# ISO 9276-3-2008

Reference numberISO 9276-3:2008(E)©ISO 2008INTERNATIONAL STANDARD ISO9276-3First edition2008-07-01Representation of results of particle size analysis — Part 3: Adjustment of an experimental curve to a reference model Représentation de données obtenues par analyse granulométrique — Partie 3: Ajustement d une courbe expérimentale à un modèle de référence Copyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---ISO 9276-3:2008(E) PDF disclaimer This PDF file may contain embedded typefaces. In accordance with Adobe s licensing policy, this file may be printed or viewed but shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In downloading this file, parties accept therein the responsibility of not infringing Adobe s licensing policy. The ISO Central Secretariat accepts no liability in this area. Adobe is a trademark of Adobe Systems Incorporated. Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below. COPYRIGHT PROTECTED DOCUMENT © ISO 2008 All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or ISO s member body in the country of the requester. ISO copyright office Case postale 56 • CH-1211 Geneva 20 Tel. + 41 22 749 01 11 Fax + 41 22 749 09 47 E-mail copyright@iso.org Web www.iso.org Published in Switzerland ii © ISO 2008 – All rights reservedCopyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---ISO 9276-3:2008(E) © ISO 2008 – All rights reserved iiiContents Page Foreword iv Introduction v 1 Scope . 1 2 Normative references . 1 3 Symbols and abbreviated terms . 2 4 Adjustment of an experimental curve to a reference model 3 4.1 General. 3 4.2 Quasilinear regression method. 3 4.3 Non-linear regression method. 3 5 Goodness of fit, standard deviation of residuals and exploratory data analysis 6 6 Conclusions 7 Annex A (informative) Influence of the model on the regression goodness of fit. 9 Annex B (informative) Influence of the type of distribution quantity on the regression result . 11 Annex C (informative) Examples for non-linear regression. 15 Annex D (informative) χ2-Test of number distributions of known sample size 17 Annex E (informative) Weighted quasilinear regression 20 Bibliography . 23 Copyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---ISO 9276-3:2008(E) iv © ISO 2008 – All rights reservedForeword ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing International Standards is normally carried out through ISO technical committees. Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee. International organizations, governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization. International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2. The main task of technical committees is to prepare International Standards. Draft International Standards adopted by the technical committees are circulated to the member bodies for voting. Publication as an International Standard requires approval by at least 75 % of the member bodies casting a vote. Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights. ISO 9276-3 was prepared by Technical Committee ISO/TC 24, Sieves, sieving and other sizing methods, Subcommittee SC 4, Sizing by methods other than sieving. ISO 9276 consists of the following parts, under the general title Representation of results of particle size analysis: ⎯ Part 1: Graphical representation ⎯ Part 2: Calculation of average particle sizes/diameters and moments from particle size distributions ⎯ Part 3: Adjustment of an experimental curve to a reference model ⎯ Part 4: Characterization of a classification process ⎯ Part 5: Methods of calculation relating to particle size analyses using logarithmic normal probability distribution The following part is under preparation: ⎯ Part 6: Descriptive and quantitative representation of particle shape and morphology Copyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---ISO 9276-3:2008(E) © ISO 2008 – All rights reserved vIntroduction Cumulative curves of particle size distributions are sigmoids, therefore fitting to a model distribution function or rendering statistical intercomparison is difficult. These disadvantages can, however, be remedied by transforming these sigmoids into straight lines by means of appropriate coordinate systems, e.g. log-normal, Rosin-Rammler or Gates-Gaudin-Schuhmann (log-log). Target size distributions in particle technology industries can also be described in terms of distribution models. In such systems, a classic linear regression assumes that the squares of the deviations between the experimental points and the theoretical straight line are, on average, equal. This is only valid in the transformed cumulative distribution value system, but not in their linear representation, and therefore named a quasilinear regression. In particular, the scale extension makes the values of the squares of the deviations at the extremities of the graph vary by several orders of magnitude. In addition, the sum of the squares of the deviations obtained by this method is not related to any simple distribution and does not allow any statistical test. Key Q3(x) cumulative distribution by volume or mass x particle size Y quantiles of the standard normal distribution 1 quasilinear regression full line • quasilinear fit point Q3(x) data point Figure 1 — Example of a functional paper with log-normal plot (cumulative distribution values plotted on a normal ordinate against particle size on a logarithmic abscissa with inverse standard normal distribution transformed) and quasilinear regression full line Copyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---ISO 9276-3:2008(E) vi © ISO 2008 – All rights reservedThe experimental data in Figure 1 are taken from ISO 9276-1:1998[1], Annex A and represent a sieve-measuring result example between 90 µm and 11,2 mm. The mathematical treatment, corresponding to non-linear coordinate systems, mentioned above, agrees with a quasilinear regression. Here the non-linear transformation of the Y-axis results in a non-linear transformation of the Y-deviations, e.g. another consideration of deviations at the tails of a distribution than at their centre. One possibility to compensate for the non-linear transformation of the Y-differences, in the result of the non-linear transformation of the Y values, is the introduction of weighting factors in the quasilinear regression (see Annex E). Moreover, a non-linear regression delivers the best adjustment and allows the most flexibility, such as statistical tests on number distributions, the adjustment of truncated or multimodal distributions or any other arbitrary models, but it requires a start approximation and a numerical mathematical procedure. The standard deviation of residuals between experimental points and the model in the non-transformed scale allows the quantification of the degree of alignment and the statistical comparison of experimental distributions. A value of greater than e.g. 0,05 indicates a non-adequate reference model. Copyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---INTERNATIONAL STANDARD ISO 9276-3:2008(E)© ISO 2008 – All rights reserved 1Representation of results of particle size analysis — Part 3: Adjustment of an experimental curve to a reference model 1 Scope This part of ISO 9276 specifies methods for the adjustment of an experimental curve to a reference model with respect to a statistical background. Furthermore, the evaluation of the residual deviations, after the adjustment, is also specified. The reference model can also serve as a target size distribution for maintaining product quality. This part of ISO 9276 specifies procedures that are applicable to the following reference models: a) normal distribution (Laplace-Gauss): powders obtained by precipitation, condensation or natural products (pollens); b) log-normal distribution (Galton MacAlister): powders obtained by grinding or crushing; c) Gates-Gaudin-Schuhmann distribution (bilogarithmic): analysis of the extreme values of the fine particle distributions; d) Rosin-Rammler distribution: analysis of the extreme values of the coarse particle distributions; e) any other model or combination of models, if a non-linear fit method is used (see bimodal example in Annex C). This part of ISO 9276 can substantially support product quality assurance or process optimization related to particle size distribution analysis. 2 Normative references The following referenced documents are indispensable for the application of this document. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies. ISO 9276-2, Representation of results of particle size analysis — Part 2: Calculation of average particle sizes/diameters and of moments from particle size distributions ISO 9276-5, Representation of results of particle size analysis — Part 5: Methods of calculation relating to particle size analyses using logarithmic normal probability distribution Copyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---ISO 9276-3:2008(E) 2 © ISO 2008 – All rights reserved3 Symbols and abbreviated terms a straight line intercept (equation of a straight line) b slope (gradient) of the straight regression line (equation of a straight line) d′ intercept parameter of RRSB distribution GGS (Gates-) Gaudin-Schuhmann distribution LND logarithmic normal probability distribution, defined in ISO 9276-5 n number of size classes nFdegrees of freedom, which is the number of data points, n, minus the number of fit model parameters N number of particles in the measured sample p set of model parameters, vector q density of particle size distribution Q(x) observed cumulative distribution, total of the particles finer than x, between 0 and 1 Q*(x; p) model estimation, theoretical cumulative distribution depending on the reference model with parameters, p r type of quantity of a size distribution, r = 0: number, r = 3: volume or mass RRSB Rosin-Rammler (Sperling and Bennet) distribution (derived from Weibull-distribution) s standard deviation of LND, logarithm of geometric standard deviation [ISO 9276-5] sqlmean square deviation of the quasilinear regression in the transformed scale sresstandard deviation of the residuals, square root from residual variance x particle size x50,rmedian particle size of distribution with type of quantity, r, intercept parameter of LND xmax,rintercept parameter of GGS distribution with type of quantity, r X(x) transform of x plotted on the x-axis [X = x for a normal distribution and X = ln x or lg x for a log-normal, Rosin-Rammler or bilogarithmic (log-log) distribution], X is equivalent to ξ in ISO 9276-1 and ISO 9276-5 Y(Q) transform of Q plotted on the y-axis (Y = inverse of standard normal distribution for a normal distribution, see Table 1 for other model types) Y* = a + bX general expression of the equation for the straight regression line of a model cumulative particle size distribution z dimensionless normalization variable in LND [ISO 9276-5] α slope parameter of GGS distribution ζ integration variable, based on z, in LND ν exponent of RRSB distribution ω weighting coefficient Copyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---ISO 9276-3:2008(E) © ISO 2008 – All rights reserved 34 Adjustment of an experimental curve to a reference model 4.1 General The estimation of parameters to be used in the regression equations appearing in this part of ISO 9276 are calculated from either particle size distribution values, Q, fractions of these particle size values, dQ, or density values, q. These particle size distribution parameters may also be used as parameters for other regression equations. Generally a certain distribution model Q*(x; p) = Q*(x; a,b…) should be adjusted to measuring data: [xi, Qi= Q(xi)] i = 1,., n The intention and capability of the regression equation is to find the optimum parameters p = a, b. such that the mean square deviation between measured Q values, Q(x), and the model, Q*(x; p), will be minimized: ()22*11(;) () minniiisQxQxn=⎡⎤=−⎯→⎣⎦∑ppp(1) 4.2 Quasilinear regression method The non-linear (or rather non-linear) optimization problem in Equation (1) can be transformed by Y to a linear Equation (2) for the various statistical models used in this part of ISO 9276. The values of X are the transformed particle size values obtained from any particle size distribution. Y* = Y*(Q*) = a + bX (2) The solution and optimization using a linear regression with Equation (2) in the transformed state, delivers an approximation for Equation (1), which can be replaced with the following quasilinear regression Equation (3): ()22ql11() minniisbXaQxn==+−⎯→⎡⎤⎣⎦∑pp (3) T