On the ordinality of temperature.

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.

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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 \Omega be the set of all outcomes of a random experiment and let X (\omega ) be a random variable defined for each outcome \omega \in \Omega. If g is a real-valued function, then the composition Y=g(X(\omega)) defines a new random variable Y often referred to as the transformed random variable.

    \[\xymatrix@!=3pc{\Omega \ar[r]^X \ar[rd]_{g\circ X}& \RR \ar[d]^g\cr& \RR}\]


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 X is a random variable with mean \mu and variance \sigma, then g(x)=(x-\mu)/\sigma transforms X to a “standardized” random variable Z with mean 0 and variance 1. Geometrically, Z\left(\omega\right) is just the signed distance between X(\omega) and the mean \mu measured in standard deviations.

If g(t)=(t-\mu)^2, then Y(\omega)=g(X(\omega)) is a random variable whose mean EY is just the variance of X. If X is a continuous random variable and g a step function approximating the identity function, then the transformed variable Y will be a discrete variable with parameters and characteristics not too different from those of X, a fact useful in the grouping of experimental data and its depiction by, say, a histogram. In Physics, g 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 g that may be used to transform the given variable. For example, if X(B) represents the length in feet of a body B and g(x) = 12x, then Y =g(X(B)) is the length in inches. However, if g(x) = \sqrt{x} then the variable Y(B)=g(X(B)) is not admissible as a measure of length. We now show, subject to a continuity requirement, that if the length of a concatenation B \oplus B ^{\prime} of two bodies B and B^{\prime} is equal to the sum of their lengths, i.e.,

(1)   \begin{equation*} X(B \oplus B^{\prime} )=X(B)+X(B^{\prime})\end{equation*}

then the only admissible changes of scale for length are linear functions Y = cX with slope c>0. In (1), the binary operation \oplus of concatenation does not combine numbers but rather A \oplus B represents the new body formed by the juxtaposition of the two objects A and B. Now, if

    \[Y(B) =g(X(B))\]

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

    \[Y(B\oplus B^{\prime})=Y(B)+Y(B^{\prime} )\]

That is,

(2)   \begin{equation*} g\left[X(B\oplus B^{\prime} )\right]=g\left[X(B)\right]+g\left[X(B^{\prime} )\right]\end{equation*}

From (1) and (2) we have

    \[g\left[X(B)+X(B^{\prime} )\right]=g\left[X(B)\right]+g\left[X(B^{\prime} )\right]\]

If this is to hold for bodies B and B^{\prime} of arbitrary length, then the function g must satisfy:

(3)   \begin{equation*} g(x+y)=g(x)+g(y)\end{equation*}

for all x, y \ge 0. As is well known, the additivity condition (3) is not enough to ensure the linearity of g. 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 g the additional condition that

(4)   \begin{equation*} \lim_{h \rightarrow 0^{+}}g(h)=0\end{equation*}

i.e., that g 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 g must be linear g(x) = cx. Moreover, since length is non negative

    \[c = g(1) > 0\]


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 Y(\omega)=g(X(\omega)) where g is affine:

(5)   \begin{equation*} g(t) = at+b, \quad a >0.\end{equation*}

For example, if X = X(P) is the coordinate of a point P on a length scale and X_0 is a fixed point on that scale, then X-X_0 is a length which we already know to be a ratio variable. It follows that if Y and Y_0 are the corresponding points after a change of scale, then Y-Y_0 = a (X-X_0) where a > 0. Thus, Y(P) = g (X(P)) where g is affine of the form of equation (5). This proves that position X on a length scale is an interval variable in Stevens’ taxonomy. Physically, the change of scale x \mapsto ax+b corresponds to a change of unit of length x\mapsto ax followed by a change of origin x\mapsto x+b.

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 \prec “hotter than” on the class of homogeneous fluid bodies. Since temperature X is a measure of degree of hotness, this implies that for any two such bodies B and B^{\prime},

    \[B \prec B^{\prime} \Leftrightarrow X(B) < X(B^{\prime})\]

Moreover, if it be demanded that this hold in any other temperature scale Y(B) = g(X(B) we must also have

    \[B \prec B^{\prime} \Leftrightarrow Y(B) < Y(B^{\prime} )\]

and so

    \[X(B) < X (B^{\prime} ) \Leftrightarrow g ( X(B) ) < g ( X(B^{\prime} ) )\]

If this is to hold for fluid bodies B and B^{\prime} of arbitrary temperature, the change of scale g 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

    \begin{equation*}X=\begin{cases} 1 & \text{if type $= \Omega^-$,}\\2 & \text{if type $ = K$ meson,}\\3 & \text{if type$= \Sigma$ hyperon. }\end{cases}\end{equation*}

in which no particular order relation is specified on the set of particles. The variable X is now merely an indicator of category and the values of X are completely arbitrary. A change of scale such as g(1) = 2, g(2) = 1 which transposes the types \Omega^- and K is perfectly admissible as indeed would any arbitrary bijection. Thus, in Stevens’ taxonomy, X would be nominal.

© Philip Pennance 01-30-2005

Bibliography

  1. C. Essex, R. McKitrick, B. Andresen: Does a Global Temperature Exist?; Journal of Non-Equilibrium Thermodynamics, vol. 32, p. 1-27 (2007).
  2. Philip Pennance, A Fable of Global Warming, 2005
  3. Philip Pennance, On the non existence of a global mean surface temperature I.
  4. Philip Pennance, On the non existence of a global mean surface temperature II , 2005
  5. S. S. Stevens. On the theory of scales of measurement. Science, 103:677–680, 1946.
  6. Clifford Truesdell, The Tragicomical History of Thermodynamics 1822-1854., Springer Verlag, 1980.
  7. Clifford Truesdell, Rational Thermodynamics, Springer- Verlag, 1984.
  8. C. Truesdell and S. Bharatha. Classical Thermodynamics as a Theory of Heat Engines. Springer Verlag, 1977.