1、| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | BRITISH STANDARD BS 6200 : Part 5 : 1997 I
2、CS 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 proceduresBS 6200 : Part 5 : 1997 This British Standard, having been prepared under the direction of the Engineering Sector
3、 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.
4、Date Text 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 In
5、dustry (Laboratory of the Government Chemist) Ministry of Defence The Welding InstituteBS 6200 : Part 5 : 1997 BSI 1997 i Contents Page Committees responsible Inside front cover Foreword v Section 1. General 1.1 Scope 1 1.2 References 1 1.3 Definitions 1 Section 2. Theoretical considerations 2.1 Acc
6、uracy, trueness and precision 2 2.2 Distribution of errors 3 Section 3. Principles of statistical evaluation 3.1 Degrees of freedom 11 3.2 The t-test 11 3.3 Repeatability and reproducibility limits r and R 11 3.4 Within-laboratory reproducibility limit R w 12 3.5 Standardization testing and precisio
7、n testing 12 3.6 Regression analysis 12 Section 4. Experimental design 4.1 Planning 13 4.2 Standardization tests 13 4.3 Precision tests 16 4.4 Reporting of results 16 4.5 Data truncation 16 4.6 Rounding of results 17 4.7 Choice of test samples 17 Section 5. Analysis of variance: method 1 5.1 Effect
8、of replication 18 5.2 Nomenclature 18 5.3 Analysis of variance applied to an example 19 5.4 Calculation of variance and standard deviation from accepted results 22 5.5 Tolerance 22 Section 6. Analysis of variance: method 2 6.1 Direct mathematical method 23 6.2 Summary of results from a test programm
9、e 23 6.3 Calculation of sums of squares: alternative procedure 23 6.4 Degrees of freedom 24 6.5 Table of variances 24 6.6 Calculation of repeatability and reproducibility 25 6.7 Statement of precision 25 Section 7. Analysis of variance: method 3 7.1 Simplified method using data transformation 26 7.2
10、 Example using transformed data 26 7.3 Recovery of statistical results from transformed data 26BS 6200 : Part 5 : 1997 ii BSI 1997 Page Section 8. Intermediate precision 8.1 Introduction 28 8.2 Intermediate precision conditions 28 8.3 Estimation of time-different intermediate precision 28 8.4 Format
11、ion of the data table 29 8.5 Calculation of sums and squares 30 8.6 Separating the sums of squares 30 8.7 Preparation of analysis of variance table, ANOVA 31 8.8 Summary of analysis of variance 32 Section 9. F-variance ratio test 9.1 Theoretical considerations 33 9.2 Applications for the F-statistic
12、 33 9.3 Examples of the use of the F-statistic 34 Section 10. The t-test 10.1 General introduction 37 10.2 Comparison of two means 37 10.3 Confidence limits, confidence interval 38 10.4 Comparison of two methods 40 10.5 Method of checking the correct application of a procedure 42 10.6 Application to
13、 regression analysis 42 Section 11. Tests for outliers and stragglers 11.1 Theoretical considerations 43 11.2 Cochrans test for excessive variance 43 11.3 Grubbs test for outlying single results or laboratory means 44 Section 12. Regression and correlation 12.1 Regression analysis 46 12.2 Examples o
14、f regression calculations 46 12.3 Correlation coefficient 47 12.4 Confidence limits of regression lines 48 12.5 Probability limits for a regression line 50 Section 13. Graphical presentation 13.1 95 % confidence limits of regression lines 51 13.2 New statistical tests 52 13.3 Youden diagrams 54 Sect
15、ion 14. Analytical limits 14.1 Uncertainty factors for estimated standard deviations 57 14.2 Practical measurements of the r/R ratio 58 14.3 Limit of detection 58 14.4 Choice of factor k for limit of detection 58 14.5 Limit of determination 60 14.6 Precision of determination 61 Section 15. Tests for
16、 a normal distribution 15.1 Introduction 62 15.2 Non-normal distributions 62 15.3 Use of normal probability graphs 62 15.4 Example of a normal probability plot 64 BSI 1997 iii BS 6200 : Part 5 : 1997 Page Annexes A (normative) Summary of statistical terms and symbols 66 B (informative) Statistical t
17、ables 71 Tables 1 Nickel content determinations 3 2 Theoretical probabilities 6 3 Convergence of standard deviation 8 4 Distribution of 1000 normal standard deviates 10 5 Relationship between degrees of freedom and t 11 6 Example of rounding rules 17 7 Results for variance analysis 18 8 Laboratory a
18、verages, standard deviations and totals 20 9 Collated results for 11 laboratories (p = 11) 21 10 Evaluations from collated results 21 11 Statistical summary 22 12 Manganese content of five alloys 23 13 Sample C manganese 23 14 ANOVA analysis of variance 24 15 Precision data for five manganese alloys
19、 25 16 The use of data transformations 27 17 Comparison of transformed data with original unmodified results 27 18 Data and statistical totals 29 19 ANOVA table (offset vanadium content) 31 20 F-ratios 34 21 Mean, repeatability and reproducibility from two standard methods 35 22 Variances and number
20、s of degrees of freedom 36 23 Ratios for F-test 36 24 Manganese content of steel bars 37 25 Offset equivalents for manganese content results 37 26 Determination of vanadium content by flame atomic absorption spectrometry 39 27 Significance of differences in pairs of means 40 28 Standard deviations a
21、nd squares for Cochrans and Grubbs tests 43 29 Cobalt content in steel in ascending order 44 30 Repeatability limits for vanadium content 46 31 Statistical parameters for regression analysis 47 32 Repeatability limits, predicted from regression equations 47 33 Minimum correlation against probability
22、 48 34 Repeatability limits at 95 % probability based on logarithmic regression 50 35 Values of the uncertainty factors for estimates of repeatability and reproducibility standard deviations (and of the corresponding repeatability and reproducibility limits) 57 36 Probability statistics for transfor
23、med results 64 B.1 Critical values for the t-test (two-tailed) 71 B.2 Critical values for the F-variance ratio test (one-tailed) at the 0.05 significance level 72BS 6200 : Part 5 : 1997 iv BSI 1997 Page B.3 Critical values for the F-variance ratio test (one-tailed) at the 0.01 significance level 73
24、B.4 Critical values for the F-variance ratio test (two-tailed) at the 0.05 significance level 74 B.5 Critical values for the F-variance ratio test (two-tailed) at the 0.01 significance level 75 B.6 Critical values for Cochrans test 76 B.7 Critical values for Grubbs test 77 B.8 Indicators for Mandels
25、 h and k statistics at the 1 % significance level 78 B.9 Indicators for Mandels h and k statistics at the 5 % significance level 79 Figures 1 Results of 57 determinations of 3.19 % nickel 4 2 Illustration of tendency of histogram to approach normal curve as intervals decrease and replicates increase
26、 5 3 Ratio of area between limits y 1 and y 2 to total area 6 4 Distribution of 1000 random normal deviates 9 5 Uncertainty of precision measure s r at 0.95 probability 14 6 Uncertainty of precision measure s R at 0.95 probability 15 7 Linear and logarithmic regression lines; confidence limits 49 8
27、Logarithmic relationship between vanadium content (% (m/m) and repeatability, r, or reproducibility, R and R w 51 9 Mandels h and k statistics for precision data for vanadium 53 10 Vanadium analysis data 55 11 Youden diagram, two-sample plot 56 12 Frequency distribution for means two standard deviat
28、ions apart 59 13 Frequency distribution for means 3.29 standard deviations apart 60 14 The calibration curve, decision limit, detection limit and capability of detection 61 15 Density functions and normal probability graphs of cumulative distributions 63 16 Vanadium analyses plotted on normal probab
29、ility graph paper 65 List of references Inside back coverBS 6200 : Part 5 : 1997 BSI 1997 v 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
30、. 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 Ste
31、el 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 an
32、alysis 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 p
33、rocedures 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 explainin
34、g 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 conf
35、er 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.vi blankBS 6200 : Part 5 : 1997 BSI 1997 1 Section 1. General 1.1 Scope This Part of BS 6200 gives guidance on the st
36、atistical 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
37、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 ref
38、erences 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
39、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 a
40、t 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 th
41、e statistical terms and symbols used in this Part of BS 6200 is given in annex A.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 measurem
42、ent 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
43、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
44、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 a
45、ssessment 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. Inte
46、rfering 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 a
47、s 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 pur
48、e 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 analysis. 2.1.5 Certified reference material A certified reference material, which is accompanied by a certificate, has one or more p
49、roperties certified by a procedure which establishes its traceability to an accurate realization of the unit in which the property values are expressed, and for which each certified value is accompanied by an uncertainty at a stated level of confidence. Testing for stoichiometry should always be part of the development of a new method where the primary aim is accuracy. 2.1.6 Repeatability limit The term repeatability limit r is used to denote the statistical result derived from interlaboratory precision experiments using the same method. It is often called repeatabilit
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