Abstract
Suppose that you go to your physician and that he measures your blood pressure. He will probably repeat the measurement a few times and will usually not obtain exactly the same result through such repetitions.
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- 1.
Henceforth, we need the notion of probabilistic or random variable (we prefer the former term, although the latter is more common). Though we assume that the reader has a basic knowledge of probability theory, for the sake of convenience, we present a brief review of the probability notions used in this book in Sect. 4.1. Note in particular the notation, since we often use a shorthand one. We do not use any special conventions (such as capital or bold characters) for probabilistic variables. So the same symbol may be used to denote a probabilistic variable or its specific value. For example the probability density function of \(v\) can be denoted either as \(p_{v}(\cdot )\) or, in a shorthand notation, as \(p(v)\). For notational conventions, see also the Appendix at the end of the book, in particular under the heading “Generic probability and statistics”.
- 2.
A definition of probability distribution, also (more commonly) called the probability density function for continuous variables, is provided in Sect. 4.1.8.
- 3.
In general the “hat” symbol is used to denote an estimator or an estimated value. If applied to the measurand, it denotes the measurement value.
- 4.
We will discuss loudness measurement in some detail in Chap. 8. Readers who are unfamiliar with acoustic quantities may consult the initial section of that chapter for some basic ideas.
- 5.
In the practical implementation of the experiment, there are different ways of varying the stimulus, either through series of ascending or descending values, or as a random sequence. The variation can be controlled by the person leading the experiment or by the test subject [9, 10] . In any case, such technicalities do not lie within the sphere of this discussion.
- 6.
- 7.
In fact the variance of the sum (or of the difference) of two independent probabilistic variables equals the sum of their individual variances. Thus, in our case, \(\sigma _{u}^{2}=\sigma _{x_{b}}^{2}+\sigma _{x_{a}}^{2}=2\sigma ^{2}\).
- 8.
The device of using the abscissae of the standard normal distribution, usually called z-points, is widely used in probability and statistics and, consequently, in psychophysics too.
- 9.
- 10.
The resolution of a measurement scale is the minimum variation that can be expressed with that scale (see also the glossary, in the Appendix, at the end of the book).
- 11.
This formulation is somewhat qualitative but sufficient for the purpose of this informal discussion.
- 12.
The BIPM and the CIPM are two of the main bodies in the international system of metrology and were established when the Metre Convention was signed (1875). A concise introduction to the organisation of the system is made in Sect. 3.7.4. and additional details on how it works are given in Sect. 10.1.
- 13.
We do not use the GUM’s notation here, since we wish to be consistent with the notation used in this book. See the Appendix for further details.
- 14.
In the GUM, the expected value of a quantity is regarded as a “best estimate” of that quantity.
- 15.
This is usually called “loading effect” in the technical literature.
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Rossi, G.B. (2014). Uncertainty . In: Measurement and Probability. Springer Series in Measurement Science and Technology. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8825-0_2
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