BS-6200-5-1997.pdf

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | BRITISH STANDARD BS 6200 : Part 5 : 1997 ICS 77.080 NO COPYING WITHOUT BSI PERMISSION EXCEPT AS PERMITTED BY COPYRIGHT LAW Sampling and analysis of iron, steel and other ferrous metals Part 5. Guidelines on statistical procedures Licensed Copy: sheffieldun sheffieldun, na, Wed Dec 06 15:09:13 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6200 : Part 5 : 1997 This British Standard, having been prepared under the direction of the Engineering Sector Board, was published under the authority of the Standards Board and comes into effect on 15 July 1997 BSI 1997 The following BSI references relate to the work on this standard: Committee reference ISE/18 Draft for comment 94/304216 DC ISBN 0 580 26455 6 Amendments issued since publication Amd. No.DateText affected Committees responsible for this British Standard The preparation of this British Standard was entrusted to Technical Committee ISE/18, Sampling and analysis of iron and steel, upon which the following bodies were represented: BCIRA British Steel Industry Department of Trade and Industry (Laboratory of the Government Chemist) Ministry of Defence The Welding Institute Licensed Copy: sheffieldun sheffieldun, na, Wed Dec 06 15:09:13 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6200 : Part 5 : 1997 BSI 1997i Contents Page Committees responsibleInside front cover Forewordv Section 1. General 1.1Scope1 1.2References1 1.3Definitions1 Section 2. Theoretical considerations 2.1Accuracy, trueness and precision2 2.2Distribution of errors3 Section 3. Principles of statistical evaluation 3.1Degrees of freedom11 3.2The t-test11 3.3Repeatability and reproducibility limits r and R11 3.4Within-laboratory reproducibility limit Rw12 3.5Standardization testing and precision testing12 3.6Regression analysis12 Section 4. Experimental design 4.1Planning13 4.2Standardization tests13 4.3Precision tests16 4.4Reporting of results16 4.5Data truncation16 4.6Rounding of results17 4.7Choice of test samples17 Section 5. Analysis of variance: method 1 5.1Effect of replication18 5.2Nomenclature18 5.3Analysis of variance applied to an example19 5.4Calculation of variance and standard deviation from accepted results22 5.5Tolerance22 Section 6. Analysis of variance: method 2 6.1Direct mathematical method23 6.2Summary of results from a test programme23 6.3Calculation of sums of squares: alternative procedure23 6.4Degrees of freedom24 6.5Table of variances24 6.6Calculation of repeatability and reproducibility25 6.7Statement of precision25 Section 7. Analysis of variance: method 3 7.1Simplified method using data transformation26 7.2Example using transformed data26 7.3Recovery of statistical results from transformed data26 Licensed Copy: sheffieldun sheffieldun, na, Wed Dec 06 15:09:13 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6200 : Part 5 : 1997 ii BSI 1997 Page Section 8. Intermediate precision 8.1Introduction28 8.2Intermediate precision conditions28 8.3Estimation of time-different intermediate precision28 8.4Formation of the data table29 8.5Calculation of sums and squares30 8.6Separating the sums of squares30 8.7Preparation of analysis of variance table, ANOVA31 8.8Summary of analysis of variance32 Section 9. F-variance ratio test 9.1Theoretical considerations33 9.2Applications for the F-statistic33 9.3Examples of the use of the F-statistic34 Section 10. The t-test 10.1 General introduction37 10.2 Comparison of two means37 10.3 Confidence limits, confidence interval38 10.4 Comparison of two methods40 10.5 Method of checking the correct application of a procedure42 10.6 Application to regression analysis42 Section 11. Tests for outliers and stragglers 11.1 Theoretical considerations43 11.2 Cochran's test for excessive variance43 11.3 Grubbs' test for outlying single results or laboratory means44 Section 12. Regression and correlation 12.1 Regression analysis46 12.2 Examples of regression calculations46 12.3 Correlation coefficient47 12.4 Confidence limits of regression lines48 12.5 Probability limits for a regression line50 Section 13. Graphical presentation 13.1 95 % confidence limits of regression lines51 13.2 New statistical tests52 13.3 Youden diagrams54 Section 14. Analytical limits 14.1 Uncertainty factors for estimated standard deviations57 14.2 Practical measurements of the r/R ratio58 14.3 Limit of detection58 14.4 Choice of factor k for limit of detection58 14.5 Limit of determination60 14.6 Precision of determination61 Section 15. Tests for a normal distribution 15.1 Introduction62 15.2 Non-normal distributions62 15.3 Use of normal probability graphs62 15.4 Example of a normal probability plot64 Licensed Copy: sheffieldun sheffieldun, na, Wed Dec 06 15:09:13 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BSI 1997iii BS 6200 : Part 5 : 1997 Page Annexes A(normative) Summary of statistical terms and symbols66 B(informative) Statistical tables71 Tables 1Nickel content determinations3 2Theoretical probabilities6 3Convergence of standard deviation8 4Distribution of 1000 normal standard deviates10 5Relationship between degrees of freedom and t11 6Example of rounding rules17 7Results for variance analysis18 8Laboratory averages, standard deviations and totals20 9Collated results for 11 laboratories (p = 11)21 10Evaluations from collated results21 11Statistical summary22 12Manganese content of five alloys23 13Sample C manganese23 14ANOVA analysis of variance24 15Precision data for five manganese alloys25 16The use of data transformations27 17Comparison of transformed data with original unmodified results27 18Data and statistical totals29 19ANOVA table (offset vanadium content)31 20F-ratios34 21Mean, repeatability and reproducibility from two standard methods35 22Variances and numbers of degrees of freedom36 23Ratios for F-test36 24Manganese content of steel bars37 25Offset equivalents for manganese content results37 26Determination of vanadium content by flame atomic absorption spectrometry39 27Significance of differences in pairs of means40 28Standard deviations and squares for Cochran's and Grubbs' tests43 29Cobalt content in steel in ascending order44 30Repeatability limits for vanadium content46 31Statistical parameters for regression analysis47 32Repeatability limits, predicted from regression equations47 33Minimum correlation against probability48 34Repeatability limits at 95 % probability based on logarithmic regression50 35Values of the uncertainty factors for estimates of repeatability and reproducibility standard deviations (and of the corresponding repeatability and reproducibility limits)57 36Probability statistics for transformed results64 B.1Critical values for the t-test (two-tailed)71 B.2Critical values for the F-variance ratio test (one-tailed) at the 0.05 significance level72 Licensed Copy: sheffieldun sheffieldun, na, Wed Dec 06 15:09:13 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6200 : Part 5 : 1997 iv BSI 1997 Page B.3Critical values for the F-variance ratio test (one-tailed) at the 0.01 significance level73 B.4Critical values for the F-variance ratio test (two-tailed) at the 0.05 significance level74 B.5Critical values for the F-variance ratio test (two-tailed) at the 0.01 significance level75 B.6Critical values for Cochran's test76 B.7Critical values for Grubbs' test77 B.8Indicators for Mandel's h and k statistics at the 1 % significance level78 B.9Indicators for Mandel's h and k statistics at the 5 % significance level79 Figures 1Results of 57 determinations of 3.19 % nickel4 2Illustration of tendency of histogram to approach normal curve as intervals decrease and replicates increase5 3Ratio of area between limits y1and y2to total area6 4Distribution of 1000 random normal deviates9 5Uncertainty of precision measure srat 0.95 probability14 6Uncertainty of precision measure sRat 0.95 probability15 7Linear and logarithmic regression lines; confidence limits49 8Logarithmic relationship between vanadium content (% (m/m) and repeatability, r, or reproducibility, R and Rw51 9Mandel's h and k statistics for precision data for vanadium53 10Vanadium analysis data55 11Youden diagram, two-sample plot56 12Frequency distribution for means two standard deviations apart59 13Frequency distribution for means 3.29 standard deviations apart60 14The calibration curve, decision limit, detection limit and capability of detection61 15Density functions and normal probability graphs of cumulative distributions63 16Vanadium analyses plotted on normal probability graph paper65 List of referencesInside back cover Licensed Copy: sheffieldun sheffieldun, na, Wed Dec 06 15:09:13 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6200 : Part 5 : 1997 BSI 1997v Foreword This Part of BS 6200 has been prepared by Technical Committee ISE/18. BS 6200 is a multipart British Standard covering all aspects of the sampling and analysis of iron, steel and other ferrous metals. A general introduction is given in BS 6200 : Part 1 : 1991, which lists the contents of all the existing and projected Parts, Sections and Subsections. This Part of BS 6200 supersedes BS 4237 : 1967, which is withdrawn. BS 4237 was based upon a report prepared on behalf of the British Iron and Steel Research Association Methods of Analysis Committee by an ad hoc panel which has been set up to examine the difficulties experienced in obtaining and expressing adequately the reproducibility of procedures used in ferrous metallurgical analysis. The report related particularly to the methods of analysis originally published as Parts of BS 1121 and later incorporated in BSI Handbook No. 19 : 1970, which has now been replaced by BS 6200. This Part of BS 6200 revises and considerably expands the contents of BS 4237, taking into account the latest developments in the application of statistical procedures to methods of analysis. It relates in particular to the provisions of BS ISO 5725. BS ISO 5725-1, -2, -3, -4 and -6 have been published; BS ISO 5725-5 is in preparation. This Part of BS 6200 is not intended to replace the general standards on statistics, but to complement them by explaining the theory and illustrating with practical worked examples the use of statistical tests in relation to the methods of analysis given in BS 6200 : Part 3 (and associated BS ENs) for the benefit of those without expert statistical knowledge. Compliance with a British Standard does not of itself confer immunity from legal obligations. Summary of pages This document comprises a front cover, an inside front cover, pages i to vi, pages 1 to 80, an inside back cover and a back cover. Licensed Copy: sheffieldun sheffieldun, na, Wed Dec 06 15:09:13 GMT+00:00 2006, Uncontrolled Copy, (c) BSI viblank Licensed Copy: sheffieldun sheffieldun, na, Wed Dec 06 15:09:13 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 6200 : Part 5 : 1997 BSI 19971 Section 1. General 1.1 Scope This Part of BS 6200 gives guidance on the statistical procedures used during the development of methods for the analysis of iron and steel, in particular those methods described in BS 6200 : Part 3 and associated BS ENs. It explains the significance of the statistical data given in these standards. This Part of BS 6200 also gives guidance on the planning of standardization and precision tests and covers some aspects of the preparation and use of certified reference materials. 1.2 References 1.2.1 Normative references This Part of BS 6200 incorporates, by dated or undated reference, provisions from other publications. These normative references are made at the appropriate places in the text and the cited publications are listed on the inside back cover. For dated references, only the edition cited applies; any subsequent amendments to or revisions of the cited publication apply to this Part of BS 6200 only when incorporated in the reference by amendment or revision. For undated references, the latest edition of the cited publication applies, together with any amendments. 1.2.2 Informative references This Part of BS 6200 refers to other publications that provide information or guidance. Editions of these publications current at the time of issue of the standard are listed on the inside back cover, but reference should be made to the latest editions. 1.3 Definitions For the purposes of this Part of BS 6200, the definitions given in BS ISO 5725 and in BS ISO 3534-1 : 1993 apply. NOTE. For ease of reference, a summary of the statistical terms and symbols used in this Part of BS 6200 is given in annex A. Licensed Copy: sheffieldun sheffieldun, na, Wed Dec 06 15:09:13 GMT+00:00 2006, Uncontrolled Copy, (c) BSI 2 BSI 1997 BS 6200 : Part 5 : 1997 Section 2. Theoretical considerations 2.1 Accuracy, trueness and precision 2.1.1 Trueness and precision The terms trueness and precision are used to describe the accuracy of a measurement method (see A.38 and A.23). Trueness refers to the closeness of agreement of the average of a large number of test results to the true result, though in practice the absolute result is seldom known, and the accepted reference value may be substituted (see A.1). Precision refers to the closeness of agreement between test results. 2.1.2 Accuracy The general term accuracy is now used to refer to trueness and precision combined. It is a measure of the displacement of a test result from the reference value due to a combination of random and systematic errors (see A.2). NOTE. The term bias (see A.6) has been in use for statistical matters for a very long time and is still used. Because it causes certain philosophical objections among members of some professions (such as medical and legal practitioners), the positive aspect is now emphasized by the use of the term trueness. 2.1.3 Trueness assessment There is no mathematical means of assessing whether values obtained are true. This has to be done by analysing samples or synthetic solutions made from elements, or compounds of known formulae, of high purity and in which the amount of the element to be determined is accurately known. Interfering elements should be absent from all reagents. One of the following alternative methods may be selected. a) A standardized method may be agreed to yield a true value. b) A reference material may be given an accepted value by consensus. 2.1.4 Reference material A reference material is defined as a material or substance one or more of whose property values are sufficiently homogeneous and well established to be used for: a) the calibration of an apparatus; b) the assessment of a measurement method; c) for assigning values to materials. NOTE. A reference material can be in the form of a pure or mixed gas, liquid or solid. Examples are pure elements or compounds as solids or in solutions of accurately known concentrations used for calibration in chemical