ASME PTC 9.1 -201 (Revision of ASME PTC 19.1-2005) Test Uncertainty Performance Test Codes A N A M E R I C A N N AT I O N A L S TA N D A R D ASME PTC 19.1-2013 (Revision of ASME PTC 9.1 -2005) Test Uncertainty Performance Test Codes AN AM ERI CAN N AT I O N A L S TA N D A R D Two Park Avenue • New York, NY • 001 USA Date of Issuance: May 30, 201 This Code will be revised when the Society approves the issuance of a new edition ASME issues written replies to inquiries concerning interpretations of technical aspects of this Code Interpretations are published on the Committee Web page under go.asme.org/InterpsDatabase Periodically certain actions of the ASME PTC Committee may be published as Cases Cases are published on the ASME Web site under the PTC Committee Pages at go.asme.org/PTCcommittee as they are issued Errata to codes and standards may be posted on the ASME Web site under the Committee Pages to provide corrections to incorrectly published items, or to correct typographical or grammatical errors in codes and standards Such errata shall be used on the date posted The PTC Committee Page can be found at go.asme.org/PTCcommittee There is an option available to automatically receive an e-mail notification when errata are posted to a particular code or standard This option can be found on the appropriate Committee Page after selecting “Errata” in the “Publication Information” section ASME is the registered trademark of The American Society of Mechanical Engineers This code or standard was developed under procedures accredited as meeting the criteria for American National Standards The Standards Committee that approved the code or standard was balanced to assure that individuals from competent and concerned interests have had an opportunity to participate The proposed code or standard was made available for public review and comment that provides an opportunity for additional public input from industry, academia, regulatory agencies, and the public-at-large ASME does not “approve,” “rate,” or “endorse” any item, construction, proprietary device, or activity ASME does not take any position with respect to the validity of any patent rights asserted in connection with any items mentioned in this document, and does not undertake to insure anyone utilizing a standard against liability for infringement of any applicable letters patent, nor assumes any such liability Users of a code or standard are expressly advised that determination of the validity of any such patent rights, and the risk of infringement of such rights, is entirely their own responsibility Participation by federal agency representative(s) or person(s) affiliated with industry is not to be interpreted as government or industry endorsement of this code or standard ASME accepts responsibility for only those interpretations of this document issued in accordance with the established ASME procedures and policies, which precludes the issuance of interpretations by individuals No part of this document may be reproduced in any form, in an electronic retrieval system or otherwise, without the prior written permission of the publisher The American Society of Mechanical Engineers Two Park Avenue, New York, NY 001 6-5990 Copyright © 201 by THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS All rights reserved Printed in U.S.A CONTENTS Notice Foreword Committee Roster Correspondence With the PTC Committee Section Introduction 1-1 General 1-2 Uncertainty Classifications 1-3 Advantages of Using This Code 1-4 Applications 1-5 Disclaimer Section Object and Scope 2-1 Object 2-2 Scope Section Nomenclature and Glossary 3-1 Nomenclature 3-2 Glossary Section Fundamental Concepts 4-1 Assumptions 4-2 Measurement Error 4-3 Measurement Uncertainty 4-4 Pretest and Post-Test Uncertainty Analyses Section Defining the Measurement Process 5-1 Overview 5-2 Selection of the Appropriate “True Value” 5-3 Identification of Error Sources 5-4 Categorization of Uncertainties 5-5 Comparative Testing Section Uncertainty of a Measurement 6-1 Random Standard Uncertainty of the Mean 6-2 Systematic Standard Uncertainty of a Measurement 6-3 Classification of Uncertainty Sources 6-4 Combined Standard and Expanded Uncertainty of a Measurement Section Uncertainty of a Result 7-1 Propagation of Measurement Uncertainties Into a Result 7-2 Sensitivity 7-3 Random Standard Uncertainty of a Result 7-4 Systematic Standard Uncertainty of a Result 7-5 Combined Standard Uncertainty and Expanded Uncertainty of a Result Section Additional Uncertainty Considerations 8-1 Correlated Errors 8-2 Nonsymmetric Systematic Uncertainty 8-3 Fossilization of Calibrations 8-4 Spatial Variations 8-5 Analysis of Redundant Means 8-6 Regression Uncertainty 8-7 Simplified Uncertainty Analysis for Calibrations iii vi vii viii ix 1 1 3 4 6 6 11 13 13 13 13 14 15 17 17 18 18 19 21 21 21 22 22 23 24 24 28 32 33 34 35 37 Step-by-Step Calculation Procedure 10-1 10-2 10-3 10-4 10-5 10-6 General Considerations Calculation Procedure Examples Combined Cycle Performance Uncertainty Taylor Series Propagation for Random Errors That Are Not Independent Flow Measurement Using Pitot Tubes Post-Test Uncertainty Analysis Example: HRSG High Pressure Steam Flow Humidity Measurement Periodic Comparative Testing References Additional References 40 40 40 42 42 52 54 59 62 71 79 81 4-2-1 4-2-2 4-3.1-1 4-3.3-1 5-3.1-1 5-4.3-1 8-2.1-1 8-2.1-2 8-2.1-3 8-2.1-4 8-5-1 10-1.1-1 10-3.1-1 10-5.1-1 10-5.3.3-1 10-6.1-1 10-6.1-2 10-6.1-3 10-6.2-1 Illustration of Measurement Errors Measurement Error Components Population Distribution Uncertainty Interval Generic Measurement Calibration Hierarchy “Within” and “Between” Sources of Data Scatter Gaussian Distribution for Nonsymmetric Systematic Errors Rectangular Distribution for Nonsymmetric Systematic Errors Triangular Distribution for Nonsymmetric Systematic Errors Triangular Distribution of Temperatures Three Cases of Redundant Means Train Thermal Performance Test Four Circumferential and Ten Radial Traverse Locations Schematic of Moisture Absorption and Removal System Example for Estimating Sensitivity From a Chart Installed Arrangement Pump Design Curve With Factory and Field Test Data Shown Comparison of Test Results With Independent Control Conditions Comparison of Test Results Using the Initial Field Test as the Control 7 12 14 16 29 29 29 31 34 43 55 67 70 72 73 73 77 6-4.1-1 6-4.1-2 8-1 8-2.1-1 8-2.1-2 8-6.4.5-1 9-2-1 9-2-2 10-1.6-1 10-1.6-2 10-1.6-3 10-1.6-4 Circulating Water-Bath Temperature Measurements Systematic Standard Uncertainty of Average Circulating Water-Bath Temperature Measurement Burst Pressures Expressions for q for the Gaussian, Rectangular, and Triangular Distributions in Figs 8-2.1-1 Through 8-2.1-3 Systematic Standard Uncertainties, b , for the Gaussian, Rectangular, and Triangular Distributions in Figs 8-2.1-1 Through 8-2.1-3 Systematic Standard Uncertainty Components for Yˆ Determined From Regression Equation Table of Data, Independent Parameters Summary of Data, Calculated Result Combined Cycle Net Facility Electrical Output Uncertainty Including the Correlated Uncertainties Combined Cycle Net Facility Heat Rate Uncertainty Including the Correlated Uncertainties Combined Cycle Net Facility Electrical Output Uncertainty Excluding the Correlated Uncertainties Combined Cycle Net Facility Heat Rate Uncertainty Excluding the Correlated Uncertainties 20 20 25 29 30 37 41 41 48 49 50 51 Section 9-1 9-2 Section 10 Section 11 Section 12 Figures Tables ? X ns iv 10-2.2-1 10-2.5-1 10-3.2-1 10-3.3-1 10-3.3-2 10-3.6-1 10-3.7.1-1 10-3.7.1-2 10-3.9-1 10-4.3.2-1 10-4.3.3-1 10-4.4-1 10-4.5-1 10-4.5-2 10-4.5-3 10-5.2-1 10-5.4-1 10-5.5-1 10-6.1-1 10-6.1-2 10-6.2-1 10-6.2-2 10-6.2-3 10-6.3-1 10-6.3-2 Test Pressure Data for an Orifice Measuring Flow in a Pipe Individual Delta Pressure Measurements Average Values at Each Traverse Circumferential and Radial Location Standard Deviations at Each Traverse Circumferential and Radial Location Summarized Average Circumferential Velocities and Grand Average Velocity Random Standard Uncertainty for Both Average Circumferential Velocities and Grand Average Velocity Systematic Standard Uncertainty for Average Circumferential Velocities Systematic Standard Uncertainty Due to Instrumentation Sources for Grand Average Circumferential Velocities Components of Uncertainty in the Grand Average Pipe Velocity Identification of Parameters Calculated Sensitivities of Each Parameter Random Standard Uncertainty Estimated for Each Parameter Systematic Standard Uncertainty Estimated for Each Parameter Identified Elemental Uncertainty Components and Values That Are Assumed to Be the Same for Each Parameter Combined Standard Uncertainty and Expanded Uncertainty of the Result Measured Values and Statistical Properties Estimated Values, Standard Uncertainties, and Calculated Sensitivities Standard Uncertainties, Combined Standard Uncertainty, and Expanded Uncertainty of the Result Pump Design Data ( Tc p 20°C) Summary of Test Results Uncertainty Propagation for Comparison With Independent Control Summary Uncertainties Summary of Results for Each Test Uncertainty Propagation for Comparative Uncertainty Analysis Correlated Terms for Comparative Uncertainty Analysis Nonmandatory Appendices A B C D E Statistical Considerations Guidelines for Degree of Freedom and Confidence Intervals Propagation of Uncertainty Through Taylor Series The Central Limit Theorem General Regression Uncertainty v 52 54 56 56 56 57 58 58 60 61 63 64 65 65 66 68 70 72 72 72 76 76 76 77 78 83 93 95 99 100 NOTICE All Performance Test Codes must adhere to the requirements of ASME PTC , General Instructions The following information is based on that document and is included here for emphasis and for the convenience of the user of the Code It is expected that the Code user is fully cognizant of Sections and of ASME PTC and has read them prior to applying this Code ASME Performance Test Codes provide test procedures that yield results of the highest level of accuracy consistent with the best engineering knowledge and practice currently available They were developed by balanced committees representing all concerned interests and specify procedures, instrumentation, equipment-operating requirements, calculation methods, and uncertainty analysis When tests are run in accordance with a Code, the test results themselves, without adjustment for uncertainty, yield the best available indication of the actual performance of the tested equipment ASME Performance Test Codes not specify means to compare those results to contractual guarantees Therefore, it is recommended that the parties to a commercial test agree before starting the test and preferably before signing the contract on the method to be used for comparing the test results to the contractual guarantees It is beyond the scope of any Code to determine or interpret how such comparisons shall be made vi FOREWORD In March 1979 the Performance Test Codes Supervisory Committee activated the PTC 19.1 Committee to revise a 1969 draft of a document entitled PTC 19.1 “General Considerations.” The PTC 19.1 Committee proceeded to develop a Performance Test Code Instruments and Apparatus Supplement which was published in 1985 as PTC 19.1-1985, “Measurement Uncertainty,” and which was intended, along with its subsequent editions, to provide a means of eventual standardization of nomenclature, symbols, and methodology of measurement uncertainty in ASME Performance Test Codes Work on the revision of the original 1985 edition began in 1991 The two-fold objective was to improve the usefulness to the reader regarding clarity, conciseness, and technical treatment of the evolving subject matter, as well as harmonization with the ISO “Guide to the Expression of Uncertainty in Measurement.” That revision was published as PTC 19.1-1998, “Test Uncertainty,” the new title reflecting the appropriate orientation of the document The effort to update the 1998 revision began immediately upon completion of that document The 2005 revision was notable for the following significant departures from the 1998 text: (a) Nomenclature adopted was more consistent with the ISO Guide Uncertainties remained conceptualized as “systematic” (estimate of the effects of fixed error not observed in the data), and “random” (estimate of the limits of the error observed from the scatter of the test data) Both types of uncertainty were defined at the standard-deviation level as “standard uncertainties.” The determination of an uncertainty at some level of confidence was based on the root-sumsquare of the systematic and random standard uncertainties multiplied times the appropriate expansion factor for the desired level of confidence (usually “2” for 95%) This same approach was used in the 1998 revision but the characterization of uncertainties at the standard-uncertainty level (“standard deviation”) was not as explicitly stated The new nomenclature was expected to render PTC 19.1-2005 and subsequent revisions more acceptable at the international level (b) There was greater discussion of the determination of systematic uncertainties (c) Text was added on a simplified approach to determine the uncertainty of straight-line regression The preparation of this 2013 revision began immediately upon publication of PTC 19.1-2005 The main distinguishing characteristics of this revision compared to its immediate predecessor are the following: O The most significant new feature of this revision is the inclusion of several new examples in Section 10; these were presented in a simplified format O The methodology is given, in subsection 8-1, for including the effects of correlated random errors in the uncertainty determination O Subsection 8-2 on nonsymmetric systematic uncertainty has been updated to include distributions other than Gaussian O A simplified uncertainty analysis for calibrations is presented in subsection 8-7 O A new Nonmandatory Appendix E is included covering general regression uncertainty It is much more comprehensive than the treatment of this subject in the earlier versions of this Code ASME PTC 19.1-2013 was approved by the PTC Standards Committee on October 15, 2013, and was approved as an American National Standard by the ANSI Board of Standards Review on November 26, 2013 vii ASME PTC COMMITTEE Performance Test Codes (The following is the roster of the Committee at the time of approval of this Code.) STANDARDS COMMITTEE OFFICERS P G Albert, Chair J W Milton, Vice Chair J H Karian, Secretary STANDARDS COMMITTEE PERSONNEL S P Nuspl, Consultant R R Priestley, Consultant S A Scavuzzo, The Babcock & Wilcox Co T C Heil, Alternate, The Babcock & Wilcox Co J A Silvaggio, Jr., Siemens Demag Delaval Turbomachinery, Inc W G Steele, Mississippi State University T L Toburen, T2E3 G E Weber, Midwest Generation EME, LLC W C Wood, Duke Energy R L Bannister, Honorary Member, Consultant W O Hays, Honorary Member, Consultant R Jorgensen, Honorary Member, Consultant F H Light, Honorary Member, Consultant P M McHale Honorary Member, McHale & Associates, Inc R E Sommerlad, Honorary Member, Consultant P G Albert, General Electric Co R P Allen, Consultant J M Burns, Burns Engineering W C Campbell, True North Consulting, LLC M J Dooley, Alstom Power G J Gerber, Consultant P M Gerhart, University of Evansville R E Henry, Sargent & Lundy J H Karian, The American Society of Mechanical Engineers D R Keyser, Survice Engineering T K Kirkpatrick, McHale & Associates, Inc S J Korellis, EPRI M P McHale, McHale & Associates, Inc J W Milton, Chevron Global Power Co PTC 19.1 COMMITTEE — TEST UNCERTAINTY R H Dieck, Chair, Ron Dieck Associates, Inc W G Steele, Vice Chair, Mississippi State University G Osolsobe, Secretary, The American Society of Mechanical Engineers J F Bernardin, Pratt & Whitney W G Davis, McHale & Associates, Inc R S Figliola, Clemson University N E Prudencio, Quanta Renewable Energy Services M Soltani, Bechtel Power Corp M J Wilson, Aerojet Rocketdyne viii ASME PTC 19.1-2013 Table A-3.3-1 Example of Use of Modified Thompson Method 26 −79 58 24 1 03 −1 21 −220 −1 −1 37 20 24 29 −38 25 −60 48 −52 −21 12 −56 89 −29 −1 07 20 −40 40 10 66 26 −72 79 41 27 −35 334 −555 GENERAL NOTE: 334 an d −555 are suspected outliers To illustrate the calculations for determining whether −555 is an outlier, the following steps are taken: ? ) p 1.1 Mean ( X Sample standard deviation ( S X) p 141 Sample size ( N) p 40 By the above equation for ? , ? p ? −555 − 1.1 ? p random standard uncertainties to the combined standard uncertainty in descending order of size Derivation of the percentage contribution of an elemental source of standard uncertainty to the combined uncertainty is computed as the ratio of the square of the combined standard uncertainty that would be computed if the elemental source were the only source of standard uncertainty to the square of the combined standard uncertainty computed when accounting for all sources of uncertainty Using the uncertainty model, expressions for the percentage contributions from elemental sources of standard uncertainty were derived and are presented below The percentage contribution of each elemental source of random standard uncertainty ( s X? i) to the combined standard uncertainty ( u R) is 556 From Table A-3.2-1, ?S X p 1.924 ? 141 p 271 Since ? > ?S X, we conclude that −555 is a possible outlier Repeating the above procedure for 334, ? p ? 334 − 1.1 ? p 333 (? is X? i) Since ? exceeds ?S X for the suspected point, 334, we conclude that 334 also is a possible outlier This procedure should be repeated for all remaining data points u R2 ? 100 The percentage contribution of each elemental source of systematic standard uncertainty ( bX? i) to the combined standard uncertainty is A-4 PARETO DIAGRAMS A-4.1 General (? ib X? i) u R2 It is often useful to display the relative sizes of the components of a whole with a bar chart One particular type of bar chart is called a Pareto diagram after Vilfredo Pareto, an Italian economist who used this type of diagram in his studies of the unequal distribution of wealth Most of the uses today, with extensive activity in the area of quality control, are attributed to Joseph Juran who defined the general principle known as the “Pareto Principle” — the “Vital Few, Trivial Many.” Mathematics were developed that described the distribution, but for the purposes illustrated here, the diagram can be defined as a bar chart with the bars arranged in descending order of size In order to apply this to a test uncertainty example, the first step is to define the individual systematic and random standard uncertainties of the mean in terms of their relative individual percentage contributions to the combined standard uncertainty of a test result, u R The second step is to create a bar chart that depicts the percentage contributions of individual systematic and ? 100 The percentage contribution of each correlated source of systematic standard uncertainty ( bX? X? 2) to the combined standard uncertainty is 2? ? 2bX? uR ? X 2 ? 100 A-4.2 Example A compressor performance example is used to illustrate the application of Pareto diagrams The values for parameter sensitivities, systematic standard uncertainties, random standard uncertainties, and correlated standard uncertainty are summarized in Table A-4.2-1 For bP? : b P? % contribution ~ to u R p p 90 (−0.0409 ? p 0.021) (0.00386) 5.0% ? ( ? P1 bP? ) u R2 100 ? 100 ASME PTC 19.1-2013 Table A-4.2-1 Values for Parameter Sensitivities, Systematic Standard Uncertainties, Random Standard Uncertainties, and Correlated Standard Uncertainty Symbol Value bP bP bT bT bT T sP sP 0.021 0.1 98 0.646 0.792 0.0625 0.030 0.1 70 sT sT 0.300 0.600 −0.0409 0.00629 0.00367 −0.00203 0.00386 ?P ?P ?T ?T uR Similarly: Similarly: b P? p 10.4%, b T? p 37.7%, b T? p s P? 17.3% % contribution ~ to u R ? T p p ? p ? T? ? T? 2bT? uR ? T 2 2(0.00367)(−0.00203)(0.0625) (0.00386) ? ? 100 ? 100 ? 100 −6.3% For s P? : s P? % contribution ~ to u R p p p (−0.0409 ? 0.030) (0.00386) 10.1% ? ( ? P? s P? ) u R2 p 7%, s T? p %, s T? p 0% Figure A-4.2-1 illustrates the relative contributions to combined standard uncertainty of individual-parameter systematic and random standard uncertainties in terms of a Pareto chart As can be seen from Fig A-4.2-1, the combination of bT? , bT? , and b T? T? is the largest contributor to the com1 2 bined standard uncertainty, u R, and bP? the smallest The ideal end result of an analysis such as this would be to take corrective action, if possible, to reduce the contribution of major factors in the combined standard uncertainty through changes in methods, instrumentation, or both Although this application of Pareto diagrams has been used to determine the relative contributions to combined standard uncertainty of systematic and random standard uncertainties, the method can be applied just as easily to the individual estimates of the elemental errors that contribute to systematic and random standard uncertainties For bT? T? 2: b T? 100 91 ASME PTC 19.1-2013 Percentage of Combined Standard Uncertainty Fig A-4.2-1 Pareto Chart for Random and Systematic Standard Uncertainties 40.0 35.0 30.0 25.0 20.0 5.0 0.0 37.7 7.3 0.4 0.1 0.0 8.1 7.7 5.0 0.0 -5.0 -1 0.0 bT1 5.0 -6.3 bT2 b P2 s P1 sT2 92 sT1 s P2 b P1 bT1 T2 Uncertainties ASME PTC 19.1-2013 Nonmandatory Appendix B Guidelines for Degree of Freedom and Confidence Intervals This Nonmandatory Appendix describes how to accommodate any degrees of freedom and/or any desired confidence level in the ASME PTC 19.1, Test Uncertainty, model This methodology was first developed by the ISO as described in [2] It is presented here in terms of systematic and random uncertainties Note that to avoid subscript confusion, the X has been dropped from b and from s throughout this Nonmandatory Appendix X The estimate of the random error for a parameter is the random standard uncertainty of the mean, or the estimate of the error associated with repeated measurements of a particular parameter The random standard uncertainty of the mean for each parameter is determined from N measurements as i X si p ?Ni 1⁄ (B-1.2) ? i (B-1.3) j Ni i i R p R?X1 ,X2, .,X (B-1.4) I? The ISO Guide combined standard uncertainty of the result adapted to PTC 19.1 nomenclature is u 2R I I −1 I I 2+2 p? ?? b ?p p?+ ? ? b + ?p ?? s p1 i i? i i where k i k ik i ?i p ∂R ∂ Xi i i i? (B-1.5) (B-1.6) The first two terms on the right side of eq (B-1.5) represent the systematic standard uncertainty of the result, b [eq (7-4.1) including the correlated terms], and the third term is the random standard uncertainty of the result, s [eq (7-3.1)] The covariance of the random errors is assumed to be zero The covariance of the systematic errors, b , is determined by summing the products of the elemental systematic standard uncertainties for parameters i and k that arise from the same source and are therefore perfectly correlated [11] (see subsection 8-1) In order to obtain the expanded uncertainty in the result, U , at a specified confidence level, the ISO Guide recommends that the combined standard uncertainty of the result be multiplied by a coverage factor The coverage factor is the value from the t distribution for the R R ik i i k j For the case of a single measurement (N p 1), previous information must be used to calculate s [4] Consider an experimental result that is determined from I measured variables as ik i ?p X j i i k i j Xp i i i i? i ik K i N The uncertainty model requires estimates of the uncertainties for each of the elemental error sources for each parameter in the data reduction equation These estimates are combined to calculate the band about the experimental result where the true result is thought to lie with C% confidence The estimates of the elemental errors fall into two categories: (1) systematic standard uncertainties, b , for the systematic errors, and (2) random standard uncertainties of the mean, s , for the random errors, where i represents each parameter The systematic standard uncertainties are related to those systematic errors that remain after all calibration corrections are made Systematic uncertainties can be estimated through manufacturer information, calibrations, and, in most cases, through sound engineering judgment There will usually be a set, K , of elemental systematic standard uncertainties for each parameter, i As discussed in para 4-3.2, each elemental estimate, b , is taken to be the prediction of the standard deviation for a particular distribution of possible errors for that particular error source Typically, these error distributions are assumed to be Gaussian (normally distributed) or rectangular (uniformly distributed) For most engineering predictions of systematic uncertainty, the 95% limits of the possible distribution are estimated rather than the standard deviation of the distribution Obtaining the standard uncertainty from the 95% estimate is simply a matter of dividing the estimate by the appropriate distribution factor [i.e., 2.0 for Gaussian, 1.65 p (1.73)(0.95) for rectangular] The systematic standard uncertainty for parameter i, b , is determined from the estimates for the K elemental error sources for that parameter as p ?p b2 N where B-1 GENERAL UNCERTAINTY ANALYSIS MODEL b 2i ?X − X ?N − ? p1 R (B-1.1) 93 ASME PTC 19.1-2013 Table B-1 Values for Two-Sided Confidence Interval Student’s t Distribution [9] 0.900 0.950 C 0.990 6.31 2.920 2.353 2.1 32 2.01 2.706 4.303 3.1 82 2.776 2.571 63.657 9.925 5.841 4.604 4.032 10 943 895 860 833 81 2.447 2.365 2.306 2.262 2.228 3.707 3.499 3.355 3.250 3.1 69 4.31 4.029 3.833 3.690 3.581 5.959 5.408 5.041 4.781 4.587 11 12 13 14 15 796 782 771 761 753 2.201 2.1 79 2.1 60 2.1 45 2.1 31 3.1 06 3.055 3.01 2.977 2.947 3.497 3.428 3.372 3.326 3.286 4.437 4.31 4.221 4.1 40 4.073 16 17 18 19 20 746 740 734 729 725 2.1 20 2.1 2.1 01 2.093 2.086 2.921 2.898 2.878 2.861 2.845 3.252 3.223 3.1 97 3.1 74 3.1 53 4.01 3.965 3.922 3.883 3.850 21 22 23 24 25 721 71 71 71 1 708 2.080 2.074 2.069 2.064 2.060 2.831 2.81 2.807 2.797 2.787 3.1 35 3.1 3.1 04 3.090 3.078 3.81 3.792 3.768 3.745 3.725 26 27 28 29 30 706 703 701 699 697 2.056 2.052 2.048 2.045 2.042 2.779 2.771 2.763 2.756 2.750 3.067 3.057 3.047 3.038 3.030 3.707 3.690 3.674 3.659 3.646 40 60 20 684 671 658 645 2.021 2.000 980 960 2.704 2.660 2.61 2.576 2.971 2.91 2.860 2.807 3.551 3.460 3.373 3.291 v ? 0.995 0.999 27.321 4.089 7.453 5.598 4.773 636.61 31 598 2.924 8.61 6.869 where ?si is either ? bi ?R p ? i? p1 I ?p ? i ? ? is i ? ?si + ? ? is i? ? K i + ?p k ? ? ? ib i ? k p ?bi ? bi k −2 k (B-1.9) ? where the quantity in parentheses is an estimate of the relative variability of the estimate of b ik For instance, if one thought that the estimate of bik was reliable to within ±25%, then ? bi k p −2 ? 0.25 ? p (B-1.10) With ?R known, the proper t value is obtained from Table B-1 for C% confidence and multiplied by u R from eq (B-1.5) to obtain the overall uncertainty in the result, UR, at a C% confidence level UR,C +2 I− p I k I ?p p? i I ? tC ? ? ? ib i? ip i + ? i? kb ik + ?p i ? ? is i? (B-1.11) 1⁄ ? B-2 LARGE SAMPLE UNCERTAINTY ANALYSIS APPROXIMATION The method described in subsection B-1 is the strict ISO method It has been shown [34–36] that for most engineering applications, when the degrees of freedom for the result from eq (B-1.7) is or greater, t95 can be taken as to a good approximation (93% to 95% coverage) Therefore, for large degrees of freedom in the result UR, 95 +2 I− p I I ?p p? i I ? 2? ? ? ib i? ip 1 k i + ? i? kb ik + ?p i ? ? is i? (B-2.1) 1⁄ ? The first two terms in the brackets in eq (B-2.1) are the systematic standard uncertainty of the result, b R [eq (7-3.1) including the correlated systematic standard uncertainty terms], and the third term in the brackets is the random standard uncertainty of the result, s R [eq (7-3.1)] The large sample uncertainty expression given in eq (7-5.2) is then obtained as (including the correlated systematic uncertainty terms) 2 (B-1.8) k required confidence level corresponding to the effective degrees of freedom in the result, ?R The values for t are given in Table B-1 To find ? R , the Welch-S atterthwaite formula is adapted as ? ? ? ib i? Ni − or the degrees of freedom of the previous information if s i is estimated [4] The degrees of freedom of the elemental systematic standard uncertainties, ?bi , may be known from previous information or estimated The ISO Guide recommends the approximation GENERAL NOTE: Given are the values of t for a fiden ce level, C, and number of degrees of freedom, v I p ?si (B-1.7) k ?bi ? UR, 95 k 94 p ?b2R + s 2R? 1⁄ (B-2.2) ASME PTC 19.1-2013 Nonmandatory Appendix C Propagation of Uncertainty Through Taylor Series The expected value of a function ƒ(X1 , X2, ,Xn) of the random variables X1 , X2, , Xn is given by C-1 INTRODUCTION Experimental results are not always directly measured It is quite common for an experimental result, r(X1 , X2, , Xn), to be defined as a function of certain variables, X1 , X2, , Xn, that are directly measured The aim of this Appendix is to provide a method by which the variance of an experimental result that is not directly measured, r(X1 , X2, ,Xn), can be expressed in terms of the variances and covariances of its arguments, X1 , X2, , Xn, which are directly measured The approach will be to relate the deviations in r(X1 , X2, , Xn) to deviations in (X1 , X2, ,Xn) by means of a first order approximation to the Taylor series expansion for r(x1 , x2, , xn) in the neighborhood of the point (?X1 , ?X2, , ?Xn), where ?Xi is the true value of the measured variable Xi In order to facilitate this project, the function r(x1 , x2, , xn) will be assumed to be continuous with continuous partial derivatives in the neighborhood of the point (?X1 , ?X2, , ?Xn) ? ? E?ƒ?X1 ,X2,… ,Xn? ? p ? ? … −? −? ? ? −? ƒ ? x1 , x , … , xn p?x1 ,x2,… ,xn dx1 dx2 … dxn ? ? where p(x1 , x2,… , xn) is the joint probability density function for X1 , X2,… , Xn The covariance ?X1 X2 of the random variables X1 , X2 presents the special case where ƒ(X1 , X2) p (X1 − ?X1 )(X2 − ?X2), i.e., ? ? E ?ƒ?X1 ,X2? ? p ? ? ?x1 − ?X1 ? −? −? ? x2 − ? X2 ? p ? x1 , x2 ? dx1 dx2 p ? X1 X2 C-3 PRELIMINARY CONSIDERATIONS If any random variable, Y, can be expressed as a linear combination of random variables, Xi, then the mean and variance of Y can be expressed in terms of the means, variances, and covariances of the variables Xi Suppose ? p a0 + a1 X1 + a2X2 + … + anXn Then E ?? p E ?a0 + a1 X1 + a2X2 + … + anXn or, using the definition of the mean value of a random variable, ?? p a0 + a ?X1 + a ?X2 + … + a n ?Xn The variance of ? will be given as follows: C-2 DEFINITIONS The primary goal of Nonmandatory Appendix C is to present an expression for the variance in r in terms of the variances and covariances of X1 , X2, , Xn The definitions below for “mean,” “variance,” and “covariance” will be used throughout this Appendix The expected value of a function ƒ(X) of a random variable X is given by ? ? E?ƒ?X? ? p ? ƒ?x? p?x? dx −? where p(x) is the probability density function for X The mean (or expected value) ?X of the random variable X presents the special case where ƒ(X) p X, i.e., ? ?2? p E ?? ? − ? ?? 2? p E??a1 ?X1 − ?X1 ? + a2 ?X2 − ?X2? + … + an ?Xn − ?Xn? ?2? ? E?X? p ? x p?x? dx p ?X −? The variance ?2X of the random variable X presents the special case where ƒ(X) p (X − ?X) 2, i.e., For example, if n p 3, then ?2? p ? E ? ?a1 ?X1 − ?X1 ? + a2 ?X2 − ?X2 + a3 ?X3 − ?X3 ?2? E ??X − ?X? 2? p ? ?x − ?X? p?x? dx p ?X2 −? ? 95 ? ASME PTC 19.1-2013 ?2? p Consequently E ?a21 ?X1 − ?X1 ? + a22 ?X2 − ?X2? + a23 ?X3 − ?X3? + 2a1 a2 ?X1 − ?X1 ? ?X2 − ?X2? + 2a1 a3 ?X1 − ?X1 ? ?X3 − ?X3? + 2a2 a3 ?X2 − ?X2? ?X3 − ?X3? ? ?r p r??X1 ,?X2, … , ?Xn? and ?2r ≈ or + a22 ?2X2 + a23 ?2X3 + 2a1 a2 ?X1 X2 + 2a1 a3 ?X1 X3 + 2a2a3 ?X2X3 X2, … , Xn are all independent we get or more generally, the variance of ? will be given by n n C-4 PROPAGATION OF UNCERTAINTY/ERROR THROUGH TAYLOR SERIES C-4.2 Assessing the Validity of the First Order Approximation In para C-4.1 we assumed that the Taylor series expansion for r could be reasonably approximated through the first-order terms In this paragraph we will briefly assess the conditions under which this approximation is meaningful Let’s first consider the simplest case where r(x) Then r( x ) p r( ? x ) + ? x ( x − ? x ) + (1/2!) ?xx (x − ?x) + higher-order terms where ∂ 2r ? xx p ∂x In this case the ratio of the second-order term in the series to the first-order term in the series is given by In subsection C-3 we found that if a random variable, ?, could be expressed as a linear combination of the random variables, Xi, the mean and variance of ? can be expressed in terms of the means, variances, and covariances of the variables Xi Now suppose an experimental result, r, is defined as a function of certain measured variables, X1 , X2, … , Xn If the function r(X1 , X2, … , Xn) can be expressed as a linear combination of the measured variables X1 , X2, … , Xn, by means of a Taylor series approximation to r(X1 , X2, … , Xn) in the neighborhood of ?X1 , ?X2, … , ?Xn, then using the results in para C-3 we will be able to express the mean and variance of r(X1 , X2, … , Xn) in terms of the means, variances, and covariances of the variables X1 , X2, … , Xn C-4.1 The First Order Approximation Suppose r p r(x1 , x2, … , xn) If we expand r through a Taylor series in the neighborhood of ?X1 , ?X2, … , ?Xn we get r?x1 ,x2, … , xn p r??X1 ,?X2, … , ?Xn + ? X1 ?x1 − ?X1 + ? X2 ?x2 − ?X2 + … + ? Xn ?xn − ?Xn + higher-order terms where ? xi are the sensitivity coefficients given by ∂r ? Xi p ∂ xi Now, suppose that the arguments of the function r(X1 , X2, … , Xn) are the random variables X1 , X2, … , Xn Furthermore, assume that the higher-order terms in the Taylor series expansion for r are negligible compared to the first-order terms Then in the neighborhood of ?X1 , ?X2, … , ?Xn we have r?X1 ,X2, … , Xn p r??X1 ,?X2, … , ?Xn + ? X1 ?X1 − ?X1 + ? X2 ?X2 − ?X2 + … + ? Xn ?Xn − ?Xn ? ? Rp ? ( x − ? x) ?x ? (xj − ?xj) (xk − ?xk) + higher order terms where ? ? ? xx So that for this case the assumption that R ? reduces to the condition that ?xx (x − ?x) ? ? x More generally, if the second-order terms in the Taylor series expansion are retained, the Taylor series expansion for r in the neighborhood of ?x1 , ?x2, … , ?xn becomes r(x1 , x2, … , xn) p r(?x1 , ?x2, ?xn) n + i? ? xi ( xi − ? xi) p1 n n + ?1/2! j? ? ?xjxk p kp ? ? ? n ? ?Xi2 ?Xi2 ip Table C-4 presents some useful formulas for propagating variance through the first-order approximation to the Taylor series for an experimental result r ?2r p ? a2i ?2Xi + jp?i + aiaj ?XiXj? ip ? ? n For the special case where the random variables X1 , ?2? p a 21 ?2X1 ?2? p n ? ?2Xi ?2Xi + jp?i + ? Xi? Xj ?XiXj? ip ? ? xjxk ? 96 p ∂x∂j∂rxk ASME PTC 19.1-2013 Table C-4 Taylor Series Variance Propagation Formulas Function p r r p Variance (in Absolute Units) and Absolute Sensitivities f(x, y) S2r Ax + By p ∂ p r p A S2x + B2 S2y p A; ?y p r p p r p p y ; ?y (x + y) S2r p x2 S2r p ?x p − x (x + y) p (b) Vy p (c) Vr p Ax ; ? Ax + By y (1 + x) p ?x ′ p p y; ? y p v2r p v2y ?y′ p −1 p p V2x ′ ? x′ p V2x + V2y p ; ?y′ V2r p p p S2r p S2x 4x V2r p V2x ?x p ′ S2x ?x p x V2r x2 akybxa-1 ; ? y p p ?x ′ bkxayb-1 Sx x? Sy y? Sr r? 97 V2r p ?x ′ p V2x ?x ′ y x+ y 1 + x 2x x1 /2 − (1 + x) p ?x V2r By Ax + By (x + y) p p r/ r y/ y y2 (V2x + V2y) ′ x2 S2x S2r V ∂ ∂ p ?x p r/ r (Ax + By) V2r x p y ; ? x+ y y (1 + x) ∂ ? ∂y/ y y? A x2 V2x + B2 y2 V2y p V2r (ySx) + (xSy) p ′ ′ GENERAL NOTES: (a) Vx p (akyb xa-1 Sx) + (bkxayb-1 Sy) p p S2x p S2r ln x V2r y2 ?x S2r r/ r ; ? x/ x y y xSy ? (x + y) ? + ? (x + y) ? xy kxayb ′ + ∂ ∂ − ySx x1 /2 p ?x p ?x r B V p ′ S2r ?x r p r/ r ?x ∂ ∂ ∂ ? ∂x/ x x? S2y p S2r p p V2r r y p x + x p S S2r ∂ ?x r r ∂ y x x+ y p r ; ? x y ∂ ? ∂ y y? p ?y r + ?x ?x r r S ? ∂x x? Variance (Dimensionless) and Relative Sensitivities /2 Vx ? ln x? ln x (aVx) + (bVy) p a ; ?y ′ p b ASME PTC 19.1-2013 The second-order terms in this expansion may be compared directly to the first-order terms in order to assess their significance Assuming the systematic error to be independent of the random component of error this becomes ?2? p ?2? C-4.3 The Limitation of the Present Approach Xi p (?i − ??i) + ?Xi then E[Xi] p E?(Xi − ?Xi) 2? p E ?(Xi − ?Xi)(Xj − ?Xj) ? p ?XiXj p ??i?j + ??i?j The total variance associated with a measured variable, Xi, can be expressed as a combination of the variance associated with a fixed component and the variance associated with a random component of the total error in the measurement In this paragraph we will relate these two sources of variance in the measured variables, X j , to the variance in an experimental result r(X1 , X2, … , Xn) In subsection 4-2, the total error in a measurement was given by These results can be combined with those in para C-3.1 to yield ?2r ≈ n ? ?? n + ? ? ? xi2? i2 + ? ? Xi ? Xj ??i?j? ip jp i+1 ? For no correlation among the random errors, with the use of sample statistics, and for a 95% confidence level, this equation leads to eq (B-1.11) so that C-6 THE PROBABILITY DENSITY FUNCTION OF A RESULT E[ ?] p E[ ? + ?] The preceding paragraphs make no assumptions about the joint probability density function of the measured variables Xi Assuming that the first order approximation to the Taylor series expansion for r is adequate and that the measured variables Xi are jointly normally distributed, the experimental result r will also be normally distributed with mean ?r and variance ?2r ?? p ? ? also E ? ( ? − ? ? ) 2? p E ? ? ( ? − ? ? ) + ? ? 2? ? n ? ?xi2? i2 + jp?i+1 ? Xi ?Xj ? i j? ip ? n p ?+? + ?2 + 2?? ?Xi ??i2 p ?? i2 + ??i2 p ?Xi2 Assuming fixed errors to be independent of random errors, it can also be shown that C-5 PROPAGATION OF SYSTEMATIC AND RANDOM COMPONENTS OF UNCERTAINTY ?2? p ?2? ? Now, if we define the random variable Xi as follows: If the higher-order terms in the Taylor series expansion are not small compared to the first-order terms, then there is no way to express the variance in r(X1 , X2,… , Xn) directly in terms of the variances and covariances for X1 , X2,… , Xn for an arbitrary joint probability density function p(x1 , x2,… , xn) ? + ?2 ? 98 ASME PTC 19.1-2013 Nonmandatory Appendix D The Central Limit Theorem Y p c1 X1 + c2X2 + … + cNXN p N ? ciXi ip Then the distribution of Y will be approximately normal (Gaussian) with expectation N N E(Y) ciE(Xi) and variance ?2(Y) ci2?2(Xi), where E(Xi) is the expectation of Xi ip ip and ? 2( Xi) is the variance of Xi, provided that Xi are independent and ? 2( Y) is much larger than any single component ci2?2 ( Xi) from a non-normally distributed Xi Although not mathematically precise, this is a reasonable statement of the central limit theorem for application purposes It is a more general version of the central limit theorem than what one would find in elementary statistics textbooks The Central Limit Theorem [2]: Suppose p? p? 99 ASME PTC 19.1-2013 Nonmandatory Appendix E General Regression Uncertainty can also can be systematic standard uncertainty in the ˆ value This X value used in the curve-fit to find a Y curve-fit X value will be called Xnew to distinguish it from the X data points, and the systematic standard uncertainty for Xnew is b new The Yˆ value determined from the regression equation for a value of Xnew is E-1 INTRODUCTION Often in an experimental program, the final outcome is a plot of the results showing the variation of Y with X (or a more complicated expression of X such as a polynomial) Least-squares regression is usually the method used to determine the best relationship for Y versus X However, both the Y and X data used to generate the regression equation can have random and systematic uncertainties that will influence the “goodness” of the results obtained from the equation This may need to be included in the uncertainty analysis In subsection 8-6, the case was considered for a straight-line, or simple linear regression, where the assumption was made that there was no random standard uncertainty in the X values and the random standard uncertainty in the Y values was constant over the range of the curve-fit The assumption was also made that the systematic standard uncertainty for the Y values and/or the X values were constants (i.e., percent of full scale) and there were no correlated elemental systematic errors between the X and Y values In this Nonmandatory Appendix, a general approach to regression uncertainty with many degrees of freedom (greater than 29) is presented (Ref [1] in subsection E-6) i X ˆ( Y Xnew mX +c N N b Yˆ2 (E-2.1) m p N i i i N ?p N i i ? Xi ? i i i p1 Xi c p N i? i i N i i N N N ?p X2 − ? ?p X ? i ? i? i? i Yi N i i N k X i i i i k YiYk N k i i k XiXk N i k XiYk k (E-3.1) i X Xi i X Yi The total uncertainty in the Yˆ value is then (E-2.3) UYˆ p 2? s + b ? ˆ Y ˆ Y 1/2 (E-3.2) where if the restrictions in subsection 8-6 apply for the random uncertainty, s can be determined from eq (8-6.5) There can be systematic standard uncertainty in the Y data and in the X data, b and b , respectively There i , N E-3 SYSTEMATIC UNCERTAINTY i N X i i i i i i , N N i Y N− i ? and the intercept c as ?p ?X2 ?p Y − ?p X ?p ?X Y N i i N (E-2.2) N − ?? N− i N i N i N ∂Yˆ b + ?1 ? ∂Yˆ ∂Yˆ b p? ? ∂ p1 Y ? p p +1 ? ∂Y ?? ∂Y ? ˆ ˆ ˆ + ? ? ∂∂XY ? b + ? ? ? ∂∂XY ?? ∂∂XY ? b p1 p p +1 ˆ ∂Yˆ ∂ Y + ? ? ? ∂X ?? ∂Y ? b p1 p1 ˆ + ? ∂X∂Ynew? b 2new ˆ ˆ + ? ? ∂X∂Ynew?? ∂∂XY ? b new p1 ∂Yˆ ∂Yˆ b +2? ? p ∂Xnew?? ∂Y ? new N i N , N where the X , Y data pairs are used to determine the slope m as N? X Y − ? X ? Y p1 p1 p1 i +c i For a straight-line, or a simple linear regression, the curve-fit expression is p mXnew Now m and c are functions of the N sets of X , Y data pairs [eqs (E-2.2) and (E-2.3)], so m p m ( X1 , X2 … , X , Y1 Y2, … , Y ) and c p c ( X1 , X2 … , X , Y1 Y2, … , Y ) Therefore, Yˆ has the general expression ˆ p Yˆ (X1 , X2 … , X , Y1 Y2, … , Y , Xnew) Y The systematic standard uncertainty for the Yˆ determined from the regression equation (E-2.1) is determined as E-2 LEAST-SQUARES ˆ Y )p ˆ Y Xi 100 ASME PTC 19.1-2013 The random and systematic standard uncertainty expressions for the Yˆ value from the regression equation used by eq (E-3.2) are functions of the Xi, Yi data values and Xnew The X variable, Xnew, has a fixed range for the regression equation To simplify the use of the regression uncertainty expressions, a set of data points can be generated for (Xnew, UYˆ) These data can then be used to produce an expression for UYˆ (Xnew) which can be used to determine the uncertainty in a curve-fit value of Yˆ for a given Xnew [1] For a more complete discussion of reporting regression uncertainties, see [1] then be combined with the systematic standard uncertainty using eq (E-3.2) to obtain the total uncertainty For the case of uniform random uncertainty in the Yi variables, the first term on the right-hand side of eq (E-4.1) gives the same value as eq (8-6.5) E-5 HIGHER ORDER REGRESSION EQUATIONS The methodology presented above for simple linear regressions applies also for higher-order linear regressions For the general n th order polynomial curve-fit model Yˆ (Xnew) p a0 + a1 Xnew + a2X2new + … + anXnnew the regression coefficients, ai, can be determined with a least-squares technique The expressions for the systematic and random standard uncertainties of the Yˆ value found using the polynomial regression model would be the same as those for the simple linear case, eqs (E-3.1) and (E-4.1), respectively These standard uncertainties can then be combined and expanded using eq (E-3.2) to obtain the total uncertainty E-4 GENERAL APPROACH TO LINEAR REGRESSION UNCERTAINTY When the Yi variables in the regression expression have variable random uncertinty and/or the Xi variables have random uncertainty, eq (8-6.5) for the random uncertainty of the curve-fit result is not appropriate In this more general case, the random standard uncertainty for each Yi data point, sYi, for each Xi data point, sXi, and for the new X variable, sXnew, must either be determined or estimated Then the random standard unertainty of the regression equation result is determined as E-6 REFERENCE ∂Yˆ 2s Y2 + ?N ∂Yˆ 2sX2 + ∂Yˆ s X2 (E-4.1) Yˆ p i? p ? ∂Yi? i ip ? ∂Xi? i ? ∂Xnew? new This value for the random standard uncertainty in the Yˆ value determined from the regression equation would s2 N [1] Coleman, H W and W G Steele, Experimentation, Validation and Uncertainty Analysis for Engineers , 3rd Edition, John Wiley & Sons, New York, 2009 (Revised [2010]) 101 I N TE N TI O N ALLY LE FT B LAN K 102 ASME PTC 9.1 -201 D0451