Hand (2005 )
Bachelor Eindproject (WB3BEP-16)
Technische Universiteit Delft
Aanbevolen voor jou
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june
focus
81
SSize matters—how measurement ize matters—how measurement
ddefines our worldefines our world
“Measurement is the contact of reason with nature”—Henry Margenau
We live in a world of measurements, says David Hand. We talk about the
weight of ingredients in cooking, the scores of students in tests, the infl a-
tion rate, the distance to the moon, the strength with which an opinion is
held, and so on and so on. In brief, we see the world through the specta-
cles of quantifi cation. But, occasionally, distortions or fractures occur in
the lenses of quantifi cation which cast doubt on our understanding of the
world—doubts which sometimes raise questions about the reality we be-
lieve we are perceiving.
Here is a very simple example. A few years ago, an article in The Times said: “Temperatures in London were still three times the February aver- age, at 55°F (13°C) yesterday.” Given this, one might reasonably ask: what is the February aver- age? That’s easy enough. It is 55/3=18 1 / 3 ºF. Or, perhaps alternatively, it is 13/3=4 1 / 3 ºC. But wait a moment: 18 1 / 3 ºF is below freezing, whereas 13/3=4 1 / 3 ºC is above freezing. Both of these can- not be right. We appear to have a contradiction. Of course, this is a particularly simple exam- ple, and readers will no doubt immediately see why the contradiction arose (even if they will not be able to decide which answer is correct). But nevertheless it raises all sorts of questions. Do other, less obvious contradictions arise, which we do not notice? What if only one of the answers is given—how do we know it is the correct one? What mathematical or statistical operations oth- er than averaging might lead to contradictions? How should we resolve such contradictions when they do arise? And, of course, more generally, what does it mean to “measure” something? In short, what is measurement? A key feature of measurement is that it serves to represent relationships between objects by re- lationships between numbers: we compare the height of the Empire State Building with the length of a (1-foot) unit ruler—and fi nd that one is 1250 times the other. At the simplest lev- el, these relationships will be in terms of a single characteristic or attribute of the objects—their weight, length, intelligence, brightness or what- ever. Measurement, then, establishes a mapping from the empirical system of the objects to a numerical system. The mapping of the length of sticks to numbers representing those lengths is a very simple example. We can place the ends of
two parallel sticks against a wall and see which of their other ends projects further into the room. If we call the sticks A and B, then we can use the numbers x(A) and x(B) to represent their lengths, and we can choose numbers such that x(A) > x(B) whenever AZB, where AZB means that stick A projects further than stick B. Of course, this mapping is not unique. Any monotonic increasing transformation of the numbers x, to numbers y, say, will also preserve the empirical relationship, so that AZB means that y(A) > y(B). In fact, we can go further with this example. There are also other empirical relationships be- tween the sticks, and we might try to fi nd num- bers which preserve those relationships as well. For example, if we place one end of stick A against the wall, and then put stick B at the other end of A, in a straight line with it, then we can fi nd an- other stick C which has the same empirical length as this combination of A and B. And we can fi nd numbers to represent the lengths of the three sticks A, B and C such that the number assigned to C is the sum of the numbers assigned to A and
“Statistics, derived from properly
measured attributes, add to the
richness, depth and understanding of
life, deepening our appreciation of it,
with the potential for making it better”
82 june
B. In fact, of course, these numbers would be our usual, familiar, numerical lengths of the sticks. But, as we all know, even these numbers are not unique. An arbitrary rescaling would yield alternative legitimate numerical representations of the lengths. For example, we can multiply by 2 to change the representation in inches to that in centimetres. The formal exploration of the use of nu- merical systems for representing relationships between objects began around the end of the 19th century, at a time when so much else about our understanding of the universe was changing. Atomic theories were beginning to be accepted as a description of reality, and not merely as a convenient mathematical fi ction. Relativity and quantum mechanics were about to shake the very foundations of physics. And the axioms of formal theories of measurement began to be laid. In doing so, however, they also laid the foundations for a major contro- versy which was to dog measurement, and in particular its relationship to statistics, for most of the 20th century. The problem was that all of these earlier axiomatisations assumed as a basic starting point that there was some em- pirical operation, equivalent to our end-to-end positioning of sticks, which could be mapped to addition. That’s great for length, weight (balance two rocks on one pan of a weighing scale by a single rock on the other) and other simple physical systems, but cracks appear in this approach when we try to measure more complicated things. In particular, the problem seemed especially acute in psychology: it is not
matical work has explored how many scale types there are, in terms of mappings to real numbers which preserve the relationships between ob- jects in the system being measured. It turns out that Stevens was basically right: with a few ad- ditions, his scales are all that can exist. In an ordinal scale, for example, any strictly monotonic transformation of the numerical representation of the relationship between the objects leads to another which also preserves the relationships between objects. In a ratio scale, any rescaling transformation will do. Transformations which preserve structure in this way are called admis- sible transformations. Based on these ideas, the philosopher Duncan Luce coined a principle for scientifi c laws. He said that a scientifi c law had to satisfy two proper- ties:
(a) admissible transformations of the independent variables had to lead only to admissible transformations of the dependent variables; and (b) the mathematical structure of theories should be invariant under admissible transformations.
In fact, Luce’s principle also stimulated consider- able controversy, and Luce eventually retracted the term “principle”. But, regardless of what one calls it, it is an immensely powerful idea. It un- derlies the ideas of dimensional analysis widely used in physics and engineering, and also used in economics and other areas. It allows one to spot, almost at a glance, certain kinds of errors in formulae: to detect the kind of creases in the apparent fabric of reality that I mentioned at the start. For example, a textbook which will remain nameless gives the following for the probability density function of the sample variance:
However, it only takes a moment to see that the units in which this is measured are not the cor- rect units for the density of a variance. In fact, even when dealing with empirical ad- dition operations, confusion can arise. An empiri- cal operation can be mapped to addition (think of placing electrical conductors in series, and map- ping to numbers called “resistance”) but it can also be mapped to other numerical operations. For example, we could map the empirical opera- tion of placing electrical conductors in series to the numbers which combined not by addition
but by the operation
111 xy x y . The numbers resulting from this mapping are called “conductance.” Although the numbers re- sulting from the two mappings will be different, there must obviously be a 1–1 mapping between them (since, for example, the order relationship is preserved in both cases). In fact, as many readers will know, the relationship is given by the reciprocal transformation:
1232 122 exp 1 2 2
n n n const s n s V V
u§· ªº ̈ ̧ ¬¼ ©¹
“The formal exploration of the use of
numerical systems for representing
relationships between objects
began around the end of the 19th
century, at a time when so much
else about our understanding of the
universe was changing”
.
at all obvious what “empirical addition” opera- tion can be made for perceived loudness (is the loudness of sound C the same as the “com- bined loudness” of sounds A and B?). And as for things such as anger, pain, depression, and so on, well, things seemed hopeless. This controversy stimulated some sharp ex- changes. The physicist Norman Campbell re- marked: “Why do not psychologists accept the natural and obvious conclusion that subjective measurements of loudness in numerical terms ... are mutually inconsistent and cannot be the basis of measurement?”. But he did not restrict his criticisms to psychologists, also saying: “The most distinguished physicists, when they at- tempt logical analysis, are apt to gibber, and probably more nonsense is talked about meas- urement than about any other part of physics.” The psychophysicist Samuel Stevens remarked that Campbell “seems to contribute his fair share” to this nonsense. When this controversy later spilled over into whether and how measure- ment issues should impact on statistical analy- sis, others were equally confi dent—“This par- ticular question admits of no doubt whatsoever” (Norman Anderson); “The height of absurdity” (John Gaito)—even if they were not always on the same side of the argument. Psychologists reacted to the assault on the foundations of their discipline in various ways. Some ignored it. Others adopted an explicitly operational perspective: provided that a well- defi ned procedure had been used to map the objects to numbers then it represented measure- ment. But Stevens presented a deeper analysis. He argued that measurement need not hinge on the notion of an empirical addition operation, but that other kinds of operation and relation- ship could be used, and that these produced different kinds of measurement scales. This led to the nominal, ordinal, interval and ratio scales with which some statisticians and all psycholo- gists are now very familiar. Stevens’s work, carried out in the 1930s and 1940s, was fairly intuitive, but recent mathe-
( )
Hand (2005 )
Vak: Bachelor Eindproject (WB3BEP-16)
Universiteit: Technische Universiteit Delft
Aanbevolen voor jou
Andere studenten bekeken ook
- Dynamic positioning of an Unmanned Surface Vehicle
- Digital Twins for Autonomous Greenhouse Control
- Dimensionality Reduction for Big Data- Molecular Imaging with Imaging Mass Spectrometry
- Calibration of advanced wind turbine controllers with iterative learning algorithm
- Small scale Motion compensation platform
- DCSC BEP Paper Guidelines