CHAPTER 1
MEASUREMENT MODELS AND UNCERTAINTY

ALESSANDRO FERRERO1 and DARIO PETRI2

1Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano, Italy

2Dipartimento di Ingegneria Industriale, Università degli Studi di Trento, Povo, Italy

1.1 INTRODUCTION

Nowadays, we are surrounded by measuring instruments and we use them several times a day, very often unconsciously and unaware of their complexity and accuracy. To realize how instruments have become a big part of our life, just think of how many times we read the speed indicator when we drive our car, the fuel indicator to know when we have to refill the tank, or, when we eventually refill it, the meter on the fuel pump.

Interestingly enough, we usually don't pay too much attention to the accuracy of the instruments we use, even if we rely on their indication to make important decisions, such as, for instance, driving safely or paying the right amount of money for the quantity of good we purchase. Even more strangely, the only instrument we generally adjust to a reference is our watch, which is probably the most accurate instrument we use in our everyday life: even the cheapest ones don't lag or lead for more than one second a day, which means that their relative accuracy is in the range of 1  ·  10- 5!

The above examples give us clear evidence that we do use and read instruments, but they still leave an important question open: are we also making a measurement? Trying to answer this question opens also another fundamental question: which is the difference between reading an instrument and making a measurement?

This chapter is aimed at providing an answer to this question.

1.2 MEASUREMENT AND METROLOGY

To understand what measuring means, let's start from the definition of measurement, taken from the International Vocabulary of Metrology (VIM) [1].

Measurement

Process of experimentally obtaining one or more quantity values that can reasonably be attributed to a quantity.

So, a measurement process provides, as a part of the measurement result, one or more quantity values that can be attributed to a quantity intended to be measured, that is also called, always according to the VIM [1], measurand.

To fully understand this definition, we have to refer to the definition of quantity. We can find it again in the VIM.

Quantity

Property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference.

The VIM states that a reference can be a measurement unit, a measurement procedure, a reference material, or a combination of such.

When physical properties are considered, the reference is generally a measurement unit, whilst, when chemical measurement are considered, the reference is quite often a reference material.

The quantity values provided by the measurement are therefore a number and a reference together expressing the magnitude of a quantity [1].

Is this the measurement result? Or, better, can a measurement result be expressed only by a number and a reference? As we will see later in section 1.5 of this chapter, a measurement procedure cannot provide the "true" value of a measurand, due to a number of factors that we will thoroughly discuss later. This means that a measurement result can only provide a finite amount of information about the measurand, and we must know if that amount is enough for the intended use of the measurement result. Otherwise, the measurement result would be meaningless.

Therefore, any measurement result has to be provided with an attribute capable of quantifying how close to the measurand's value the obtained quantity value is. This attribute is called uncertainty, and the correct definition of measurement result, as provided by the VIM, is as follows.

Measurement result

Set of quantity values being attributed to a measurand together with any other available relevant information.

In a note to this general definition, the VIM states that:

A measurement result is generally expressed as a single measured quantity value and a measurement uncertainty.

The above general definitions have introduced a number of concepts (quantity value, reference, relevant information, uncertainty), that will be covered in the next Sections, and show that a measurement is a definitely more complex procedure than simply reading an instrument.

The science that includes all theoretical and practical aspects of measurement, regardless to the measurement uncertainty and field of application, is called metrology [1]. Its definition, as provided by the VIM, is as follows.

Metrology

Science of measurement and its application

As many other sciences that have a deep impact on human life, metrology finds its roots back into the ancient times, and the evolution of the human needs had a significant impact on its development, and its present formulation. Therefore, before analyzing the fundamental concepts of nowadays metrology, let us find an answer to two basic questions: why do we measure? And for what do we use our measurement results?

A quick glance into the past will help us to find the answers to these questions.

1.3 MEASUREMENT ALONG THE CENTURIES

At the beginning of the human adventure, the measurement concept did not exist and the experimental activity was confined to direct observations. Instruments did not exist, and our senses were the only available tools to observe and somehow quantify the reality.

The first physical quantity that was barely measurable with our senses and probably started metrology was time. We can only imagine that this interest was related to the impact that the night and day cycle, the lunar period and the rotation of seasons have on human life and the human ability of predict them to optimize all activities aimed at providing food, from hunting to agriculture and cattle breeding.

As a matter of fact, the archeological findings from prehistory show with little doubt that the first conceived measuring instruments were aimed at the measurement of time, through the observation of the stars or the sun. Although the most famous among these instruments is the Stonhenge Circle, its "operating principle" is not yet clear. On the other hand, many remains have been found from a later period, across prehistory and history, that proved how time was measured through the observation of the displacement of the shadow of a vertical device (the gnomon) created by the sun on the ground [2].

However, at that time, this activity was more of a religious and prophetic kind, rather than aimed at increasing knowledge. We had to wait until the organized social structures appeared to meet awareness of knowledge and a systematic discussion about its meaning.

1.3.1 Measurement in Ancient Greece

There is no doubt that ancient Greece was the cradle of philosophy, and philosophy was considered the most important means of knowledge. We have also evidence, from documents and archeological findings, that instruments were used, mainly for length and capacity measurements. However, the great philosophers of that time did not consider instruments as tools to advance knowledge.

The motivation can be found in the dichotomy between the world of philosophical abstractions and the empirical world of observations, clearly expressed by Socrates in the fifth century BC.

In Socrates' philosophy, we can build our knowledge by defining abstract models that allow us to explain, through suitable logical steps, what happens to us. In this way, we can build our own world of philosophical abstractions.

On the other hand, there is a physical world around us, that shows up through a number of occurrences, facts and events, and that, at Socrates' time, we could perceive only through our senses. This is the so-called empirical world and, according to Socrates, the abstractions were aimed at explaining the true essence of the empirical world. In his philosophy, abstract logical constructions were the only way to knowledge, and observations had the simple role of triggering speculation. The use of observation results as a way to validate abstract models was not part of Socrates' approach, and was not part of the scientific method at least until Galileo's time. The subordinate role assigned to observations with respect to logics has lasted for so many centuries that even now the word "empiricism" implies a negative connotation.

1.3.2 Measurement in the Roman Empire

We have to wait until the Roman Empire to discover, from the many available historical documents and texts, that they had a rather modern approach to measurement. Indeed, we know that they had standards (at least for length, mass, and capacity), these standards were approved, kept and maintained by the central government, and secondary standards were disseminated in the provinces of the empire.

Evidence of the importance assigned by Romans to measurement is given by the fact that the primary standards were kept in the temple of Iuno Moneta.1 Only valuable and important objects were kept in the house of a God. And only money-related objects were kept by the God of Money!

According to these facts, we expect to find several traces, in the Latin...