As noted in [4], the non existence of a global mean surface temperature follows from the fact that temperature is an ordinal random and not interval as claimed in some statistics texts. As illustrated in [3] the same data set can consistent with both warming and cooling. The purpose of this post is review some of the mathematics governing changes of scale and in passing to deduce the ordinality of temperature.

## The Stevens’ taxonomy

Although well known, restrictions on changes of scale are often not given sufficient importance in science education. It, indeed is, ironic to find more discussion of this matter in Philosophy, Psychology, and Sociology texts than in Natural Science texts. In fact, it was a Harvard psychologist, S. S. Stevens [6], who produced a well known taxonomy of measurement scales based on the classes of transformation admissible under a change of scale. For the convenience of the reader, we include a brief outline of Stevens’ taxonomy. In this classification a variable is called *ratio*, *interval*, * ordinal* or *nominal* according to the class of transformations permitted under a change of scale. The classes of admissible transformations are summarized in Table 1.

By means of examples from Physics we illustrate how each of these categories arises. We begin by briefly examining the transformation of random variables in general.

## Transformations of Random Variables

Let be the set of all outcomes of a random experiment and let be a random variable defined for each outcome . If is a real-valued function, then the composition defines a new random variable often referred to as the *transformed* random variable.

Such transformations have important mathematical and physical applications. Common statistical applications include standardization, grouping of data, changing the form of a distribution, and changes of frame and scale. For example, if is a random variable with mean and variance , then transforms to a “standardized” random variable with mean 0 and variance 1. Geometrically, is just the signed distance between and the mean measured in standard deviations.

If , then is a random variable whose mean is just the variance of . If is a continuous random variable and a step function approximating the identity function, then the transformed variable will be a discrete variable with parameters and characteristics not too different from those of , a fact useful in the grouping of experimental data and its depiction by, say, a histogram. In Physics, can represent a Galilean or Lorentzian change of frame or a change of scale — say from feet to inches. What mainly concerns us here are changes of scale.

## Linear Changes of Scale

When the transformation of a random variable represents a change of scale of some quantity, then the axioms governing that quantity can impose restrictions on the functions that may be used to transform the given variable. For example, if represents the length in feet of a body B and , then is the length in inches. However, if then the variable is not admissible as a measure of length. We now show, subject to a continuity requirement, that if the length of a concatenation of two bodies and is equal to the sum of their lengths, i.e.,

(1)

then the only admissible changes of scale for length are linear functions with slope . In (1), the binary operation of concatenation does not combine numbers but rather represents the new body formed by the juxtaposition of the two objects and . Now, if

is a new measure of length (in the rest frame) of a body , we require that in the new scale:

(2)

If this is to hold for bodies and of arbitrary length, then the function must satisfy:

(3)

for all . As is well known, the additivity condition (3) is not enough to ensure the linearity of . However, if the lengths of a sequence of bodies tend to zero in one scale, then they tend to zero in any scale. This additional physical axiom leads us to impose upon the additional condition that

(4)

i.e., that is continuous from the right at the origin. As is well known, see for example Truesdell [3], assumptions (3) and (4) are sufficient to prove that must be linear . Moreover, since length is non negative

This proves that length is a ratio variable in Stevens’ taxonomy.

Ratio variables arise whenever objects can be concatenated in such a way that equation (1) holds. Examples of such variables in Physics are time intervals, angle, mass, etc. However, variables such as position on a time scale and position on a length scale are not ratio variables since their admissible class of changes of scale is larger.

## Affine Changes of Scale

A random variable is called *interval* if it admits only changes of scale where is affine:

(5)

For example, if is the coordinate of a point on a length scale and is a fixed point on that scale, then is a length which we already know to be a ratio variable. It follows that if and are the corresponding points after a change of scale, then where . Thus, where is affine of the form of equation (5). This proves that position on a length scale is an interval variable in Stevens’ taxonomy. Physically, the change of scale corresponds to a change of unit of length followed by a change of origin .

In general, if the distance of a point from the origin of some scale is a variable of ratio type, then the corresponding position of that object on the scale will be a variable of interval type. Thus, for example, the time of occurrence of an event is an interval variable whereas the time elapsed during some process is a ratio variable.

## Ordinal Changes of Scale

In classical thermodynamics, the only restriction on a change of temperature scale is that it must preserve the order relation “hotter than” on the class of homogeneous fluid bodies. Since temperature is a measure of degree of hotness, this implies that for any two such bodies and ,

Moreover, if it be demanded that this hold in any other temperature scale we must also have

and so

If this is to hold for fluid bodies and of arbitrary temperature, the change of scale must be a strictly increasing function. Since there are no other restrictions implied by the usual thermodynamic axioms, it follows that temperature is an ordinal random variable and not, as frequently claimed, interval. In particular, average temperature becomes a scale dependent object.

## Arbitrary Changes of Scale

Finally, consider a quantity such as particle type

in which no particular order relation is specified on the set of particles. The variable is now merely an indicator of category and the values of are completely arbitrary. A change of scale such as , which transposes the types and is perfectly admissible as indeed would any arbitrary bijection. Thus, in Stevens’ taxonomy, would be nominal.

© Philip Pennance 01-30-2005## Bibliography

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