Uncertainty, Calibration and Probability

The Statistics of Scientific and Industrial Measurement

C.F Dietrich

• Publisher: Routledge
• Year: 2017
• Print ISBN: 9780750300605
• eBook ISBN: 9781351406277
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

All measurements are subject to error because no quantity can be known exactly; hence, any measurement has a probability of lying within a certain range. The more precise the measurement, the smaller the range of uncertainty. Uncertainty, Calibration and Probability is a comprehensive treatment of the statistics and methods of estimating these calibration uncertainties. The book features the general theory of uncertainty involving the combination (convolution) of non-Gaussian, student t, and Gaussian distributions; the use of rectangular distributions to represent systematic uncertainties; and measurable and nonmeasurable uncertainties that require estimation. The author also discusses sources of measurement errors and curve fitting with numerous examples of uncertainty case studies. Many useful tables and computational formulae are included as well. All formulations are discussed and demonstrated with the minimum of mathematical knowledge assumed. This second edition offers additional examples in each chapter, and detailed additions and alterations made to the text. New chapters consist of the general theory of uncertainty and applications to industry and a new section discusses the use of orthogonal polynomials in curve fitting. Focusing on practical problems of measurement, Uncertainty, Calibration and Probability is an invaluable reference tool for R&D laboratories in the engineering/manufacturing industries and for undergraduate and graduate students in physics, engineering, and metrology.

1

Uncertainties and Frequency Distributions

Introduction: Calibration

1.01 It is only in this century and to a large extent in the last fifty years or so that the real importance of calibration measurements has come to be realized. In our industrialized society it is essential that instruments used in quality control and for the measurement of precision workpieces be properly calibrated. Interchangeability between mass produced parts, a need which is a cornerstone of present-day industry, can only be satisfactorily achieved if measuring equipment is properly calibrated.

1.02 The need for action at national level was realized some fifty years ago by Australia, with its National Australian Testing Authority (NATA), and a little later, after World War II, by the Japanese. In Great Britain the British Calibration Service (BCS) was set up in late 1966 with a remit to establish a national calibration service. This government-backed organization decided that it would be too expensive to set up a chain of government calibration laboratories throughout the country and chose instead to organize a scheme whereby existing laboratories in industry, the universities and in the government sector would be approved by BCS to issue authenticated calibration certificates, having gone through a strict assessment exercise. Measurements were divided between different fields e.g. electrical DC and LF mechanical, optical, flow, thermal, radiological etc. Each field was supervised by a committee of experts, who drew up a series of criteria documents covering each field. These criteria documents dealt with laboratory equipment, laboratory environment, laboratory personnel and also specific types of measurement within each field. The service was run by a headquarters staff of experts, who assessed each laboratory seeking approval, in conjunction with outside experts.

1.03 It was realized early on for a laboratory seeking approval that the most important factor governing its approval was proof that it could measure to a given accuracy. This was achieved by submitting to each applicant laboratory a number of workpieces or instruments which had been previously calibrated at the National Physical Laboratory (NPL). The applicant laboratory’s measurements were then compared with those of the NPL, and if the differences were sufficiently small, the laboratory was granted approval to issue authenticated certificates for those measurements. Further, it was also realized that an approved laboratory would need to be supervised, to see that its equipment was kept in calibration, its laboratory environment maintained etc., and most important of all that its measurement capability was maintained. This was achieved by setting up a so-called audit measurement scheme, which entailed the sending round to approved laboratories a series of workpieces and instruments, previously calibrated at the NPL, for measurement by the laboratories.

1.04 Another crucial factor was realized, namely that all measurements have a degree of uncertainty associated with them, and that if the measurement values given on certificates were to have any validity, the degree of this uncertainty would need to be carefully assessed and stated. From statistical theory, the error or particular uncertainty of measurement has a probability associated with it. Generally the larger the error, the smaller the probability of such an error occurring. It was decided that appropriate criteria for calibration should be that the correct value of the quantity measured should lie between equal plus and minus limits about a mean value with a probability of 0.954 or, put another way, that there was a 95.4% chance that the correct value should lie between the stated limits. Approximately, this means that there is a 1 in 20 chance of the correct value lying outside these limits. In special cases different probability limits may be chosen, for example 0.9973 which gives the probability of the correct value lying outside these limits as approximately 1 in 400.

1.05 The subject of uncertainty of measurement raises many problems and it is the purpose of this book to provide an understanding of these problems and how to deal with them.

1.06 The British Calibration Service (BCS) became part of the National Physical Laboratory in 1977 and has now been integrated with another NPL-based organization, the National Testing Laboratory Accreditation Scheme (NATLAS), to form a new organization known as the National Measurement Accreditation Service (NAMAS). The two sections of the service will cover calibration (BCS) and testing (NATLAS).

1.07 Many countries of the EC have now followed the example of the United Kingdom and have set up their own national calibration services, which in general are closely modelled on the British scheme. The arrival of national calibration schemes in other countries of Europe has now led to the signing of reciprocal agreements between some of these countries, which cover the recognition of each other’s national calibration scheme certificates. This enables a purchaser of a piece of equipment or of an instrument to obtain an authenticated certificate, traceable to international standards, which gives a factual truthful assessment of the equipment’s specification or measurement capability.

1.08 The origin of calibration goes back to the early eighteenth century, when the words ’calibre’ and ’calliper’ were different versions of the same word which described an instrument for measuring or comparing bores. The version ’calibre’ became associated with guns and is now synonymous with the internal diameter of a gun barrel, whilst the transitive verb ’to calibrate’ became associated with the actual measurement of gun bores and also with a so-called calibre scale. On this scale the distances of graduations from an initial line were proportional to the cube roots of the natural numbers. The first interval represented the diameter of a cannon ball of one pound weight, and successive intervals from zero thus gave the diameters of balls whose weights were successive multiples of one pound weight.

1.09 From its earliest known use ’to calibrate’ was thus associated with measurement and with scales, and later in the mid-nineteenth century the term became associated with graduations on thermometer tubes, barometer tubes and pressure measuring devices, the graduations being spaced so as to correct for any irregularities in the bores of the tubes. The term had now become linked not only with measurement but with the idea of correcting errors, that is by making the graduation intervals of unequal size. In modern times, where the emphasis is on mass production and standardization, the term ’to calibrate’ has become associated with instruments which already have scales, and a set of calibration measurements usually takes the form of a table of corrections to be appliced to the instrument readings in order to allow for errors of one sort or another. In its modern use the term ’to calibrate’ means to make measurements which are traceable to the national standards. Thus, if a piece of measuring equipment is used to measure a series of workpieces that have been previously measured on calibration equipment, that is equipment that has very small errors compared with the measuring equipment under test, then the differences between corresponding measured quantities will give the errors in the measuring equipment. The sizes of the calibrated workpieces, obtained from the calibration equipment, give a close approximation to the most probable values of these sizes. A set of calibration measurements thus gives what may be thought of as the most probable value of a measurement quantity to be associated either with a component or an instrument.

Uncertainty of Measurement

1.10 No measurement is perfect; a single measurement if repeated several times may give as many slightly differing results, whilst the results if repeated on another day may all show a slight shift. This idea of an uncertainty in measurements may seem disconcerting, but it is nevertheless something that has to be accepted and as far as possible allowed for. In fact it is the main aim of this book to show how the degree of uncertainty of calibration readings can be estimated. A general definition of uncertainty of measurement is that it is the residual error which may exist in an instrument or workpiece after calibration corrections have been made. In other words, all measurements have a certain indeterminancy associated with them, however small, and the phrase ’uncertainty of measurement’ is used to describe the magnitude of indeterminancy or error which may be present in a set of calibration measurements. A much more precise definition will be given later after consideration has been given to the concept of distributed uncertainties.

Sources of Uncertainties

1.11 Before considering the nature of uncertainties more precisely, we shall consider the various types of uncertainty which may be present in any calibration measurement and which add their contribution to the total uncertainty. The following catalogue of uncertainties is not exhaustive but covers most of the principal sources of uncertainty:

1. Uncertainties in standards or in calibration equipment;
2. Uncertainties due to operator error;
3. Resolution or discrimination uncertainties;
4. Environmental uncertainties, including variation of temperature, of pressure, of flow rate, of power supplies, etc.;
5. Lack of repeatability-instability uncertainties;
6. Functional uncertainties, caused by the malfunctioning of equipment;
7. Uncertainties caused by lack of cleanliness;
8. Uncertainties due to poor quality surface texture and incorrect geometry;
9. Uncertainties associated with lapse of time, which produces changes in equipment or workpieces.

The last type of uncertainty can be assessed only from several sets of measurements taken over an extended period of time. When assessing the total uncertainty of measurement of a set of measurements every possible source of uncertainty should be accounted for, each source making its contribution to the total uncertainty.

Uncertainties as Distributions

1.12 We must examine more closely what we mean by an uncertainty. One very common concept is that an uncertainty is the amount by which a measurement differs from some true value. This uncertainty is often thought of as having a maximum value, so that we speak of the magnitude of any measurement plus or minus some figure which specifies the uncertainty of measurement or error. It might also be thought that if the uncertainties from different sources are known, then the total uncertainty is the sum of the component uncertainties. As we shall see, this simple idea of uncertainties, if adopted, would lead to completely erroneous results.

1.13 Let us consider a simple example of a source of error. Consider an angle measuring device such as a rotary table, fitted with a precise circular scale and microscope capable of reading the angular orientation of the table top to a second of arc. A piece of glass with a straight line engraved on it is mounted horizontally on the table top so that the prolongation of the line passes through the axis of rotation of the table top. A second piece of glass, with two parallel lines engraved on it, spaced about three times the width of the first line, is mounted just above the first piece of glass with its two lines radiating approximately from the centre of rotation of the table. The second piece of glass is mounted rigidly and independently of the table top. If both sets of lines are now viewed through a microscope mounted above the table top, the single line can be moved, so as to fit centrally between the two lines, by rotating the fine rotation screw of the table. The orientation of the table top in degrees, minutes and seconds is now noted. If the table top is now rotated a little and the single line again positioned centrally between the two lines it is very unlikely that the second angular orientation of the table top will be the same as the first. If a fairly large number of such readings are taken, say one hundred and ten, then many of these readings will be different.

1.14 Let us now consider these one hundred and ten readings and attempt to analyse them. Firstly we take the sum of all the readings and divide by the number of readings to obtain the mean. The readings are now arranged in ascending order of size and the difference between the largest and smallest readings is divided into eleven equal sub-ranges. In general the number of sub-ranges is usually taken between eight and sixteen, depending on the number of readings taken and their distribution. A table can now be made of the number of readings occurring in each sub-range, and a diagram constructed, consisting of a series of cont...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright
• Dedication
• Table of Contents
• Preface
• 1 Uncertainties and Frequency Distributions
• 2 The Gaussian Distribution
• 3 General Distributions
• 4 Rectangular Distributions
• 5 Applications
• 6 Distributions Ancillary to the
• 7 A General Theory of Uncertainty
• 8 The Estimation of Calibration Uncertainties
• 9 Consistency and Significance Tests
• 10 Method of Least Squares
• 11 Theorems of Bernoulli and Stirling and the Binomial, Poisson and Hypergeometric Distributions
• Appendix I Tables
• Appendix II
• Appendix III
• Appendix IV The uncertainty in the difference in height between points on a surface (viz. a surface plate)
• Appendix V Confidence and tolerance limits of y for given x, that is for y|x for a mean line
• Appendix VI Computational formulae
• Appendix VII A suite of formulae for the computation on a computer of the approximate or exact values of various statistical functions
• Bibliography
• Index