Statistical treatment of test results

Transcription

1 SCAN-G :07 Revied 007 Pulp, paper ad board Statitical treatmet of tet reult 0 Itroductio The value of tatitical method lie i the fact that they make it poible to iterpret tet reult accordig to trictly objective criteria. A tatitical aalyi of tet data doe ot icreae the experimetal preciio by traformig ucertai reult to certaitie, but it doe make it poible to expre umerically, i the form of defiite probabilitie, the igificace to be attached to the cocluio draw from the reult. The purpoe of thi Guidelie i to give a brief decriptio of thoe tatitical method which are recommeded for ue i the treatmet of tet data derived accordig to SCAN-tet Method ad to promote uiformity i the ue of tatitical term ad ymbol, ad i the mode of expreig tet reult. Thi SCAN-tet Guidelie replace SCAN-G :63. Thi revied verio differ from the earlier verio i that certai equatio which were coidered helpful before computeried help wa available have ow bee omitted. I additio, the ectio relatig to the calculatio of ucertaitie have bee exteded, ad iformatio i provided a a aid to the developmet of preciio tatemet i SCAN-tet Method. Thi Guidelie ca with advatage be read i cojuctio with SCAN-G 6:00 Ucertaity of reult from phyical tetig. The ISO Techical report ISO/TR 4498 Paper, board ad pulp Etimatio of ucertaity for tet method may alo be helpful. Scope Thi SCAN-tet Guidelie give a brief decriptio of the imple tatitical method commoly ued i the treatmet of tet data ad provide the eceary equatio. It doe ot give detailed derivatio of the equatio, or doe it provide proof of the theorem preeted. The equatio are preeted i a maer which eable pero without tatitical traiig to apply the method, but it i recommeded that a pero with tatitical traiig ad experiece be coulted before cocluio are draw from the reult. A umber of umerical example are preeted i Aex A a a illutratio of how the method are to be applied. Term ad defiitio. Meauremet proceig of a igle tet piece or tet portio Note The umber of meauremet required for a tet i uually tated i the method.. Reult of a meauremet value of the property obtaied by a igle meauremet.3 Tet complete procedure, icludig preparatio of the tet material, performig the umber of meauremet required, ad makig the eceary calculatio

2 SCAN-G :07 Page.4 Reult of a tet value (e.g. mea, tadard deviatio etc) reported for the tet ad calculated from the reult of all the meauremet.5 Populatio fiite or ifiite amout of material or umber of uit Note It i ormally ot practical to meaure every uit i a populatio, ad a ample mut therefore be take..6 Sample limited amout of material take from a populatio Note The material i take for the purpoe of providig a fiite umber of tet piece o which a certai property i to be meaured, i order to obtai a etimate of the value of that property i the material from which the ample i take. Note I thi Guidelie, the term ample i ued i it tatitical ee. I phyical or chemical tet method thi word may have aother meaig..7 Radom ample limited amout of material elected at radom from a populatio, i.e. i uch a maer that ay uit i the populatio ha a equal probability of beig elected, o that the ample may be coidered to be fully repreetative of the populatio from which it i take.8 Tet piece piece or quatity of material take from the ample for ue i a igle meauremet i a chemical or phyical tet.9 Statitic igle value calculated from ad ued to repreet a et of meauremet reult, uually a a etimate of ome parameter of a populatio from which a ample ha bee take.0 Mea, arithmetic mea, μ or x a tatitic decribig the populatio calculated a the um of the idividual meauremet reult divided by the umber of meauremet Note the ymbol μ i ued to deote the mea of the populatio, ad the ymbol x i ued to deote the mea of a et of meauremet, ad thu the etimated mea of the populatio.. Media, x M tatitic decribig a et of meauremet choe o that 50 % of the value are above ad 50 % of the value are below x M. Rage differece betwee the larget ad the mallet value of a et of meauremet reult.3 Diperio meaure of the extet to which the reult of the idividual meauremet are cattered about the mea.4 Variace of a populatio, σ um of the quare of the deviatio of the idividual value of the property from the calculated mea divided by the umber of meauremet,.5 Variace of a ample, um of the quare of the deviatio of the idividual value of the property from the calculated mea divided by a factor equal to oe le tha the umber of meauremet, ( ) Note Diviio by thi factor eure that the calculated variace of the ample i a ubiaed etimate of the variace of the populatio from which the ample i take..6 Stadard deviatio, σ or quare root of the variace Note Thi i the mot commoly ued meaure of the diperio. The ymbol σ i ued to deote the tadard deviatio of the populatio, ad the ymbol i ued to deote the tadard deviatio of a et of meauremet, ad thu the etimated tadard deviatio of the populatio..7 Coefficiet of variatio, CoV tadard deviatio divided by the mea Note The coefficiet of variatio i expreed a a percetage. It i ometime referred to a the relative tadard deviatio..8 Straggler member of a et of value which i uuually high or uuually low ad i icoitet with the other member of that et at a 5 % probability level Note If the reult of a tatitical tet how that a reult i a traggler but ot a outlier, it hould ormally ot be rejected..9 Outlier member of a et of value which i extremely high or extremely low ad i icoitet with the other member of that et at a % probability level Note If the reult of a tatitical tet how that a reult i a outlier, thi reult hould be excluded from the ubequet tatitical calculatio..0 Sigificace extet to which the data idicate that the oberved effect ha a give probability of ot beig due olely to chace

3 SCAN-G :07 Page 3 Note The level of igificace i uually expreed either a the probability p that the give reult i due to chace or, more commoly, a the probability ( p) that the reult i ot due to chace. Sigificace tet are frequetly applied for p < 0,05, i.e. for ( p) > 0,95, commoly called the 95 % igificace level. I critical ituatio, a igificace of 99 % (p <0,0) may be required.. Degree of freedom umber of idepedet compario which ca be made betwee the member of a ample Note The umber of degree of freedom i uually - where i the umber of idepedet meauremet of tet beig coidered.. Cofidece limit limit which defie with a give probability the rage withi which a give tatitic i etimated to lie Note The cofidece limit mot commoly referred to are thoe aociated with the etimated mea of a populatio calculated from meauremet made o a limited ample take from the populatio..3 Studet t-ditributio probability of a cotiuou radom variable ued to ae the igificace of a meaured tatitic. Note Value of the ditributio provided i tabular form for differet degree of freedom are ued i the calculatio of cofidece limit. Note Thi ditributio wa publihed i 908 by Studet, the peudoym of W S Goet..4 Ucertaity meaure of the ucertaity aociated with a give tatitic Note I the cae of the mea, the imple ucertaity i equal to the etimated tadard deviatio of the populatio. Note Iformatio about the calculatio of combied ucertaitie aociated with differet ource of ucertaity i give i SCAN-G 6..5 Expaded ucertaity, U imple ucertaity multiplied by a coverage factor, k, o that U = k Note I the cae of the mea, the expaded ucertaity i aalogou to the cofidece limit..6 Repeatability coditio coditio where idepedet tet reult are obtaied with the ame method o idetical tet item i the ame laboratory by the ame operator uig the ame equipmet withi a hort iterval of time Note I the tetig of pulp, paper ad board, it i ot poible to make meauremet trictly uder repeatability coditio if, a i ofte the cae, the tet i detructive..7 Repeatability tadard deviatio, r tadard deviatio of meauremet reult obtaied uder repeatability coditio.8 Repeatability limit, r value le tha or equal to which the abolute differece betwee two tet reult obtaied uder repeatability coditio i expected to be with a give probability.9 Reproducibility coditio coditio where the tet reult are obtaied with the ame method o idetical tet item i differet laboratorie with differet operator uig differet equipmet. Note I the tetig of pulp, paper ad board, it i ot poible to make meauremet trictly uder reproducibility coditio if, a i ofte the cae, the tet i detructive.30 Reproducibility tadard deviatio, R tadard deviatio of tet reult obtaied uder reproducibility coditio.3 Reproducibility limit, R value le tha or equal to which the abolute differece betwee two tet reult obtaied uder reproducibility coditio i expected to be with a give probability 3 Ditributio I ay populatio of differet value of a variable, the differet value occur with differet frequecie. The ature of the populatio ca thu be decribed by the ditributio of the frequecie with which the differet value of the variable occur. Thi frequecy ditributio ca be illutrated graphically by a frequecy curve. Whe tet are carried out o a umber of tet piece take from a ample of the populatio, the value obtaied are alo ditributed with differet frequecie, which are etimate of the probabilitie that thee value occur with thee frequecie i the populatio. The probability ditributio thu decribe the probable frequecie with which differet evet are expected to occur i a ample take at radom from the populatio.

4 SCAN-G :07 Page 4 Thi probability ditributio ca be decribed by a probability curve which will reemble the frequecy curve of the total populatio. The primary purpoe of a tatitical aalyi i (a) to calculate uitable tatitic to decribe the ditributio ad (b) to ae the reliability of thee tatitic a a decriptio of the populatio. 3. Hitogram If a et of meauremet ha bee made o tet piece take from a ample from the populatio, it i poible to obtai a approximate idea of the appearace of the frequecy curve of the populatio by cotructig a hitogram, Figure, where meauremet reult are grouped i clae idicated by the figure at the bae of each rectagle. The height of each rectagle repreet the umber of meauremet reult aiged to the iterval. Number of meauremet reult Figure. Hitogram Magitude Figure i a hitogram for a et of 5 meauremet, the magitude of the reult of which are betwee ad 7. Two of the reult lie betwee ad 3, four betwee 3 ad 4, five betwee 4 ad 5, three betwee 5 ad 6, ad oe betwee 6 ad 7. If thi tet i exteded by icreaig the umber of meauremet ad if the cla iterval i decreaed, the tepped form of the hitogram will approach a mooth frequecy curve. Such a frequecy curve ca ofte be give a defiite mathematical form. 4 Meaure of locatio To decribe a ditributio i imple term, ome meaure of the locatio of the ditributio i required, i.e. ome meaure of the cetre of the ditributio. 4. Mea The (arithmetic) mea or average value of a et of reult of meauremet of a give variable i calculated a the um of the idividual meauremet divided by the umber of meauremet: 4. Weighted mea x ( x + x + + x ) =... x = [] If the data are orted ito group with differet umber of value i the differet group, thi mut be take ito coideratio whe the mea i calculated. The mea of a frequecy ditributio i calculated a the weighted mea where the differet value recorded are multiplied by their repective frequecie ad the um i the divided by the total umber of meauremet: Σfixi x = [] Σ f where f i are the frequecie of the value x i. 4.3 Media I ome cae, particularly whe a ditributio i extremely kew, it i iappropriate to calculate the mea ice thi may be uduly affected by the magitude of the extreme value. I thi cae it may be more appropriate to calculate the media x m which i defied a: 3 m + + i x, x, x... x < x < x, x, x... x [3] where x, x, x 3 are the reult of the idividual meauremet arraged i acedig order. If the umber of meauremet i odd, i.e. if the highet umber i x +, the media i calculated a xm x + = [4] If the umber of meauremet i eve, i.e. if the highet umber i x, the media i calculated a ( x + x + x m = ) [5] Note It i poible to exted thi procedure to divide the data ito quartile, where 5 % of the value are larger tha the upper quartile ad 75 % of the value are larger tha the lower quartile etc. 5 Meaure of diperio The mea aloe i ot ufficiet to decribe the character of a ditributio. Some meaure of the degree of diperio aroud the mea i alo required. The

5 SCAN-G :07 Page 5 meaure of diperio motly ued i the tadard deviatio σ, which i the poitive quare root of the variace. From a et of meauremet reult x x...x, the variace σ of the populatio i etimated by calculatig accordig to the expreio: = ( x x ) [6] The term - i ued itead of, ice diviio by ha bee how to give a biaed etimatio of σ, particularly for low value of. A meaure of the tadard deviatio i the populatio σ i coequetly obtaied by calculatig: = ( x x ) [7] I additio to the tadard deviatio, it i ometime of iteret to report the relative tadard deviatio, i.e. the magitude of the tadard deviatio i relatio to the mea value. Thi i called the coefficiet of variatio, CoV, ad it i uually expreed a a percetage: CoV = 00 [8] x 6 Tet for ad rejectio of outlier Occaioally oe member of a et of meauremet value may appear to differ abormally from the other. The quetio the arie a to whether or ot thi differece i becaue the abormal value belog to a differet populatio. Several tatitical criteria have bee uggeted for awerig thi quetio. It i clear that a value which doe ot properly belog to the tet erie hould ot be icluded i the calculatio of the mea or i ay ubequet aalyi, but it mut be emphaied that extreme cautio hould be exercied before a value i rejected. A value hould preferably be rejected oly if it ha bee etablihed that a actual mitake ha bee made. A uuually high or low value may ofte be due to the atural variatio i the material ad, if thi value i rejected, the remaiig value will give a icorrect picture of the ditributio ad particularly the tadard deviatio will be too mall. I order to tet whether a value i ureaoably far from the mea, Grubb' tet (cf. ISO 575) ca be applied to check whether a exceptioally high or a exceptioally low value x i i to be regarded a a outlier. Grubb' tatitic, G, i calculated a: xi x G = [9] Critical value for Grubb' tet at the % probability level are give i Table. If the value obtaied for G i higher tha the value give i the table, x i i a outlier ad ca be rejected. Note that thi procedure hould be ued oly oce for each et of data. Table. Critical value for Grubb' tet No of Outlier value p<0,0 Straggler p<0,05 5,764,75 6,973,887 7,39,00 8,74,6 9,387,5 0,48,90,564,355,636,4 3,699,46 4,755,507 5,806,549 6,85,585 7,894,60 8,93,65 9,968,68 0 3,00,709 3,03,733 3,060, ,087,78 4 3,,80 5 3,35,8 7 The ormal ditributio Experiece ha how that the reult of et of meauremet are ofte ditributed i a maer which how a cloe agreemet with the mathematically wellkow ormal ditributio which ha a bell-haped form. The equatio for the frequecy curve i a fuctio of the mea μ ad the tadard deviatio σ, viz.: y = e σ π where y i the relative frequecy; x i the meaured quatity. ( x μ ) σ [0]

6 SCAN-G :07 Page 6 T The frequecy curve i how i Figure, where the meaured quatitie x are give a multiple of σ. The curve reache it maximum at x = μ, ad i ymmetrical aroud thi poit. Relative frequecy y 0,5 0,4 0,3 0, 0, 0-3σ -σ -σ μ σ σ 3σ Meaured quatity x Figure. Normal frequecy curve. I the particular cae where μ = 0 ad σ =, the equatio i reduced to: x y = e [] π Thi equatio ca be itegrated up to a certai value, or betwee certai limit. I thi way it i poible to calculate the probability that the reult of a igle meauremet, x, will fall above or below a certai value or that x will occur betwee two give limit. It i ueful to ote that the probability that a igle value of a ormally ditributed variable deviate from it mea by more tha i 3,7 %, by more tha σ i 4,6 %, ad by more tha 3σ i 0,3 %, ad that a deviatio of more tha 4σ occur o average oly oce i meauremet. Alteratively, it ca be aid that approximately 68 % of all value lie withi ± σ from the mea, ad that 95 % of all value lie withi ± σ. 8 Accuracy ad preciio It i importat i the preetatio of tet reult to ditiguih betwee accuracy ad preciio. The accuracy of a tet reult i a tatemet of the extet to which it coform to the true value. The preciio of a tet reult i a tatemet of the extet to which the et of meauremet reult o which the tet reult i baed are dipered about the mea. The preciio i expreed a the cofidece limit, or a the ucertaity of the mea. Both the accuracy ad the preciio ca be reported i either abolute or relative meaure. I ay report it i importat to tate which meaure ha bee ued, epecially if the reult are expreed a percetage. 9 Sytematic ad radom error All error ca be divided ito two type, ytematic ad radom error. A ytematic error affect the accuracy of the method ad of the value meaured, but the exitece ad magitude of a ytematic error are ormally ukow, ice they ca be etimated oly by compario of the calculated mea for a et of meauremet with the true mea, which ha bee calculated or determied i a idepedet maer. If the magitude of a ytematic error i ideed kow, tep will ormally be take to elimiate thi error. A radom error i a meaure of the preciio or reproducibility of a method, the magitude beig idicated by the diperio. A ytematic error i regular ad may be due, for example, to a defect i the meaurig device ued, wherea a radom error i irregular ad i due to variatio by chace i the meaured reult. 0 Radom variatio I a tatitical cotext, the term radom error i ofte ued to decribe variatio which are ot error but which are true variatio i the material. If a meauremet ca be repeated o a igle tet piece, the reult of the repeat meauremet will vary withi a certai rage (ad the variatio will uually be ormally ditributed) ad thi variatio i the radom error aociated with the method or the itrumet. If meauremet are made o a et of differet tet piece, the variatio i the reult i due ot oly to ay radom error i the method or itrumet but alo to a real variatio withi the ample. Thee two poible caue caot ormally be eparated. Ule iformatio to the cotrary i available, it i uually correct i a pulp, paper or board cotext to aume that the differece amog meauremet reult are ideed due to variatio i the material. The tadard deviatio i thu ofte a importat material property which hould be reported together with the mea. The cofidece iterval After havig obtaied a etimate of a certai quatity o the bai of a tet erie, it may be eceary to tate the degree of ucertaity aociated with thi etimate. Thi i uually doe by givig the limit withi which the true value of the aeed magitude i expected to lie with a certai pecified degree of igificace (00-p) %. Thee limit are called the cofidece limit, the iterval betwee them i called the cofidece iterval, ad the value of p i the level of igificace.

7 SCAN-G :07 Page 7 If meauremet have bee made o a ample from a populatio havig a ormal ditributio, ad the mea value x ad the tadard deviatio have bee calculated, it ca be how that the tadard deviatio aociated with the mea i give by m m = [] The (00-p) % cofidece limit of the mea ca the be expreed by the formula: tp ± [3] where t p i a value related to the ormal ditributio which give the probability that the true value lie withi the give limit. The appropriate value of t p i obtaied from a table of Studet t ditributio. Thi value i depedet both upo the elected level of igificace, p, ad upo the umber of meauremet, N, o which the calculatio of i baed. The umber of degree of freedom, f, i give by f =. Normally N =. The level of igificace ca be choe at will, the level mot frequetly ued beig thoe give i Table, viz: 5 %, % ad 0, %. Uually the 5 % level i recommeded. I thi cae, the expreio tp U =± [4] i alo the expaded ucertaity aociated with the mea, where t p i the coverage factor (p <0,05). It hould be oted here that the cofidece limit ca be made arrower, i.e. the preciio of the etimate of the mea value ca be icreaed, by icreaig the umber of meauremet, but thi doe ot of coure affect the variatio i the material idicated by the tadard deviatio. Required umber of meauremet With the aid of equatio [4] for the ucertaity or cofidece iterval, it i poible to calculate how may meauremet will be required i order to obtai a etimate of the mea for a certai variable with a give degree of preciio. A coditio for thi, however, i that a etimate,, of the tadard deviatio,σ, i available. Aumig that the tadard deviatio,, ha bee calculated from earlier meauremet, ad that we wih to etimate the mea of the populatio with a preciio of ± a ad (00 - p) % cofidece, (i other word, we require that the (00 - p) % cofidece iterval for the mea value hall have a width of a), the approximate umber of obervatio,, required to fulfil thee coditio i give by: tp = a [5] where the appropriate value of t i take from Table. Table Studet t-ditributio Degree of freedom, f p 5% % 0,% 4,78 4,60 8,6 5,57 4,03 6,86 6,45 3,7 5,96 7,36 3,50 5,40 8,3 3,36 5,04 9,6 3,5 4,78 0,3 3,7 4,59,0 3, 4,44,8 3,06 4,3 3,6 3,0 4, 4,4,98 4,4 5,3,95 4,07 6,,9 4,0 7,,90 3,97 8,0,88 3,9 9,09,86 3,88 0,09,84 3,85,08,83 3,8,07,8 3,79 3,07,8 3,77 4,06,80 3,74 5,06,79 3,73 6,06,78 3,7 7,05,77 3,69 8,05,76 3,67 9,05,76 3,66 30,04,75 3,65 40,0,70 3,55 60,00,66 3,46 0,98,6 3,37,96,58 3,9 p

8 SCAN-G :07 Page 8 3 Repeatability limit / reproducibility limit It i ot alway the preciio of the mea that i the matter of greatet cocer. Sometime, it i importat to ue the tatitical data available to provide a etimate of the maximum expected differece, with a give degree of probability, betwee two item take at radom from a populatio. I thi cae, if the expaded ucertaity aociated with a igle item i U = tp, the the maximum expected differece i equal to: Δ= U [6] Thi equatio ca alo be ued to ae the repeatability limit, i.e. the maximum expected differece betwee the mea of two et of meauremet made o material from the ame populatio. t p Δ= [7] For meauremet made uder reproducibility coditio i differet laboratorie, the maximum expected differece betwee two et of meauremet i equal to Δ= * [8] t p where * i a appropriately calculated tadard deviatio derived from a coideratio of the variou ource of variace i differet itrumet, differet laboratorie etc.. Thi i dicued more fully i ISO/TR 4498 ad i Claue 7. 4 Compario of two mea (t-tet) A aociated tak, ofte ecoutered i practice, i to compare the mea obtaied from two differet erie of obervatio, ad to ae whether or ot, at a give probability level, they ca be coidered to come from the ame populatio, e.g, whether two batche of omially the ame material differ igificatly i their propertie. Coider the cae where x ad x are the mea of the erie ad, l ad are the umber of idividual meauremet i each of the two erie, ad ad are the calculated tadard deviatio. If there i o great differece betwee ad ad there i o caue for aumig that there i ay eetial differece i the diperio of the populatio, a combied meaure of the tadard deviatio baed o all the meauremet reult,, ca be calculated accordig to the formula: = ( x ) ( ) x + x x [9] + The tatitic t i the calculated accordig to: t = x x ( + ) [0] Thi value of t i the compared with the value t p (Table ) correpodig to the p % level of igificace ad the umber of degree of freedom f give by: f = + [] If the value of t i greater tha the value t p, there i a igificat differece betwee x ad x ad the two ample caot be coidered to have come from the ame populatio. Note It ca ever be etablihed that the two ample do i fact come from the ame populatio; oly that if they come from differet populatio the the differece betwee the two populatio i le tha x x. 5 Additivity of variace The aemet of the igificace of a tet reult or the ucertaity aociated with the reult i ofte more complicated tha a mere calculatio of the mea ad tadard deviatio of the meauremet data. If the ucertaity i to be aeed i relatio to the reult of other tet i the ame laboratory or of tet i differet laboratorie, other ource of variatio ad ucertaity mut be coidered. The baic priciple to be oberved i uch cae i that, provided the differet ource of variace are idepedet ad ucorrelated, the variace are additive, i.e.: total = + + [] I thi cotext, it i alo importat to ote that if a mea tadard deviatio i required thi mut alway be calculated a the root mea quare: mea ( j ) = [3] j where j i the umber of item i the erie. It i ot correct merely to calculate the mea of the tadard deviatio.

9 SCAN-G :07 Page 9 6 Repeatability limit Whe two idepedet tet are carried out uder repeatability coditio, i.e. withi the ame laboratory by the ame operator uig the ame equipmet o the ame occaio, the 95% probability that the two tet reult will ot differ by a amout greater tha Δ i tp Δ= [4] provided that the oly ource of variatio i i the material. I equatio [4], i a overall tadard deviatio calculated i accordace with equatio [9]. Thi i ot however a realitic ituatio. I geeral, uder ormal laboratory coditio withi a paper mill, there are alo other ource of variatio betwee tet, which are aociated with a tadard deviatio betwee tet, bt, o that the total withi-laboratory tadard deviatio,, i give by: wl wt wl bt = + [5] where wt i the withi-tet tadard deviatio ad i the umber of meauremet i each tet. It i poible to aalye the reult ad to determie eparately the value of bt, but thi i ofte ot eceary. The repeatability tadard deviatio, r, i equal to ad the repeatability limit, r, are calculated a: wl r =,96 r which ca be implified to the expreio: r =,77 r [6] [7] Note The tadard deviatio calculated directly from the reult of differet tet carried out i thi maer i the repeatability tadard deviatio. It i ot the betwee-tet tadard deviatio which i defied a the compoet of the repeatability deviatio which i idepedet of variatio i the material. Note I SCAN-G 6, the quatity here referred to a ha the deigatio. wl bt I SCAN-G 6, thi dicuio i exteded to iclude other ource of variatio withi a laboratory where the tet i detructive ad where tet are o loger carried out uder repeatability coditio. I thi cae, if repeated tet are carried out o homogeeou material from the ame batch o differet occaio, it i poible to calculate the log-term withi-laboratory reproducibility limit i a aalogou maer if the total tadard deviatio i calculated. 7 Reproducibility limit Similarly, if imilar tet are carried out i differet laboratorie with differet item of equipmet, other ource of variatio will itroduce additioal ucertaitie. I thi cae, the total variace i called the reproducibility variace ad it may be expreed a: R = bl + bt + [8] ad the reproducibility limit, R, are give by: R R =,96 R [9] Note i ot equal to the betwee-laboratory tadard deviatio bl, which i oly oe compoet of the total reproducibility ucertaity. 8 Withi-laboratory tadard deviatio outlier Whe a compario i made ivolvig differet laboratorie, it i ometime eceary to exclude a laboratory that how a ureaoably high withilaboratory tadard deviatio. I order to determie whether a abormally large tadard deviatio i tatitically a outlier, Cochra' tet accordig to ISO 575, with a rejectio level of %, ca be applied. The tet hall be applied oly to the laboratory havig the highet deviatio. Cochra' tatitic, C, i calculated a: i.max P i i= C = [30] where P i the umber of laboratorie i the compario. Critical value for Cochra' tet are give i Table 3. A laboratory hould be excluded from the compario if the value of C obtaied i higher tha the value give i the table. Ay laboratory excluded o the bai of too high a tadard deviatio hall be completely excluded from the ubequet aalyi.

10 SCAN-G :07 Page 0 Table 3. Critical value for Cochra' tet (p<0,0) No of lab No of meauremet i each tet, ,633 0, ,564 0,43 7 0,508 0, ,463 0, ,45 0, ,393 0,8 0,366 0,343 0,4 3 0,3 4 0, ,88 0,00 6 0,74 7 0,6 8 0,49 9 0,39 0 0,9 0, ,97 0, Reportig tet reult The reult of a erie of meauremet of a property hould geerally be reported by: (a) the mea; (b) the umber of meauremet, ; (c) the tadard deviatio, ; (d) the 95 % cofidece iterval of the mea, or the expaded ucertaity of the mea; (e) whether ay outlier have bee rejected ad the criterio of rejectio that ha bee applied. Note If the data have bee traformed o that the cofidece iterval ha become aymmetrical i relatio to the mea (e.g. a i the cae of a kew ditributio), the mea value, the two cofidece limit ad the umber of meauremet hould be reported, a well a the mode of calculatio ued. Note I certai cae, uch a with kew ditributio, it may be uitable to report the media value, the rage of the variatio, ad the umber of meauremet. 0 Literature. SCAN-G 6:00 Paper, board ad pulp Ucertaity of reult from phyical tetig. ISO/TR 4498:006 Paper, board ad pulp Etimatio of ucertaity of tet method 3. ISO 575-:994 Accuracy (truee ad preciio) of meauremet method ad reult Part : Geeral priciple ad defiitio 4. ISO 575-:994 Accuracy (truee ad preciio) of meauremet method ad reult Part : Baic method for the determiatio of repeatability ad reproducibility of a tadard meauremet method

11 SCAN-G :07 Page Aex Numerical example A.0. Geeral The followig umerical example are give a a added illutratio of the applicatio idicated i the text. They alo provide data which ca be ued to check the accuracy of calculatio programme from other ource or thoe writte iterally. A.. A ample from a populatio with a ormal ditributio Fiftee idividual meauremet of a variable with a ormal ditributio have bee obtaied i a tet a how i Table A.. Table A. 4,0 4,37 4,5 4,4 4,45 4,59 4,8 4,44 4,66 4,3 4,47 4,70 4,36 4,50 4,75 Calculate i tur:. x = 66,73 x. x = = 4, x = 97,39 ( x x ) 4. = ( ) = 0, = 0, CoV = x = 4,0 % 7. f = = 4 8. t5 for f = 4 =, U = = 0,0997 A.. A populatio with a kew ditributio The followig twelve idividual meauremet of a variable with a ukow ditributio have bee obtaied i a tet, a how i Table A.. Table A.,07,79 3,37 4,40,37,84 3,4 4,84,64 3,3 3,78 6,30 Draw a hitogram i order to obtai a idea of the type of ditributio. Number of meauremet reult Magitude The hitogram idicate a kew ditributio. A ordiary calculatio of the mea ad the diperio may the be iappropriate. Try a traformatio, e.g. by takig the bae-te logarithm, a i Table A.3. Table A.3 0,36 0,446 0,58 0,64 0,375 0,453 0,533 0,685 0,4 0,509 0,577 0,799 Draw aother hitogram for the traformed variable. Number of meaured reult , 0, 0,3 0,4 0,5 0,6 0,7 0,8 0,9 log (Magitude) The ditributio obtaied by thi traformatio appear to be approximately ormal, ad the computatio ca therefore be cotiued with the logarithmic value:. x = 6,86. x = 0,538

12 SCAN-G :07 Page 3. x = 3, ( x x ) 4. = ( ) = 0, = 0,37 6. f = = 7. t5 for f = =, U = = 0,0777 Revertig to the origial cale, we obtai x' = ( atilog0,538) = 3,34 The lower 95 % cofidece limit = atilog(0,538 0, 0777) =,79 The upper 95 % cofidece limit = ati log(0, , 0777) = 4,00 The cofidece iterval i thu aymmetrical aroud thi weighted mea, x '. I thi cae, the coefficiet of variatio i difficult to iterpret, ad hece the followig i reported: A.3. x ' = 3,34 ( = ) 95 % cofidece iterval,79 to 4,00. Note The calculatio of value that had ot bee coverted to logarithm would have give a mea of 3,50, ad the cofidece limit,77 ad 4,3, i.e. a hift toward higher value which i reality are le repreetative of the ditributio. Required umber of meauremet If it i aumed that the data obtaied are related to a populatio with a ormal ditributio, it i poible to calculate the umber of meauremet required i order to reduce the ucertaity of the etimated mea of the populatio to le tha a give value. Coider the cae where it i required to ae the mea of the populatio with a preciio of ± 0,0 with 95 % cofidece. Prelimiary tet how that the tadard deviatio i approximately 0,5. How may tet piece hall be meaured? Table give, for f = t 5 =, 96 Calculate t5,96 0,5 = = a = 4,0 0,0 Approximately 5 meauremet hould thu be made. A.4. The igificace of the differece betwee two mea Set of 0 meauremet of a ormally ditributed variable have bee made o each of two ample of paper, a i Table A.4. Table A.4 Meauremet Sample Sample No Mea 6,0 4, Std.dev.,357,394 The tak i to determie whether thee two ample ca be regarded a havig bee draw from the ame populatio, i.e. whether the differece betwee the two mea may be due to chace, or whether there i a tatitically igificat differece betwee them. The calculated value of the mea ad tadard deviatio are how i the table. Sice the two tadard deviatio are very imilar, the combied tadard deviatio ca be ued. Calculate: ( x ) ( ) x + x x =,376 = Calculate: ( x x) t5 = + + =,694 The value of t 5 give i Table for f = ( + - ) = 8 i,0. Sice the calculated value of t 5 i le tha thi value, the differece betwee the two mea i ot igificat ad the two ample ca very well have bee take from the ame populatio. A.5. Rejectio of extreme value Twelve laboratorie have each carried out a tet o material upplied from a igle batch. Each tet coit of te meauremet ad the reult are how i Table

13 SCAN-G :07 Page 3 A.5, a the mea ad tadard deviatio for each laboratory. Table A.5 Laboratory Mea Std.dev. 5,6 3,5 54,4 3,7 3 54,8 3,3 4 55,6 5, 5 56, 3,8 6 56,8 3,0 7 57, 3,6 8 57,4 3, 9 58,6 3,6 0 60,0 3,5 6, 3,8 75,8 3,4 I the third colum of thi table, the value of the tadard deviatio reported from laboratory No 4 appear to be uuually high. To check whether thi i a tatitical outlier, calculate Cochra tatitic C C = ( 5,) = 0,67 Accordig to Table 3, the critical value for laboratorie with 0 meauremet i each tet i 0,4. Sice the value obtaied i le tha thi critical value, there i o eed to coider thi laboratory to be a outlier. I the ecod colum of Table A.5, the value of the mea reported by Laboratory No eem to be uuually high. To check whether thi i a tatitical outlier, calculate Grubb tatitic G ( 75,8 x ) G = =,874 Accordig to Table, the critical value for a outlier for laboratorie i,636. Sice the value obtaied i higher tha thi critical value, thi laboratory i a tatitical outlier ad hould be elimiated from ubequet calculatio. Table A.6 Laboratory Mea Std.dev. 5,6 3,5 54,4 3,7 3 54,8 3,3 4 55,6 5, 5 56, 3,8 6 56,8 3,0 7 57, 3,6 8 57,4 3, 9 58,6 3,6 0 60,0 3,5 6, 3,8 Mea 56, Std.dev.,688 I thi table, the mea i calculated a: x i =56,89 the mea tadard deviatio i calculated a: i = 3,695 ad the tadard deviatio of the mea i calculated a ( x ) i x = =,688 The reproducibility tadard deviatio i thu,69, ad the reproducibility limit are calculated a R =, 96 = 7,45 Note The reproducibility limit iclude a cotributio from the variatio withi the material. If it i of iteret to calculate the tadard deviatio betwee laboratorie diregardig the cotributio from the tet material, thi ca be calculated a: 3, 695 bl =,688 0 =,4 A.6 Calculatio of reproducibility limit After elimiatio of laboratory a idicated above, the data how i Table A.6 are obtaied.

14 SCAN-G :07 Page 4 SCAN-tet Method are iued ad recommeded by KCL, PFI ad STFI-Packfork for the pulp, paper ad board idutrie i Filad, Norway ad Swede. Ditributio: Secretariat, Scadiavia Pulp, Paper ad Board Tetig Committee, Box 5604, SE-4 86 Stockholm, Swede