Goran Kostić GUIDEBOOK TO CONCEPTS AND COMPUTATIONS IN MEASUREMENTS

Transcription

1 Goran Kostić GUIDEBOOK TO CONCEPTS AND COMPUTATIONS IN MEASUREMENTS

2 Goran Kostić PAGES FROM: Metrology Companion The principal distributor of this book is Amazon.com, Inc. Leskovac, 2018

3 CONTENTS CONTENTS 5 Abbreviations 12 Letter symbols 13 Greek alphabet 14 PREFACE 15 INTRODUCTION 19 PART I FUNDAMENTALS OF MEASUREMENTS 1 QUANTITIES AND UNITS OF MEASUREMENT Measurable quantity Quantities of the same kind System of quantities Base quantity Derived quantity Ordinal quantity Dimension of a quantity Unit of measurement Base unit of measurement Derived unit of measurement Coherent unit of measurement Value of a quantity Numerical value of a quantity Conventional value of a quantity Reference value Quantity equation Numerical value equation Unit equation Conversion factor between units System of units Future International System of Units Decimal units of measurement Off-system unit of measurement 38 Kostić 2018 Metrology Companion 5

4 1.23 Binary unit of measurement Expressing the value of a quantity in the SI Constant 40 2 MEASUREMENTS Measurement Metrology Measurand Principle of measurement Method of measurement Measurement procedure Robustness of procedure Standard operating procedure Standard measurement procedure Qualification and validation Validation of measurement procedure Accreditation About design of experiment Mathematical model Examples Simulation 58 3 MEASURING ARTIFACTS Measuring artifact Material measure End gauge Nominal value Indication of a measuring artifact Indication interval Measuring interval Response function Measuring transducer Sensor Detector Measurement signal Resolution Sensitivity Response time Limit of detection Limit of quantification Dynamic interval Dead interval Hysteresis 65 6 Kostić 2018 Metrology Companion

5 3.21 Influence quantity Stability Selectivity Transparency Operating conditions Reference operating conditions Limiting conditions Gauging of a measuring artifact Error of indication of a measuring artifact Datum error of a measuring artifact Zero error of a measuring artifact Bias of a measuring artifact Intrinsic error of a measuring artifact Accuracy class Examples Type approval of a measuring artifact Verification of a measuring artifact 70 4 MEASUREMENT RESULTS, ACCURACY AND ERRORS Measurement result Accuracy Absolute error Relative error Random error Systematic error Combined error Example Limits of error Outlier Correction Precision of measurement results Repeatability of measurement results Reproducibility of measurement results Rounding of number Truncation of digits Significant digits 80 5 MEASUREMENT STANDARDS AND CALIBRATIONS Measurement standard Reference material Primary measurement standard Secondary measurement standard Transfer measurement standard 83 Kostić 2018 Metrology Companion 7

6 5.6 Reference measurement standard Working measurement standard International measurement standard National measurement standard Relativistic effects on the values of standards Calibration Calibration report 88 PART II STATISTICAL ANALYSIS OF MEASUREMENT RESULTS 6 OUTLINES OF THE STATISTICS About statistical analysis Statistical analysis of results of repeated measurements Probability Random variable Dependent variable and independent variable Correlated variables Sample Maximum likelihood value Best estimate Biased evaluation Distribution of a variable Normal distribution Means Mode Median Arithmetic mean Experimental arithmetic mean Indicators of dispersion of values Natural Deviation Experimental standard deviation Deviation of the arithmetic mean Deviation of the standard deviation Standard deviation from a median, range and sample size Degrees of freedom Examples Effective degrees of freedom Example Degrees of freedom from the deviation of the standard deviation of the arithmetic mean Example Kostić 2018 Metrology Companion

7 6.27 Confidence level and confidence interval Example Standard deviation and its degrees of freedom from the level and interval of confidence and sample size T- distribution Uniform distribution Triangular distribution U-distribution Lognormal distribution Chi-square distribution F-distribution Indicator of asymmetry of a distribution Indicator of peakedness of a distribution Histogram Example Example Convolution Filtering Operations on distributions and central limit theorem Monte Carlo simulation Example of estimate of a value and its statistical parameters Example of estimate of a length of gauge block and its statistical parameters VARIANCES Natural variance Experimental standard variance Evaluations of grouped results Experimental standard covariance Example of compution of the standard covariance Example of compution of the standard covariance of the means Standard correlation coefficient Combined variance Example of the combined variance of an evaluation of resistance of electrical circuit element and the maximum possible variance of this variance Example of the combined variance of an evaluation of length of a gauge block and the maximum possible variance of this variance Example of the combined variance of ph value Example of the combined variance of resistance of resistors in a series 195 Kostić 2018 Metrology Companion 9

8 8 TESTS AND FUNCTIONS FITTING Detection of outliers Example Fitting Method of the least sum of squares Example P-value Goodness-of-fit test Estimating the type and parameters of a distribution Pearson chi-square test for the goodness of fit of distributions Example Shapiro-Wilk test for normality of a population Example White noise test Analysis of variance (ANOVA) Example Transformation of a variable Example MEASUREMENT UNCERTAINTY Measurement result and its uncertainty Type A standard measurement uncertainty Type B standard measurement uncertainty Example Example Combined standard measurement uncertainty Expanded standard measurement uncertainty Relative standard measurement uncertainties Evaluations from non-normally distributed results Expressing the standard measurement uncertainty Expressing the standard measurement uncertainty when corrections are not applied General guidance on reporting a measurement result Procedure for estimating the measurement result and associated data Example of calibration of a thermometer The measurement problem Least squares fitting Standard uncertainty of the predicted correction addend Elimination of the correlation between the slope and the intercept point Kostić 2018 Metrology Companion

9 9.13 Resistance, reactance and impedance, computed from the same component measurement results The measurement problem Mathematical model of the measurement Final results and their standard uncertainties, standard covariances and standard correlation coefficients - approach a Final results and their standard uncertainties, standard covariances and standard correlation coefficients - approach b Metrological traceability Example 266 REFERENCES 267 INDEX 273 Kostić 2018 Metrology Companion 11

10 Preface From a shop floor to the fundamental sciences, statistical analysis of results of repeated measurements is required. The purpose of the analysis is to determine the value of measured quantity on the basis of these results, as well as an accuracy of the determined value. It is requested that accuracy is described by a standard measurement uncertainty in order to enable traceability and ensure computing of a confidence level and confidence interval. The essence of measurement is the same in all areas. The purpose of this publication is to be a companion for people dealing with measurements, that is experiments or observations, from production to research, from physics and chemistry to biology and medicine. The publication is also intended for students. In front of you is the Companion with concisely described subjects that are indispensable in all measurements. The Companion consists of about minimal sufficient set of concepts and methods required to compute a measurement result and its measurement uncertainty. The text is in the form that enables direct applications in computations, manual or spreadsheet, and in writing software. The manner of presentation is encyclopedic. However, the order of presentation allows reading the text as a book or textbook. It seems that the form of encyclopedias is suitable for expert writings because it facilitates understanding of relations among concepts that make the essence of every exact knowledge. Most of this text is a terminologically consistent and compliant synthesis of international recommendations and regulations, articles and encyclopedic texts. References are given in brackets. Special attention was paid to the definitions of terms and their consistent use. Most of the terms are according to the second and third editions of the International vocabulary of metrology (see [VIM 2] and [VIM 3]) which was issued by the Joint Committee for Guides in Metrology (JCGM). These glossaries were not applied when it was thought that they do not correspond to what they should express, or that there are far better terms. Any such exceptions from the glossaries are noted with the references. Kostić 2018 Metrology Companion 15

11 The second part of the Companion, regarding statistical analysis of measurement results, is the basis for computation of the standard measurement uncertainty. It is intended for anyone involved in the analysis of the measurement results. The chapter about the standard measurement uncertainty is according to the instructions published by the JCGM in the Guide to the expression of uncertainty in measurement (see [GUM]). The level and interval of confidence of the measurement result can be considered as a purpose of determination of all measurement errors. An accurate evaluation of the level and interval of confidence is enabled by the standard measurement uncertainty of a traceable value. In this Companion, the traceability is described in the short, but comprehensive text given at the end of the section concerning measurement uncertainty. At the end of the Preface, attention is drawn to the following viewpoint expressed in [GUM]. The evaluation of uncertainty is neither a routine task nor a purely mathematical one; it depends on detailed knowledge of the nature of the measurand and of the measurement. The quality and utility of the uncertainty quoted for the measurement result therefore ultimately depend on the understanding, critical analysis, and integrity of those who contribute to the assignment of its value. All equations in the Companion are for the coherent units. The following marking is used in the Companion. Bold fonts are used for terms when defined. Next to the terms, in parentheses, are given the abbreviated terms that may be used when this does not lead to confusion. Underlined text indicates that in this Companion there are detailed explanations of a concept which underlined text indicates. Equations, tables and figures are marked with the number of the chapter, then by a point, and subsequently by their order number in the chapter. A grey shaded text needs special attention. For the help in the creation of the Companion, I wish to thank to engineer Siniša Hristov, professor Dr Gligorije Perović, professor Dr Dragić Banković, professor Dr Vesna Jevremović and Dr Emina Krčmar. To my longtime colleague, engineer Zlatko Sudar, I am grateful for comments which have led to an improved text. 16 Kostić 2018 Metrology Companion

12 I am grateful to professor Ljiljana Sudar and professor Dr Živomir Petronijević for their efforts and valuable remarks of the trial readers. Thanks to Ljiljana D. Kostić for her trust and material support. This Companion was created as part of the development project of measurement and control equipment, which is in progress in the laboratory for development of electronic products, Symmetry, based in Leskovac, Serbia. This is the first English edition of the Companion which has been prepared by updating and revision of the first, well-received, Serbian edition. Please send comments to the Leskovac, 27th August Goran Kostić Kostić 2018 Metrology Companion 17

13 Introduction The measurements are essential for human activity - from the trade to the fundamental sciences. International trade has been one of the important reasons for the establishment of the International Bureau of Weights and Measures (BIPM). The Bureau contributes to the reduction of technical barriers to trade, ensuring that measurements and tests carried out in different countries are considered equivalent. Scientists create their theories on a basis of measurement results which are for them facts about the material world, true within limits of measurement errors. The International System of Units was conceived after ancient discussions of European scientists about the need for a new system of measurement. The first proposal of a unified system of units, based on the metre (metric) and decimal, was made as early as 1670 by Gabriel Mouton, a priest from Lyon, France. That system replaced national and regional variants that made scientific and commercial communication difficult. The first metric system was established in France in 1799 as an important fruit of the French Revolution. The current version of the International System of Units (SI) was completed in 1971 at the 14th General Conference on Weights and Measures (CGPM). The world became metric in early 1993 when SI became primary in the US government institutions. The International Bureau of Weights and Measures was created by the Metre Convention signed in Paris on 20th May This Convention was revised in 1921 and has remained the basis of the international agreement on units of measurements. The original signatories of the Convention were representatives of 17 States, and since 7th August 2018, the Convention acceded 60 Member States and the General Conference on Weights and Measures 42 Associates. Since its establishment, the headquarters of Bureau is in Sevres near Paris. The purpose of Bureau is to provide the worldwide base of the unified system of measurements which enable traceability to the International System of Units. Therefore, the task of Bureau is conservation of international measurement standards, comparisons of national and Kostić 2018 Metrology Companion 19

14 international standards and carrying out and coordinating the measurements of physical constants relevant to activities of the Bureau. By harmonizing and providing complete coverage of all areas, the International System of Units is an outstanding tool of the modern sciences and practices which created it. [BIPM] [SI] 1.8, PP 95 [Britannica] [NIST] 20 Kostić 2018 Metrology Companion

15 1 Quantities and units of measurement 1.1 Measurable quantity Measurable quantity (quantity) is an attribute of a body, a substance, or a phenomenon, that may be distinguished qualitatively and determined quantitatively. Quantity is not a body, a substance or a phenomenon. Quantity is independent of measurement and system of quantities. The term quantity may refer to a quantity in a general sense (length, time, mass, temperature...) or to a value of a particular quantity (length of a given rod, an electric resistance of a given wire, a number of entities...). [VIM 2] 1.1 [VIM 3] 1.1 [Sonin] Quantities of the same kind Quantities of the same kind are quantities whose values may be added or subtracted, but that it has a physical representation. Examples. The addition of values of all forms of energies, such as the amount of heat, kinetic energy and potential energy, has a physical representation. Thus, all forms of energies, are quantities of the same kind. A value of energy and a value of moment of force have the same dimension but there is no physical representation of addition or subtraction of their values, so those quantities are not of the same kind. In a given system of quantities: - quantities of the same kind have the same dimension - quantities of the same dimension are not necessarily of the same kind - quantities of the different dimension are always of a different kind. Quantities of the same kind are grouped into categories of quantities. Examples of the categories of quantities are: - thickness, circumference, wavelength - heat, work, energy. [Quinn] [VIM 2] 1.1 Notes [VIM 3] 1.2, 1.7 Kostić 2018 Metrology Companion 23

16 1.4 Base quantity Base quantity, together with other base quantities of a system of quantities, enables obtaining derived quantities. The selected base quantity always has a physical representation so that it may be realized using its definition. A base quantity is an attribute for which the following mathematical operations are defined in physical terms: comparison, addition, subtraction, multiplication by a quantity of dimension one and division by a quantity of dimension one. Each of these operations is performed on physical properties of the same kind and yields a physical property of that kind. Each of these physical operation obeys the same rules as the corresponding mathematical operation for pure numbers, that is, quantities of dimension one. This makes possible to describe the physical properties by the language of mathematics. The choice of a set of base quantities is somewhat arbitrary. However, for the selected base quantities it must be possible to define the operations previously specified. It is preferred that the base quantities of a set are mutually independent. The set may be minimal, or somewhat larger, than one out of which it is possible to obtain all necessary derived quantities. Example. The base quantities may be: the length, mass, area, speed and force. A shape and color cannot be base quantities because they lack acceptable addition. Things from nature can be described only by comparison with the several well-known things among which are the base quantities. [Sonin] 2.2, 2.1, 2.8 [SI] 1.1, 1.2 Kostić 2018 Metrology Companion 25

17 1.24 Expressing the value of a quantity in the SI A numerical value of a quantity and symbol of measurement unit is printed in Roman or Greek upright type. The numerical value always precedes the unit symbol, and a space is always used to separate the number from the unit symbol. The only exceptions, in which no space is left between the number and the unit symbol, are the unit symbols for plane angle: degree,, minute,, and second,. A prefix symbol precedes the unit symbol without a space, and it is considered as a new inseparable unit symbol, as well as mathematical entity. A numerical value of a quantity with a unit symbol is regarded as the product, and they are treated using the ordinary rules of algebra. One must neither use the plural of a unit symbol, nor the period after a symbol except at the end of a sentence. A prefix symbol, formed by placing together two or more prefix symbols, are not permitted. This also applies to prefix names. A prefix symbol and a prefix name will not be used with the following symbol and name of the units of time: min, minute; h, hour; d, day. No space or hyphen is used between the prefix name and the unit name. The combination of prefix name plus unit name is a single word. In an expression, only one unit is used. An exception is in expressing the values of time, and of plane angles, using non-si units. However, for plane angles it is generally preferable to divide the degree decimally, except in fields such as navigation, cartography, astronomy, and in expressing very small angles. Example, one would write rather than Unit symbol is never qualified by further information about the property of the quantity. Extra information on the property of the quantity should be attached to the quantity symbol, and not to the unit symbol. Example, U max = 50 V, but not U = 50 V max. In forming products and quotients of unit symbols the normal rules of algebraic multiplication or division apply. Multiplication must be indicated by a space or a half-high dot ( ). Division is indicated by a horizontal line ( ), by a solidus (/), or by negative exponents. When several unit symbols are combined, care should be taken to avoid ambiguities, for example by using brackets or negative exponents. To remove ambiguities, a solidus without brackets, must not be used more than once in a given expression. Among the base units of the SI, the kilogram is the only one whose symbol and name, for historical reasons, includes a prefix ( k ). Symbol and name for the decimal unit of mass are formed by attaching prefix symbol to the unit symbol g, and prefix name to the unit name gram. 6 Example, 10 kg is written as 1 mg, not as 1 μkg. [SI] 3, 5 Kostić 2018 Metrology Companion 39

18 Table 1.2 SI coherent derived units with special names and symbols [SI] [SI Supplement] SI derived quantity Special name of SI unit Special symbol for SI unit Expressed in terms of other SI units Expressed in terms of SI base units plane angle radian rad m m 1 = 1 solid angle steradia sr m 2 m 2 = 1 frequency hertz Hz 1 s force newton N 2 kg m s pressure, stress pascal Pa N / m kg m s energy, work, amount of heat joule J N m = W s kg m 2 2 s power, radiant flux watt W J / s kg m 2 3 s electric charge, amount of electricity coulomb C F V A s electric voltage, electric potential difference, electromotive force volt V W / A kg m s A electric resistance ohm Ω V / A kg m 2 3 s A 2 electric conductance siemens S A / V kg 1 m 2 s A 2 electric capacitance farad F C / V kg 1 m 2 s A 2 electric inductance henry H Wb / A kg m s A magnetic flux weber Wb magnetic flux density tesla T V s = T m 2 Wb / m 2 = N / (A m) kg m2 2 s A 1 2 kg s A 1 Celsius temperature degree C 4) 4) Celsius K 4) luminous flux lumen lm cd sr m 2 m 2 cd = cd illuminance lux lx lm / m 2 2 m cd activity referred to a 1 becquerel Bq s radionuclide ionizing radiation absorbed dose, specific gray Gy J / kg m 2 2 s imparted energy, kerma equivalent radiation dose sievert Sv J / kg m 2 2 s catalytic activity katal kat 1 mol s 4) The units C and K are equal in magnitude. Numerical value of the Celsius temperature and numerical value of the thermodynamic temperature in kelvins are in the relation given by the equation: t [ C] = T [ K] [SI] Kostić 2018 Metrology Companion

19 2 Measurements 2.1 Measurement Measurement is a determination of a value of a quantity by its comparison with the known value of the quantity. The value of a quantity may be measured with very high accuracy except perfect. The following are necessary for measurement: a specification of measurand, a description of measurement procedure and a measuring artifact. Measurement does not apply to nominal properties. Examples: sex of a human being; ISO two-letter country code; the sequence of amino acids in a polypeptide... [Sonin] 2.3 [GUM] [VIM 2] 2.1 [VIM 3] 2.1, 1.30 Examples 2.2 Metrology Metrology is the science of measurements. Metrology includes all theoretical and practical aspects of measurements, whatever the measurement accuracy and field of application. Legal metrology is part of metrology relating to measurements specified in the law. The purpose of legal metrology is to ensure accuracy and reliability of measurements intended for: foreign and domestic trade public safety health protection environmental protection. [VIM 2] 2.2 [VIM 3] 2.2 Kostić 2018 Metrology Companion 49

20 2.3 Measurand Measurand is a particular quantity subject to measurement. The requested accuracy determine how many details should be given in a specification of a measurand. For example, if the length of a nominally one-metre long steel bar is to be determined to micrometre accuracy with relative error around , its specification should include the temperature and pressure of the environment in which is the bar. [VIM 2] 2.6 [VIM 3] 2.3 [GUM] Principle of measurement Principle of measurement (or measurement principle) is a phenomenon serving as the basis of a measurement. Example, thermoelectric effect applied to the measurement of temperature. [VIM 2] 2.3 [VIM 3] Method of measurement Method of measurement (or measurement method) is a concept of applying the principle of measurement. Primary method of measurement (or primary measurement method, primary method) is a method having the highest metrological qualities, whose procedure is completely explained and described, and for which the combined standard uncertainty traceable to the SI units is determined. Primary ratio method is a method used to determine a quotient of value of measurand and value of measurement standard (where both values are of the same kind). Primary direct method is a method in which a value of measurand and values of used measurement standards, are of different kind. [BIPM] [NIST] 50 Kostić 2018 Metrology Companion

21 3 Measuring artifacts 3.1 Measuring artifact Measuring artifact is an artifact intended to be used for making a measurement, alone or in conjunction with supplementary artifacts. A measuring artifact may be a device, a material or a substance. Measuring artifacts arranged in approximate order of increasing complexity are: part, measuring transducer, reference material, measuring device, material measure, measuring instrument, apparatus, equipment, measuring chain, measuring system and measuring installation. Measuring artifact is periodically checked, or qualified, by a user or a recognized laboratory. The checks determine whether an error of a measuring artifact, in operating conditions, is within a specified interval and whether it will be within this interval in future. Measuring device (or meter, or gauge) is a measuring artifact which provides an indication. [VIM 2] 4 introduction 3.2 Material measure Material measure (measure) is a measuring artifact intended to supply, or reproduce, one or more known values of a quantity. Examples: a weight, a measure of volume, a gauge block, a reference material, a standard electric resistor, a standard signal generator. An indication of a material measure is its assigned value. An error of a material measure is the error of the indication of the material measure. [VIM 2] 4.2, 3.2 Note 3, 5.20 Note 3 [VIM 3] 3.6, 4.1 Kostić 2018 Metrology Companion 59

22 3.3 End gauge End gauge is a measuring artifact which supplies one known value. Example. A length end gauge has the form of: parallelepipeds (gauge block), pins, blades, plugs, rings, snaps. Go - no go gauge consists of two end gauges which determine whether a value of a measurand is within an interval given by the end gauges. 3.4 Nominal value Nominal value is a rounded or approximate value of a measuring artifact that is significant to measurement. Examples a) 100 Ω as the nominal value marked on a standard resistor. b) 1 L as the nominal value marked on a volumetric flask. c) 0.1 mol/l as the nominal value for the concentration of a solution of hydrogen chloride. [VIM 2] 5.3 [VIM 3] Indication of a measuring artifact Indication of a measuring artifact (indication) is the value of a quantity displayed by the measuring artifact. An indication may refer to the value of a measurand, a measurement signal, or some other quantity used for determining the value of a measurand. The indication of a material measure is its assigned value. Direct indication is a value which is given by a displaying device. Displaying device (or indicating device) is part of a measuring artifact which displays an indication. For example, this term is used for: a label, a display or a part which is used to set a value. Scale of a measuring artifact (scale, or dial) is a set of marks which is in a specified relation to the values of a measurand. Examples: a thermometer scale; a barometer scale; an analog clock dial. [VIM 2] 3.2, 4.12, 4.17, 4.1 [VIM 3] 4.1, Kostić 2018 Metrology Companion

23 4 Measurement results, accuracy and errors 4.1 Measurement result Measurement result (or result of measurement, or estimated value) is a value assigned to a measurand on the basis of one or more results of measurements. A measurement result is determined in the manner specified in the description of the measurement procedure: taking a result of a single measurement (or particular result), computing the arithmetic mean of the results of repeated measurements, taking the value that appears most often in the results of repeated measurements (that is, taking the mode), computing on the basis of a functional relationship between the value of a measurand and component values (or component results, or component variables, or component quantities, or input quantities, or components) on which the value of the measurand depends... Results of repeated measurements of the same quantity are represented by the random variable which is characterized by statistical parameters. A complete measurement result includes information about its accuracy. [VIM 2] 3.1 [GUM] [VIM 3] Accuracy Accuracy of a value is a qualitative concept which indicates a closeness of this value to a reference value. Accuracy of a measurement result (accuracy) is a qualitative concept which indicates a closeness of this measurement result to a value of the measurand. Accuracy of a measurement result is determined in the manner specified in the description of the measurement procedure. Kostić 2018 Metrology Companion 71

24 Accuracy of a measurement result depends on errors which are classified into the two groups. The first group is random errors, whose influence on the final measurement result can be reduced by increasing the number of repeated measurements and then computing the arithmetic mean of the measurement results. The second group is systematic errors, whose influence on the final measurement result can be reduced by applying certain correction. A complete measurement result includes information about its accuracy. However, if inaccuracy is considered to be negligible for the intended purpose, the information about accuracy may be omitted. In many fields, the ordinary way of expressing a measurement result does not include the information about accuracy. In order to enable the use of one measurement result and its accuracy as component values of another measurement result and its accuracy, the information specified in 9.10 General guidance on reporting a measurement result should be provided. Results of repeated measurements of the same quantity are represented by the random variable which is characterized by statistical parameters. The description of the accuracy of the result of repeated measurements is its standard measurement uncertainty. The term precision should not be confused with accuracy. [VIM 2] 3.5 [VIM 3] 2.13, 2.15, 2.9 Note 2 [GUM] Absolute error Absolute error (error, or difference, or divergence) of a value is given by formula: ABSOLUTE_ERROR = VALUE REFERENCE_VALUE. (4.1) VALUE = REFERENCE_VALUE + ABSOLUTE_ERROR (4.2) REFERENCE_VALUE = VALUE ABSOLUTE_ERROR (4.3) ABSOLUTE_ERROR = = SYSTEMATIC_ERROR + RANDOM_ERROR (4.4) In practice, the conventional value is used as the reference value. The value and its absolute error have the same dimension. 72 Kostić 2018 Metrology Companion

25 5 Measurement standards and calibrations 5.1 Measurement standard Measurement standard (standard, or etalon) is a measuring artifact intended to realize, reproduce or define the value of a quantity to serve as a reference in measurements. Examples: 1 kg mass standard; 100 Ω standard resistor; standard ammeter; caesium frequency standard; hydrogen reference electrode; reference solutions of cortisol in human serum with specified concentration. Intrinsic measurement standard (intrinsic standard) is based on an inherent and reproducible phenomenon in nature. Examples: triple-point-of-water cell as a standard of temperature; Josephson array voltage standard; quantum Hall effect electric resistance standard. Collective standard is a set of standards that, through their combined use, constitute a standard. Group standard is a set of standards that, individually or in combination, provide a sequence of values of quantities of the same kind. [VIM 2] 6.1 [VIM 3] 5.1 in which associated measurement uncertainty is required, Reference material Reference material, RM, is material with homogeneous, stable and accurately defined properties used as a reference in measurements. Reference material may be a pure or mixed: gas, liquid or solid. Examples: a solution used in calibration of measuring instruments for chemical analyses; a color chart; a fish tissue containing a specified mass concentration of dioxin. Certified reference material, CRM, is the reference material accompanied by a certificate specifying the values of the material properties and traceable standard measurement uncertainty of these values. Kostić 2018 Metrology Companion 81

26 Example. Human serum accompanied by a certificate specifying the value of the concentration of cholesterol and the associated traceable standard uncertainty. [NIST] [VIM 2] 6.13, 6.14 [VIM 3] 5.13, Primary measurement standard Primary measurement standard (primary standard) is a standard that has the highest metrological qualities, and its value is accepted by the convention without reference to other standards of quantity of the same kind. The concept of primary standard is equally valid for both the base quantities and the derived quantities. Examples: international prototype of the kilogram; triple-point-of-water cell as a primary standard of temperature; Josephson array voltage standard; primary standard of substance concentration prepared by dissolving a known mass of a substance to a known volume of solution. [VIM 2] 6.4 [VIM 3] Secondary measurement standard Secondary measurement standard (secondary standard) is a standard whose value is assigned by comparison with the primary standard of quantity of the same kind. [VIM 2] 6.5 [VIM 3] Kostić 2018 Metrology Companion

27 6 Outlines of the statistics 6.1 About statistical analysis Statistical methods are used to determine the representative value of an investigated property for a group of elements. Elements within the group can have significantly different values of the property. Evaluation (or estimation, or statistic ) is a value estimated by the statistical method. Population is a group of all elements for which we determine the representative value of a property by the statistical method. The population has a finite or infinite number of elements. Figure 6.1. The example of four samples of values which yield: the equal arithmetic means of variables x and y, the equal deviations of x, the same linear regression function obtained by the method of least squares... [Anscombe] Kostić 2018 Metrology Companion 91

28 The investigated property of a population is determined by the statistical method usually on a basis of a sample of the population that truly represents the population. The sample truly represents the population when a number of its elements is sufficiently large and when the elements are randomly selected from the population. In order to prove that the conclusions drawn from the sample are with a certain probability valid for the population can be proved by goodness-of-fit tests. In order to reduce the possibility of errors in statistical analysis, data graph should be necessarily considered. The graph provides an overview of qualitative features of data. The diagrams in Figure 6.1 demonstrate the importance of graphical presentation and visual analysis of data. A property of a population determined by a statistical method is the property of a large number of elements of that population. On a basis of statistical evaluation, we can determine only the probability that a particular element of a population has a certain property. Statistics is subjective; statisticians try to explain or predict the material world through arbitrary, but reasonable way, using probability theory, mathematics and common sense. Unlike the statistics, probability theory for a fully defined problem gives a unique and a reproducible solution. Probability theory is the algebra of inductive reasoning. (Boolean algebra is the algebra of deductive reasoning.) Statistical methods are among the most widespread quantitative methods, with important applications in almost all human activities. [Njegić] 1.1., 1.2. [Ivković] II.1 [Britannica] Probability theory [Drake] 7-1, 7-6 [Lazić] I, 1.2 [Anscombe] 92 Kostić 2018 Metrology Companion

29 6.2 Statistical analysis of results of repeated measurements The purpose of statistical analysis of results of repeated measurements of a particular quantity is to determine the best estimated value of the quantity. And also, to determine an interval about the best estimated value which, with requested probability, encompass a true value of the quantity. Normally, for the best estimated value of the quantity is taken the maximum likelihood value of this quantity. The results of repeated measurements most often have normal distribution because the central limit theorem is most often valid. In the case of normal distribution, and after correction for all significant systematic errors, the maximum likelihood value of the quantity is an arithmetic mean of the results of measurements. For the normal distribution, the interval: ARITHMETIC_MEAN ± DEVIATION_OF_THE_ARITHMETIC_MEAN is the interval in which a true value of the quantity is with the probability of [GUM] 3, 4 Kostić 2018 Metrology Companion 93

30 6.24 Degrees of freedom Degrees of freedom, ν, of an evaluation estimated from a sample of size n is given by formula (6.64). The R is a number of evaluations estimated from this sample and used to estimate this evaluation. Also see Examples. ν = n R (6.64) The least possible sample size is 1, and from this sample it is possible to estimate an evaluation of the degree of freedom equal to 1. Evaluations with the degrees of freedom equal to 1 have the greatest possible dispersion. The dispersion is reduced by increasing the sample size, that is, the degrees of freedom, so the degrees of freedom is the indicator of reliability of an evaluation. The degrees of freedom of an evaluation is equal to the number of elements of the sample above the least number of elements necessary to compute the evaluation, plus, 1. The degrees of freedom defined by (6.64) gives a value which is a parameter of different distributions. For example, a sample from population with the normal distribution, for which (6.64) gives a degrees of freedom ν, has the T-distribution with parameter degrees of freedom equal to ν. Also, the standard uncertainty, for which (6.64) gives a degrees of freedom ν, has the T-distribution with parameter degrees of freedom equal to ν. If for a sample from a population with the normal distribution, is known its standard deviation s, and a degrees of freedom of this deviation, ν, formula (6.65) gives half-width of confidence interval, Δ, for a confidence level of an interval, P. The coefficient k is the coverage factor which can be determined from the relation among P, ν and k listed in Table 6.3 for the T-distribution. Formula (6.65) is derived from (6.77). Δ = k s (6.65) The degrees of freedom enables accurate computation of a level and interval of confidence, even in the case of small sample size. The quantities Δ and k from (6.65) can be used as an expanded standard measurement uncertainty and its coverage factor. Without giving a general definition, the concept of the degrees of freedom first started to use Sealy Gosset ( Student ) in 1908 and Ronald Fisher in Since then, the concept is based on an incomplete theory. A particular problem is the computation of the degrees of freedom. [Njegić] 5.6. [GUM] 4.2.6, G.3.2, G.3.3, G.6.4 [Eisenhauer] Kostić 2018 Metrology Companion 121

31 6.27 Confidence level and confidence interval Confidence level (or level of confidence, or coverage probability) is a probability that a value of variable is from a specified interval. Confidence interval (or interval of confidence, or two-sided confidence interval, or coverage interval) is the interval which has the specified probability that the value of variable is from it. Confidence limits are the limits of confidence interval. The number of measurement results is from a particular interval. The probability that the result is from mentioned interval is equal to the quotient of the number of results from mentioned interval, and the total number of results. The mentioned interval is the confidence interval. The mentioned probability is the confidence level. Generally, there is more than one coverage interval for a specified probability. Shortest confidence interval of a variable is a confidence interval with the narrowest width among all coverage intervals having the same coverage probability. In the case of unimodal distribution, the shortest confidence interval is to the left of the mode and to the right of the mode. Unless otherwise indicated, the confidence interval means the shortest confidence interval. A confidence level, P, of a variable x with a probability density function f(x), for confidence interval [a, b], is given by (6.76). With (6.76) there is a graphical example for the normal distribution. b P = P( a x b) = f( x) dx (6.76) For a variable with the normal distribution, it can be computed an arithmetic mean x, and standard deviation of the mean, s. For a requested confidence level, P, a confidence interval can be given in the form of: [ x k s, x + k s] or simplified x ± k s. (6.77) In the case of the normal distribution, the coefficient k can be determined from function (6.78) which gives the relation between P and k. The values in Table 6.2 are obtained from (6.78). k 2 1 P( k) = dk (6.78) 2 π k 0 e 2 The coefficient k is called the coverage factor. Product k s is a half-width of confidence interval (or one-sided confidence interval). Kostić 2018 Metrology Companion 125 a

32 A standard deviation of a variable with the normal distribution is a half-width of a confidence interval of the variable at the confidence level (of the interval) equal to One or more measurement results, from a population with the normal distribution, have the T-distribution. For details and about levels and intervals of confidence, see 6.29 T-distribution. The level and interval of confidence of the measurement result can be considered as a purpose of all determinations of measurement errors. [Ivković] PP 79, PP 81, PP 17 [GUM] 7.2.4, G.3.2, G.5.3, G.1.4 Table 6.2 For the normal distribution, N(x): relations between the confidence level, P, and coverage factor k P k Kostić 2018 Metrology Companion

33 6.38 Histogram The sample of values can be splitted into disjoint subintervals of equal widths (or cells, or bins). The histogram is the graphic representation of a number of the values that fall into each of the subintervals. The histogram is made in the following way. First, split a part of the horizontal axis which contains the values from the sample into disjoint subintervals of equal width, see Figure 6.14 b). Then draw rectangles whose bases overlap subintervals on the horizontal axis and whose heights are proportional to the number of values that fall into the subinterval of their base. The histogram appearance is significantly affected by the subintervals width and their position along the axis. The limits of subintervals can be determined according to the following two recommendations. a) If we make the histogram for a sample with a small number of possible discrete values, the subintervals should be chosen so that each discrete value corresponds to one subinterval. This is the case in Example , in which 8033 discrete measurement results have one of 16 possible discrete values, out of which only 5 values are distinguished by the number of appearances. Histograms, such as in this case, are bases for reliable estimating. b) If we make the histogram for a sample with a large number of possible values, subintervals width and position in the horizontal axis can be determined as follows. If the n values in the sample have the normal distribution and n is larger than about 25, the optimal subinterval width w, is given by formula (6.116) with the parameter C = 3.5. The standard deviation of values in the sample is denoted by s. This formula is derived from the criterion of the minimum of integrated mean squared error (IMSE). s w = C (6.116) 3 n If the values in the sample do not have the normal distribution, and n is larger than about 25, based on the experience we can also recommend using (6.116) with C = 2. In the case of discrete values, it is necessary to round the subintervals width so that each subinterval contains the same number of possible values (that is, round subintervals width to an integer multiple of a resolution). 144 Kostić 2018 Metrology Companion

34 The limits of subintervals should be chosen so that more of the values are around the middle of subintervals. In the histograms described in b), it is classified the case in Example , in which 78 discrete measurement results have one of 30 possible values. This example shows how the subintervals width affects the histogram appearance. In most statistical software a default subinterval width and position are not optimal. The height of a rectangle of a histogram is proportional to the probability that values fall into the subinterval of the rectangle base, so the histogram is the variation of the probability density function. By visual inspection of the well-made histogram, we can estimate the following properties of the distribution: a type; a mode; an existence of outliers; a dispersion of values; an existence of several maximums; a symmetry; and a peakedness. The lack of an evaluation based on the histogram is subjectivity. The approximate arithmetic mean of n values represented by a histogram, x, is given by formula (6.117) derived from (6.19). The c i is a number of values in a subinterval with the middle x i. 1 1 x ( b x + b x b x ) = b x n n = n n i i i 1 [Britannica] Statistics [Perović, communication] [Scott] [Hristov, communication] n (6.117) Kostić 2018 Metrology Companion 145

35 6.40 Filtering Filtering is a procedure to reduce random differences of values in a sequence. Filtering is most often performed by averagings. Moving average filtering uses the convolution, it is simple and optimal to reduce random differences of values in a sequence. This filtering retains sharp changes but gives the wrong idea about values and widths of peaks in the sequence. These disadvantages are worsened by decreasing a bandwidth of the filtering. In Figure 6.17 there is an example of filtering by using the convolution. The weight function (that is, window) is the Chebyshev type sequence with seven coefficients (that is, points) scaled so that their sum equals to one. (The moving average filtering using the weight function with equal coefficients.) Filtering is applied to the fluctuating discrete measurement results. These results with the corresponding histogram are also shown in Figure In addition to averaging, filtering is also implemented by fitting. Savitzki - Golay filtering uses local polynomial fitting across a moving window within the values in the sequence. The polynomial is often of the fourth degree, and it is fitted by the method of least squares. Such filtering tends to preserve the values and widths of peaks in the sequence. See [Savitzky]. [Savitzky] [Hristov, communication] [Guiñón] Figure Filtering of the sequence of discrete measurement results by using the convolution: a) the Chebyshev type weight function, b) the sequence of discrete measurement results and c) the convolution of these two functions. [Hristov, communication] 150 Kostić 2018 Metrology Companion

36 6.41 Operations on distributions and central limit theorem When a die is thrown, each number from 1 to 6 comes up with equal probability. The probability function of these numbers, P(x 1 ), is uniform and discrete, and it is represented pictorially in the following figure. Think about the probabilities to happen values which are the sum of two numbers obtained from two tosses of a die. The possible values of the sum are 2 to 12. There is only one way of making 2, two ways of making 3, etc. See the following picture. The sum 7 is the most probable because it can be achieved in the greatest number of ways. The random variable x 1, which assume values obtained from tosses of the die, has the uniform and discrete probability function. The random variable x 2, whose values are the sum of the two numbers obtained from two tosses of the die, has the triangular and discrete probability function. As an example, using the properties of probability, we can compute the probability that the sum of x 2 is 5: P(x 2 = 5) = P[(4 1) (3 2) (2 3) (1 4)] = = P(4 1) + P(3 2) + P(2 3) + P(1 4) = = P(4) P(1) + P(3) P(2) + P(2) P(3) + P(1) P(4) = = 4 (1/6) 2 = 4/36 = 1/9. We see that P(x 2 = 5) can be written as P( x = 5) = P(5 i) P( i ), which is the convolution of discrete function P with itself. Based on previous example, we can determine the probability function of the x 2, P 2 (x 2 ): 1/ 6, x1 = 1, 2,..., 6 P 1 ( x1 ) = 0, 0 x1 7 P ( x ) = P ( x i) P ( i) = P ( x ) P ( x ) i = Kostić 2018 Metrology Companion i = 1