-tistical treatment of outlier values has received increasingat tent ion in recent decades and th e various s tat is t ical tes tsthat have been proposed are described and compared in texts(2,8)and a n extensive review (9) dealing specifically with thistopic.Tests for Rejection of Grossly Deviant Values. Sta-tistical approaches used for identifying values that are affectedby deter mina te errors are based on testing the hypothesis tha t

such values can be attr ibuted to random variation alone withinsome reasonable level of probability. If the probability ofobtaining such outliers is determined to be very small, basedon the overall population variance, one may reasonably con-clude th at this hypothesis is incorrect, i.e., these values cannotbe at t r ibuted to s imple random variat ion and ar e l ikely th eresul t of a det erm inat e error. On this basis, suspected valuesmay be rejected in a statistically legitimate fashion, providedth at the confidence level chosen is a reasonable one.Confidence levels commonly used for tests involving therejection of data are 99%, 95% or 90%, the higher levelsrepresenting the more conservative approach. Operating ata high confidence level reduces the likelihood of rejecting alegitimate value containing no de termin ate error (i.e., it re-duces the so-called a risk or Type I error), bu t i t also in-creases the likelihood of retaining a value that contains adeterm inate error (Le., it increases the so-called 3 risk or TypeI1 error ) ( I O ) . The consensus of most statisticians is th at th eformer error (rejecting a good value) is considered moreserious and, therefore, the use of high confidence levels in therejection of data is greatly preferred. For example, whenopera ting at the 99% confidence level ( a = 0.01) with apopulation con taining no outliers, only 1% of the good valueswill be rejected. In contrast, when operating at the 90%confidence level ( a = 0.10), 10% of the good values will berejected a nd , for small sample sizes, the effect upon the es-tima tion of the population mean becomes significant relativeto the accuracy desired by most analyt ical procedures.[However, Dixon has argued tha t the use of rejection tests a tthe 9 0 7 ~ onfidence level is warranted in many cases ( I I ) . ]Since the use of a 99% confidence level ( a = 0.01) allowsfor the rejection of only the most extreme deviations-particularly when applied to small samp le sizes-the /3 riskof retaining a value tha t does not belong to the general samplepopulation (Le., a b ad value) is quite large, and m ost ex-perimental is ts are not sat isf ied when operat ing at th is levelin testing for the rejection of data. Operating a t the 95%confidence level ( a = 0.05)provides a reasonable compromise.If values not rejected at th e 95% confidence level are viewedwith suspicion, addit ional measurem ents are probably war-ranted . It should be noted tha t no specific test or choice ofconfidence level is ideal, however. As Natrella has no ted, theonly sure way to avoid publishing any bad results is to throwaway all results (12). The converse, of course, is alsotrue-the only sure way to avoid discarding any good resultsis to retain them all.A numb er of tests, such as those based on the Studen tst distribut ion (13 ) , require independent knowledge of thepopulation s tand ard deviat ion (cr) and the population mean( k )or the unco ntaminated sample standard deviation (s ) andthe uncontam inated sample mean (8)or their application(2). In the absenc e of such indep ende nt information, thesetests ar e of limited use since a decision must be mad e as towhether the suspected values should be included or excludedin the calculation of s and X . Ne ither decision is satisfactoryas the choice made automa tically biases the ou tcome of thetest. [Th e often-cited Chauvenets criterion for the rejectionof a deviant measurem ent (2, 7) is also based on t he sam plestandard deviation and , therefore, poses the same problemswhen applied to small samples; it has been noted th at thiscriterion is set to reject, on average, half an o bservation of gooddata per sam ple, regardless of the sam ple size (8,14).] Foranalytical measurements, where cr and g (and uncontaminatedvalues of s and X ) are general ly not known independently,tests th at do not require the use of these quan tities are greatlypreferred.

APPLICATION O F DIXONS RANGE TESTSThe Q Test and Related Subrange Ratios. In a classic


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ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991 141T h e rl0 ratio is commonly designated as "Q" an d is generallyconsidered to be the m ost convenient, legitimate, statisticaltest available for the rejection of devian t values from a smallsample conforming to a G aussian distribution. (It is equallywell suite d to larger da ta se ts if only one outlier is present.)Th e fact tha t small data sets are common in analytical testingprocedures, in com bination with the simplicity of this test,accounts for the fact that t he Q test is included in nearly allmodern statistical treatises and textbooks designed for usein analytical chemistry (1,3).A few authors (1 ), following Deming's viewpo int (videsupra) , object to the rejection of da ta from any small samplebased on s tat is t ical tes ts , claiming that the amo unt of in-formation available is insufficient to establish th e distributionpattern; recommended al ternat ives include dropping thehighest and lowest values (for a samp le containing five or morevalues) or reporting the median. Howeve r, this would appe arto be overly cautious since most series of repetitive an alyticalmeasurements follow a Gaussian distribution (18)providedtha t s is small compared toX thus, the Q test is not applicableto analyt ical measurements when operat ing close to the de-tection limits). Dixon has teste d the relative merits of thesample mean an d median as an e st imator of th e populationmean u nder various conditions (11,15) nd has concluded that,for the most pa rt , the sample m ean (af ter the reject ion of

outliers) appears to provide a better approximation than doesthe median.In his origina l calcula tion of the critical values of th e various

r criteria (1.9,Dixon was able to ob tain exact solutions onlyfor the case where n = 3 or 4. Critical values for n = 5 , 7 , 10 ,15,20,25, and 30were calculated by using numerical methods.All other values were obtained by interpolation and weregenerally judged to be accurate within fO.OO1.An irritating featu re of the Q tes t and Dixon's related su-brange ratio tests,as hey c urrently exist, is th e lack of suitablecritical values of Q (and r l l , r21,etc.) a t the 95% confidencelevel (a = 0.05), since this confiden ce level is frequently utilizedfor all other statistical tests. Th e lack of 95% confidence levelvalues arises from the fact th at D ixon generated critical values

at several s tan dard probabili ty levels (a = 0.005, 0.01, 0.02,0.05, etc.) corresponding to 99.5%, 99%, 98%, 95%, etc.,confidence levels in term s of a one-tailed t es t (15). As com-monly applied by analytical chemists an d other experimen -talists, however, these tests are used as two-tailed test s (i.e.,one is generally interested in testing ou tlier values a t both th eupper an d lower ends). Th e probabil i ty th at an individualgood value may lie outside a specified interval in either tail(Le., on eith er th e high or low end) is twice as large as th eprobability th at it lies outside th e chosen interval in only onetail. As a result, the a risk doubles for a two-tailed te st (18)and t he confidence level decreases accordingly. T hus , Dixon's99.5%, 99% , 98%, an d 95% confidence levels translate into9 9 % , 98%, 96% , and 90% levels, respectively, when consid-ering a two-tailed test, a fact not recognized by some autho rs( I b ,3 e , m ) . Since Dixon did not calculate one-tailed criticalvalues a t the 97.5% confidence level, there hav e been no valuesavailable at th e 95% level when using these tests as wo-tailedtests. There fore, abo ut half of the curr ent analytical chem istrytextbooks (3a-d ,h) listing valid critical values of Q l is t thevalues for the 99 % ,96 % , and 90% confidence levels, despitethe fa ct tha t th e 96% confidence level is somewhat unor-thodox, while the remainder (3fg,i-l) provide only 90%confidence level values. As noted earlier, the treatises t ha thave at tem pted to prese nt cri t ical values for Q a t t h e 95%confidence level ( I ) have invariably listed flawed values.

[I n tests designed to detec t the presence of a single outlier,King has argued (5 ) that the effect of running a two-tailedtest is approximately to double the a risk relative to th at for

1950 article ( 4 ) ,Dixon investigated the performance of severalstatistical tests in term s of their ability t o reject bad valuesin data sets taken f rom Gaussian populat ions. Th e testsinvestigated included both those which require independentknowledge of u or s and those which do not require suchinformation. Of the tests included in the latte r group, Dixonconcluded that tests based on ratios of the range a nd varioussubranges were to be preferred as a result of their excellentperformance an d ease of calculation. [Dixon also noted th atan oth er tes t which performs well in screening for outliers isa modified F test, in which the ratio of the stand ard deviationscalculated by including and excluding the suspected deviantvalue is compared to critical values of F; however, this la ttertest may be "masked" by a second deviant value.] Th e rangetests, all of which are closely related, include th e following(where the values are ordered such th at x1 < x 2 < ... < x,-~< x,):1. For a single outlier x1

2. For outlier x1 avoiding x,

3. For outlier x1 avoiding x,, x,-,

4. For outlier x l voiding x2

5. For outlier x1 avoiding x2 and x,

6. F or outlier x1 avoiding x2 and x,,

(Th e parenthet ical equation s are designed for tes t ing x,,the highest value rather tha n th e lowest value, xl.) In Dixon'snotation, th e first digit in the su bscript of each ratio, r i j ,refersto th e numbe r of possible suspected outliers on the same endof the da ta as t he value being tested, while the second digi tindicates the num ber of possible outliers on the opposite endof the data f rom the suspected value. Thus , the rat io rl0simply compares the difference between a single suspectedoutlier (xl or x n ) and i ts nearest-neighboring value t o theoverall range of values in the sample-in othe r words, it de-termines the f ract ion of the total range that is at t r ibu tableto one suspected outlier. Th e other ratios are similarly for-mulated except t ha t they use subranges that are specif icallydesigned to avoid the influence of additio nal outliers eitheron the opposite end of the d ata ( r l land r12),on the same endof the data ( rm) , r both (rZ1 nd rZ2) .Clearly, the latter ratiosrequire larger sample sizes to perform satisfactorily. Dixonsubs equ ently generated critical values for all of the se ratios(15) or sam ple sizes of 3I 5 30 and recommended (basedon a combination of the relative performance of each ratioand its degree of independence from othe r outlying values)that , as a general rule, the various ratios be applied as ollows(16): fo r 3 I 5 7 , us e rlo; or 8I I 0, use r l l ; fo r 11In I 3, use rZl; or n L 14, use rZ2.

n


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14 2 ANALYTICAL CHEMISTRY, VOL. 63 , NO . 2, JANUARY 15, 1991Table I . Critical Values of Dixon's r I oQ) Parameter As Applied to a Two-Tailed Test at Various Confidence Levels ,Inc luding the 9 5% Confidence Level"

confidence level80 7c 90 7c 95% 96 o/ o 98% 99 ?&

N b ( a = 0.20) ( a = 0.10) ( a = 0.05) ( a = 0.04) ( a = 0.02) ( a = 0.01)3 0.886 0.941 0.970 0.976 0.988 0.9944 0.679 0.765 0.829 0.846 0.889 0.9265 0.557 0.642 0.7 10 0.729 0.780 0.8216 0.482 0.560 0.625 0.644 0.698 0.740, 0.434 0.507 0.568 0.586 0.637 0.6808 0.399 0.468 0.526 0.543 0.590 0.6349 0.370 0.437 0.493 0.510 0.555 0.59810 0.349 0.412 0.466 0.483 0.527 0.56811 0.332 0.392 0.444 0.460 0.502 0.54212 0.318 0.376 0.426 0.441 0.482 0.52213 0.305 0.361 0.410 0.425 0.465 0.50314 0.294 0.349 0.396 0.411 0.450 0.48815 0.285 0.338 0.384 0.399 0.438 0.47516 0.277 0.329 0.374 0.388 0.426 0.46317 0.269 0.320 0.365 0.379 0.416 0.45218 0.263 0.313 0.356 0.370 0.407 0.44219 0.258 0.306 0.349 0.363 0.398 0.43320 0.252 0.300 0.342 0.356 0.391 0.42521 0.247 0.295 0.337 0.350 0.384 0.4182 2 0.242 0.290 0.331 0.344 0.378 0.41123 0.238 0.285 0.326 0.338 0.372 0.40424 0.234 0.281 0.321 0.333 0.367 0.399

25 0.230 0.277 0.3 17 0.329 0.362 0.39329 0.227 0.273 0.312 0.324 0.357 0.3882; 0.224 0.269 0.308 0.320 0.353 0.38428 0.220 0.266 0.305 0.316 0.349 0.38029 0.218 0.263 0.301 0.312 0.345 0.37630 0.215 0.260 0.298 0.309 0.341 0.372

?.

In this an d th e other accompany ing tables, the newly genera ted or corrected values are indicated in boldface. Sam ple size.Table 11. Critical Values of Dixon's r l lParameter A s Applied to a Two-Tailed Test at Various Confidence Levels , Includingth e 95 % Confidence Level

confidence level80 % 90 % 95 % 96 Yo 98% 99%

N O (456891011

-12131415161718192021222324252627282930

(ISample size.

'N = 0.20) (0.9100.7280.6090.5300.4790.4410.4090.3850.3670.3500.3360.3230.3130.3030.2950.2880.2820.2760.2700.2650.2600.2550.2500.2460.2430.2390.236

a = 0.10)0.9550.8070.6890.6100.5540.5120.4770.4500.4280.4100.3950.3810.3690.3590.3490.3410.3340.3270.3200.3140.3090.3040.2990.2950.2910.2870.283

( a = 0.05) (0.9770.8630.7480.6730.6150.5700.5340.5050.4810.4610.4450.4300.4 170.4060.3960.3860.3790.3710.3640.3570.3520.3460.3410.3370.3320.3280.324

n = 0.04) (10.9810.8760.7630.6890.6310.5870.5510.5210.4980.4770.4600.4450.4320.4200.4100.4000.3920.3840.3770.3710.3650.3590.3540.3490.3440.3400.336

cr = 0.02)0.9910.9160.8050.7400.6830.6350.5970.5660.5410.5200.5020.4860.4720.4600.4490.4390.4300.4210.4140.4070.4000.3940.3890.3830.3780.3740.369

( a = 0.01)0.9950.9370.8390.7820.7250.6770.6390.6060.5800.5580.5390.5220.5080.4950.4840.4730.4640.4550.4460.4390.4320.4260.4200.4140.4090.4040.399

a one-tailed tes t; bu t, for very small sample sizes, the effectmay be slightly less than double depend ing upon t he d i s-t r ibu t ion pa t tern of t h e d a t a since the tail containing theperceived de viant is determined by the sample rather than

by independent knowledge of the true distribution of thepopulation. This argum ent implies th at, at worst, assuminga doub ling of the a risk upon going from a one-tailed tes t toa two-tailed tes t will result in critical values th at a re slightly


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ANALYTICAL CHEMISTRY, VOL. 63, NO. , JANUARY 15, 1991 143Table 111. Critical Values of DixonsrI2 arameter As Applied to a Two-Tailed Test at Various Confidence Levels, Includingthe 9 5 % Confidence Level

confidence level80 70 90 % 95 % 96% 98 70 99 %NQ (a= 0.20) (a = 0.10) (a = 0.05) (a= 0.04) (a= 0.02) (a= 0.01)

56789 .101112131415161718192021222324252627282930

0.9190.7450.6360.5570.5040.464b0.4310.4060.3870.3690.3540.3410.3300.3200.3110.3030.2960.2900.2840.2780.2730.2680.2630.2590.2550.251

0.9600.8240.7120.6320.5800.5370.5020.4730.4510.4320.4160.4010.3880.3770.3670.3580.3490.3420.3360.3300.3240.3190.3140.3090.3050.301

0.9800.8780.7730.6920.6390.5940.5590.5290.5050.4850.4670.4520.4380.4260.4150.4050.3960.3880.3810.3740.3680.3620.3570.3520.3470.343

0.9840.8910.7910.7080.6560.6100.5750.5460.5210.5010.4820.4670.4530.4400.4290.4190.4100.4020.3940.3870.3810.3750.3700.3650.3600.355

0.9920.9250.8360.7600.7020.6550.6190.5900.564c0.5420.5230.5080.4930.4800.4690.4580.4490.4400.4320.4230.4170.4110.4050.3990.3940.389

0.9960.9510.8750.7970.7390.6940.6580.6290.602c0.5800.5600.5440.5290.5160.5040.4930.4830.4740.4650.4570.4500.4430.4370.4310.4260.420Sam ple size. In Dixons original table (13 ) , h e r1 2cri tical value a t the 80 % confidence level for n = 10 is 0.454. However, the cubicregression curve based on t he 40%, 60%, 90%, 96%, 98%, and 99 % confidence level critical values for n = 10 as well as a regression curvef i t t ed to the 80% cricital values versus sample size indicates tha t this value shou ld be 0.464. Therefore, i t i s conc luded th a t th e 80 % criticalvalue originally published for r12a t n = 10conta ins a typographica l e r ro r . The same r12critical value for 95 % confidence is obtained ei th erby using the corrected 80 % value or by omitt ing i t al together from th e regression analysis. Th e r12critical values in Dixons original table(13) or n = 13 a t t h e 98 % a n d 99 % confidence levels are 0.554 a n d 0.612, respectively. However, cubic regression curves f i t ted t o thecri t ical values at these two confidence levels as a function of sample size yield corrected values of 0.564 a n d 0.602, respectively, indicatingth at the original values contained typograp hical errors. These la t ter values were used in resolving the r I 2 ritical value for the 95 % confi-dence level a t n = 13 (a l though the sam e 95 % value was obtained by om itt ing these values and including th e cri t ical values for the 40% and60% confidence levels).

too high, and t he resul t ing decisions tha t ar e made wil l beoverly conserva tive. As shown by com paris on of Dixonsoriginal one-tailed critical values for 95% confidence (15) withhis two-tailed critical values for th e 90% confidence level (16b),it is clear that Dixon assumed a doubling of the a isk.]Interpolation of Critical Values at the 95% ConfidenceLevel. As noted above, Dixon was able to obtain exact so-lutions for the various critical values only for th e cases wheren = 3 or 4. Although th e general form of the eq uation for n2 5 has been presented ( 1 5 ) , he specific expressions for thecentral density function vary w ith each sample size, and theseexpressions have not been published. Therefore, in the cu rrentwork, appr opri ate two-tailed critical values of Q a t the 95%confidence level were initially estimated by plotting the(two-tailed) critical values for the 99%, 98 %, 96% , 90% , and80% confidence levels as generated by Dixon. T he graphicallyinterpolated values for 95% Q were then checked by usingregression analysis to d etermine the best f i t t ing empiricalpolynomial functions to the values th at Dixon published; thesefunctions were then solved for the appropriate critical valuesof Q at the 95% confidence level.Wi th the exception of cases in which t he original Dixontables contained appa rent errors (vide infra), it was found tha ta cubic funct ion provided a n optima l f i t to the Q values inthis region. Interestingly, the use of a quadrat ic functiongenerally yielded the same values of 95% Q, to thr ee signif-icant figures, as those obtained from a cubic function despitea notably poorer fit; fourth-power functions also produced th esame 95% values to within fO.001 b ut w ere less sensitive toerrors in the tabular da ta. I t was also noted tha t the cubic

functions could generally be extended to include the dat a forthe 60% and 40% confidence levels but, except as notedbelow, these values were not included in fitting the cubicregression curves since the da ta in the lower confidence levelregion did not sig nificantly affect the ca lculation of the 95%values.Based on t he foregoing analysis, th e critical values of Q a tth e 95% confidence level were the n calculated from th e cubicregression curve for each sam ple size and w ere found t o bewithin fO .OO1 of th e values obta ined graphically. Cubicfunctions were subsequently f i t ted t o the 9 9%, 98% , 96% ,90%, and 80% confidence level dat a for each of the other ratiofunctions defined by Dixon. T o check the veracity of thegenerated equations, the critical values for 96% and 90% werealso calculated and , in each case (except as noted below), werefound to be w ithin *0.001 of Dixons tabular values.In using this approach, i t was noted th at cubic equationscould no t be f i t ted to th e rz0critical values for n I 9. Afterexamining Dixons tabular da ta carefully, it was discoveredth at th e 90% confidence level critical values showed a dis-continuity in this region. T o circumvent this problem, the60% confidence level values were i n c l u d e d in fitting the rz0da ta to cubic equations. In the region 4 5 nI 8, he criticalvalues of rz0a t the 95% confidence level obtained by includingan d excluding the 60% critical values were within f0.0002,i.e., un dete ctab le to thre e significant figures (see Figure 1).Fo r n I 9, the 90% values were then omitted. In this manner,cubic expressions were generated t ha t provided excellent fitsto the da ta and permit ted both the 95% a nd new 90% criticalvalues to be comp uted. Interestingly, in comparing the newly


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14 4 ANALYTICAL CHEMISTRY, VOL. 63 , NO . 2, JANUARY 15, 1991Table IV. Crit ical Values of Dixon's r zo arameter As Applied to a Two-Tailed Test at Various Confidence Levels , Includingth e 95 % Confidence Level

confidence level80% 90 % 95 % 96 C7 c 98 % 99 %h'" ( n = 0.20) ( a = 0.10) ( n = 0.05) ( a = 0.04) (a= 0.02) ( a = 0.01)

4568910111213141516171819

2021222324252627282930

I

0.9350.7820.6700.5960.5450.5050.4740.4490.4290.4110.3950.3820.3700.3590.3500.3410.3330.3260.3200.3140.3090.3040.3000.2960.2920.2880.285

0.96;0.8450.7360.6610.6070.5650.5310.5040.4810.4610.4450.4300.4180.4060.3970.3876(0.379)0.3786(0.372L0.371(0.365)0.364b(0.358)0.358b(0.352)0.352b(0.347)0.346b(0.343)0.3426(0.338)0.33S6(0.334)0.333b(0.330)0.32g6(0.326)0.326O(0.322)

0.9830.8900.7860.7160.6570.6140.5790.5510.5270.5060.4890.4730.4600.4470.4370.427b0.41Sb0.41060.402b0.395b0.39060.383b0.37gb0.374b0.370b0.3Mb0.361b

0.9870.9010.8000.7320.6700.62;0.5920.5640.5400.5200.5020.4860.4720.4600.4490.4390.4300.4220.4140.4070.4010.3950.3900.3850.3810.3760.372

0.9920.9290.8360.7780.7100.6670.6320.6030.5790.5570.5380.5220.5080.4950.4840.4730.4640.4550.4470.4400.4340.4280.4220.4770.4120.4070.402

0.9960.9500.8650.8140.7460.7000.6640.6270.6120.5900.5710.5540.5390.5260.5140.5030.4940.4850.4770.4690.4620.4560.4500.4440.4390.4340.428

"Sample s ize . bS ta r t in g wi th n = 19 , the r.," critical values fo r both t he 90% and 95% confidence levels were calculated from the cub icregression curves f i t ted to the cri t ical values-published by Dixon ( 13 ) cor responding to the two- ta i led 60 %, 8 0 % , 9 6 % , 98% , a n d 9 9 %confidence levels (but omitt ing the p ublished 90% confidence values) . For th e 90% confidence level , th e values originally published byDixon are indicated in parentheses und erne ath the newly generated values. From a comparison of the tw o sets of values, it is obvious thatthe cri t ical values in th e original table were shif ted up one row in the column corresponding to the two-tailed 90% confidence level (see text).generate d 90% confidence level values of rz0with the originaltabular data , the results reveal that , in Dixon's original table( I j ) , he two-tailed 90% confidence level values (correspondingto Dixon's one-tailed N = 0.05) for 19 I I 0 were acci-dentally displaced upw ard by one row (see Tab le IV).In any set of data in which t he cri t ical values at the 90%an d/ or 96% confidence levels, as calculated from the cubicregression e qua tion, differed by more tha n h0.001 fromDixon 's tabular values, both the cubic equation an d the or-iginal tabula r da ta were carefully checked fo r error. In thisma nne r, f ive addition al typographical errors were uncoveredin Dixon's original tables. Correcte d values were obtaine d intwo ways: (i ) by generating a new cubic equation for thespecific sample size omitting th e suspected tab ular value (whileextending the cubic fit to include the 60%-and, in some casesthe 40%-confidence level value) ; an d (ii) by generat ing acubic or quartic equation to fit the critical values at the specificconfiden ce level as a function of sam ple size. Iden tical valueswere obtained with both a pproache s and , in all five cases, thecorrected values revealed tha t on e digit had been incorrectlytypeset in th e original paper (15) . Th e corrected values areindicated in Tables I11 a n d VI.Th e resultant critical values of Q (r l0) t the 95% confidence

0.4 0 . 5 0.6 0 . 7rz 0 Values

Figure 1. Typical plot of crit ical values as a function of the confidencelevel . T h e curve shown is the regression curve for rz0 at n = 10(including the values for 60%, 80%, 9 0 % , 96%, 98%, a n d 9 9 % )corresponding to the cubic equation: YO onfidence level = -343.75754- 1835 .0349r2 , - 2 5 4 8 . 5 7 2 2 r Z o 2+ 1 188 .477 rZo3 .level an d similarly generated critical values for the rll, r l z , a,and rZ2 unctions are included in Tables I-VI along with thevalues published by Dixon corresponding t o the 99% , 98% ,96%, 9070,and 80% confiden ce levels-all confide nce levels.shown being app lica ble t o a two-tailed t e s t . All values have


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ANALYTICAL CHEMISTRY, VOL. 63 , NO. 2, JANUARY 15, 1991 145Table V. Cri t ical Values of Dixon's r I I aram eter As Applied to a Two-Tailed Test at Various Confidence Levels, Includingth e 95 % Confidence Level

confidence level80 % 90 % 95 % 96% 98% 99 %N" (a= 0.20) (a= 0.10) (a 0.05) (a= 0.04) (a= 0.02) (a= 0.01)

56789101112131415161718192021222324252627282930

0.9520.8210.7250.6500.5940.5510.5170.4900.4670.4480.4310.4160.4030.3910.3800.3710.3630.3560.3490.3430.3370.3310.3250.3200.3160.312

0.9760.8720.7800.7100.6570.6120.5760.5460.5210.5010.4830.4670.4530.4400.4280.4190.4100.4020.3950.3880.3820.3760.3700.3650.3600.355

0.9870.9130.8280.7630.7100.6640.6250.5920.5650.5440.5250.5090.4950.4820.4690.4600.4500.4410.4340.4270.4200.4140.4070.4020.3960.391

0.9900.9240.8420.7800.7250.6780.6380.6050.5780.5560.5370.5210.5070.4940.4820.4720.4620.4530.4450.4380.4310.4240.4180.4120.4060.401

0.9950.9510.8850.8290.7760.7260.6790.6420.6150.5930.5740.5570.5420.5290.5170.5060.4960.4870.4790.4710.4640.4570.4500.4440.4380.433

0.9980.9700.9190.8680.8160.7600.7130.6750.6490.6270.6070.5800.5730.5590.5470.5360.5260.5170.5090.5010.4930.4860.4790.4720.4660.460

" Sample size.Table VI. Crit ical Values of Dixon's rZ2 arameter As Applied to a Two-Tailed Tes t at Various Confidence Levels, Includingth e 95 % Confidence Level

confidence levelN"

6789101112131415161718192021222324252627282930

80%(a= 0.20)

0.9650.8500.7450.6760.6200.5780.5430.5150.4920.4720.4540.4380.4240.4120.4010.3910.3820.3740.3670.3600.3540.3480.3420.3370.332

90,370(a = 0.10)

0.9830.8810.8030.7370.6820.6370.6000.5700.5460.5250.5070.4900.4750.4620.4500.4400.4300.4210.4130.4060.3990.3930.3870.3810.376

95 %(a= 0.05)

0.9900.9090.8460.7870.7340.6880.6480.6160.5900.5680.5480.5310.5160.5030.4910.4800.4700.4610.4520.4450.4380.4320.4260.4190.414

96 %(a= 0.04)

0.9920.9190.8570.8000.7490.7030.6610.6280.6020.5790.5590.5420.5270.5140.5020.4910.4810.4720.464b0.4570.4500.4430.4370.4310.425

98 %(a 0.02)

0.9950.9450.8900.8400.7910.7450.7040.6700.6410.6160.5950.5770.5610.5470.5350.5240.5140.5050.4970.4890.482b0.4750.4690.4630.457

99 %(a= 0.01)0.9980.9700.9220.8730.8260.7810.7400.7050.6740.6470.6240.6050.5890.5750.5620.5510.5410.5320.5240.5160.5080.5010.4950.4890.483

"Sample size. b Th er z z critical values listed in Dixon's original table (13)for n = 24 at the 96 % confidence level (two-tailed) and for n =26 at the 98 % confidence level are 0.484 and 0.486, respectively. Th e values shown in this table (0.464 and 0.482, respectively) were obtainedby a cubic regression curve fitted t o the c ritical values at these two confidence levels as a function of sample size. These same values are alsogenerated from t he cubic regression lines fitted t o th e critical values for each of thes e sample sizes omitting th e questionable values andincluding the values at the 40 % and 60 % confidence levels. It is concluded that the original tabular critical values for rz 2 for these twosample sizes and confidence levels were th e re sult of typographical errors.been carefully cross-checked, an d t he 95% critical values arejudged to be accurate w ithin fO.OO1 relative t o the accuracyof the values at th e other confidence levels (which Dixon stated

(15) ere themselves generally accurate to within *0.001). T hevalues shown cover the entire range of sam ple sizes (3 5 n I30) included in Dixon's original article.

n>=14


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Anal. Chem. 1991, 63,46It is suggested that the critical values of Q and the relatedr criteria which have been generated for the 95% confidencelevel should be used routinely by pra cticing analytical chemistsin testing for the rejection of outliers since this confidencelevel provides a reasonable compromise between ultracon-servatism and the overzealous rejection of deviant values.Thes e values should also be incorporated into futur e analyticalchemistry treatises and textbooks dealing with tests for therejection of da ta to provide a uniform set of critical valuesat th is s tand ard conf idence level.As a concluding comment, i t should be noted tha t recentstudie s on the variance of the arithm etic mean a fter rejectionof outliers suggest the sup eriority of two more recently pro-posed criteria (Huber -type skipped mean an d Shapiro-Wilkrules) for rejection decisions (1 9 ) , articu larly for larger sam-ples containing multiple outliers. Nonetheless, the simplicityof Dixon's range ratio tests argues strongly for their continueduse in many analytical applications.

ACKNOWLEDGMENTI express my appreciation to my colleagues, George Schenkand David Coleman, for helpful suggestions regarding thismanuscript .

LITERATURE CITED(a) Miller, J. C.; Miller, J. N. Statistics for Analytical Chemistry. 2nd ed.;Wiley: New York. 1988; pp 62-64, 218. (b) Anderson, R . L. PracticalStatistics for Analytical Chemlsts;Van Nostrand Reinhoid: New York,1987; pp 31-32. (c) Caulcutt, R. ; oddy, R . Statistics for AnalyticalChemists; Chapman and Hall: London, 1983; pp 66-67, 248.Barnett, V. ; Lewis, T. Outliers in StatisticalData, 2nd ed.; Wiiey: NewYork. 1984.(a) Skoog, D. A.; West, D. M.; Holler, F. J. Analytical Chemistry: AnIntroduction, 5th ed.; Saunders: Philadelphia, 1990; p 56. (b ) Skoog,D. A.; West, D. M.; Holler, F. J. Fundamentals of Analytical Chemistry,5th ed.; Saunders: New York, 1988 ; pp 13-16. (c) Hargis, L. G.Analytical Chemistry; Prentice-Hall: Englewocd Cliffs, NJ, 1988; p 56.(d) Fritz, J. S.: chenk, G. H. Quantitative Analytical Chemistry, 5thed.: Allyn 8 Bacon: Boston, 1987; pp 45-46. (e) Potts, L. W. Quan-

146-15 1titative Analysis: Theory and Pract ice; Harper 8 Row: New York,1987; pp 78-80. (f) Rubinson, K. A. Chemical Analysis; Little, Brown:Boston, 1987; pp 162-164. (9) Day, R. A., Jr.; Underwood, A. L.Quantitative Analysis, 5th ed.; Prentice-Hall: Englewood Cliffs, NJ,1986; pp 29-31. (h) Manahan, S. . Quantitative Chemical Analysis;Brooks Cole: Monterey. CA, 1986; pp 74-75. (i)Kennedy, J. H. Ana-iyticai Chemistry: Principles, 2nd ed.; Harcourt, Brace, Jovanovich:New York, 1990; pp 35-39. (i)Harris, D. C. Quantitative ChemicalAnalysis; Freeman: San Francis co, 1982; pp 51-52. (k) Ramette, R.W. Chemical Equilibrium and Analysis ; Addison-Wesley: Reading, MA,1981; pp 53-54. (I)Christian, G. D. AnalyticalChemistry; Wiley: NewYork, 1980; pp 78-79. (m) Flaschk a, H. A,; Barnard. A. J., Jr.; Stur-rock, P. E. Quantitative Analytical Chemistry; Willard Grant: Boston,1980; pp 19-20.Dixon, W. J. Ann. Math. Stat. 1950, 27. 488-506King, E. P. . Am. Stat. Assoc. 1953, 48, 531-533.Deming, W. E. Statistical Aaustment of Data; Wiley: New York,1943 (republished by Dover: New York. 1964); p 171 .Parratt, L. G. Probability and Experimental Errors in Science; Wiley:New York, 1961; pp 176-178.Hawkins, D. M. Identification of Outliers;Chapm an and Hall: London,1980.Beckman, R . J.; Cook, R . D. Technometrics 1983, 25, 119-149.(a) Burr, I. W. Applied Statistical Methods; Academic: New York.1974; pp 19 4-195. (b) Hamilton, W. C. Statistics in Physical Science;Ronald: New York. 1964; pp 45-49.Dixon, W. J. I n Contributions to Order Statistics; Sarhan, A. E.,Greenberg, B. G., Eds.; Wiley: New York, 1962; pp 299-342 (see pp314-317).Natrelia, M. G. Experimental Statistics; National Bureau of StandardsHandbook 91; NBS: Wash ington, DC, 1963 ; Chapter 17."Student". [Gossett, W. S.] Biometrika 1908, 6 , 1.Taylor, J. R . An Introduction to Error Analysis: The Study of Uncer-tainties in Physical Measurements University Science : Mill Valley,CA, 1982; pp 142-145.Dixon, W. J. Ann. Math. Stat. 1951, 22 , 68-78.(a) Dixon, W. J. Biometrics 1953, 9, 74-89. (b) Dean, R . 8.; Dixon.W. J. Anal. Chem. 1951, 23, 636-638.Peters, D. G.; Hayes, J. M.; Hieftje, G. M. Chemical Separations andMeasurements : Theory and Practice of Analytical Chemistry; Saun-ders: Philadelphia, 1974; p 36.Youmans, H. L. Statistics for Chemistry; Charles E. Merrill: Columbus,OH, 1973; p 65 ff .Hampel, F. R . Technomefrics 1985, 27 , 95-107.

RECEIVEDor review Ju ne 15, 1990. Accepted October 16,1990.

Frequency-Domain Spectroscopic Study of the Effect ofn-Propanol on the Internal Viscosity of Sodium Dodecyl SulfateMicellesM. J. Wirth,* S . - H .Chou, and D . A. PiaseckiD e p a r t m e n t of Chemis t ry a n d Biochemistry, Universi ty of Delaware, New ark , Delaware 19716

The rotational diffusion behavior of tetracene in sodium do-decyi sulfate micelles is studied as a function of the n-propanol content of the micella r solution. Fluorescence an-isotropy measurements using frequ encydom ain spectroscopyshow that tetracene reorients faster with increasing concen-tration of n-propanol. This result Is consistent with micellarliquid chrom atographic studies and lends insight into the ro leof n-propanol as a mobile-phase modifier. The componentsof the rotational diffusion tensor are determined from thedouble-exponential anisotropy decay. These confirm the va-lidity of the Debye-Stokes-Einstein model and allow caicu-iation of the viscosity of the micelle interior. The viscositydecreases from 8 to 4 CP as the concentration of n-propanolincreases from 0 to 10% (v /v) . The components of the ro-tational diffusion tensor indicate that the solvation environmentof tetracene in the micelle is structurally disordered.* T o whom correspondence should be addressed.

INTRODUCTIONThe importance of micelles in analytical chemistry hasburgeoned both in spectroscopy and in separation science.Micelles are used to enhance fluorescence ( I ) , thermal lensing(21, and room-temperature phosphorescence ( 3 ) . In liquidchromatogra phy, micelles modify the organic con tent of themobile phase ( 4 ) ,allowing more rapid gradient elution ( 5 ) .Micelles can serve as the pseudostationary phase in elec-trokinetic chromatography (6),roviding selectivity for sep-aration of polar organic solutes. Th e nature of solute inter-actions with micelles is thus a vital area of research.An imp ortant question entails the dynam ics of solutes in-teracting with micelles. One of the drawbac ks of micellarliquid chromatograp hy has been poor column efficiency dueto s low mass t ransfer between the surfactant-modif ied s ta-tionary phase and the mobile phase ( 7 ) . Dorsey et al. dem -onstrated th at the addition of at least 1% of n-pro pano l givesa 40% increase in the number of theoretical plates for thesolute benzene in micellar liquid chrom atogra phy (8). They0003-2700/91/0363-0146$02.50/0 1991 American Chemical Society

(rorabacher et al. 1991 anal chem) q-test

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