# Ratios and Meaningfulness in Measurement

Back in 1993, a writer on the STAT-L discussion list posted comments that included the following paragraph:

"Measurement level must be considered to avoid making meaningless statements. A typical example of a meaningless statement is the claim by the weatherman on the local TV station that it was twice as warm today as yesterday because it was 40 degrees Fahrenheit today but only 20 degrees yesterday. Fahrenheit is not a ratio scale, and there is no meaningful sense in which 40 degrees is twice as warm as 20 degrees. It would be just as meaningless to compute the geometric mean or coefficient of variation of a set of temperatures in degrees Fahrenheit, since these statistics are not invariant or equivariant under change of origin. There are many other statistics that can be meaningfully applied only to data at a sufficiently strong level of measurement."

A similar statement was made by Roberts (1985, p. 312):

"To illustrate the definition [of measurement as meaningful when invariant across changes in scale], we note that it is meaningful to assert that I weigh more than the elephant in the zoo. This is meaningful because it is false under all acceptable scales of weight. Meaningfulness is not the same as truth. On the other hand, it is meaningless to assert that the temperature of this room is twice the temperature outside. For this might be true under one scale of temperature, e.g., Fahrenheit, while false under another scale, e.g., Centigrade. Yet, it is meaningful to assert that I weigh twice as much as the elephant. For if this statement is true in pounds, it is also true in grams, kilograms, etc."

Both of these statements have treated the arbitrary origins of temperature scales as though they are absolute origins. Roberts explicitly compares the absolute origin of weight measures (all scales start from no weight) with the arbitrary origins of different temperature scales (only degrees Kelvin starts from the theoretical absolute origin of no temperature).

Let's take the STAT-L writer's statement that "there is no meaningful sense in which 40 degrees is twice as warm as 20 degrees." It is obviously true that just because 40 degrees Fahrenheit is twice 20 degrees Fahrenheit, we cannot expect the associated Celsius measures (4.44 and -6.67) to be in the same relation. Where we divide 40 by 20 and get 2 in Fahrenheit, we divide 4.44 by -6.67 and get -.67 in Celsius.

But this procedure is misconceived. Measures are not just numbers but representations of amount. The statement that today is twice as warm as yesterday because today it is 40 and yesterday it was 20 is meaningful because the amount of temperature represented by the difference between 0 and 20 degrees temperature is indeed half of what is represented by the difference between 0 and 40, and that ratio difference will indeed remain constant across any scales that actually measure temperature.

For instance, the differences between the relevant associated Celsius measures are as follows:

0 F is -17.78 C.

20 F is -6.67 C.

40 F is 4.44 C.

Then,

-6.67 - (-17.78) = 11.11.

4.44 - (-17.78) = 22.22.

And

22.22 / 11.11 = 2.

Just as

40 / 20 = 2.

Temperature is invariant or equivariant under changes of origin, and so is measured on an equal-interval ratio scale. Ratio relationships such as 'twice-as' have to apply as much to the attribute being measured as they do to the numbers. A temperature of 40 degrees is twice that of 20 degrees no matter which scale it is measured on.

It would seem that familiarity with a difference model of measurement, such as Rasch's, can lead to psychometric lessons on measurement for thermometrics. For more on the lessons on measurement that thermometry can offer psychometrics, see Choppin (1985).

If we choose a suitable "origin" for our current purposes, such as "sea level" for measuring the height of mountains, then "twice as high" is definitely meaningful. If 72°F (22°C) is "comfortable", then 82°F (27°C) is "hot", and 92°F (32°C) is "twice as hot" relative to our "origin" of "comfortable". This is exactly how to choose origins for Rasch measures, so that ratio statements become meaningful, e.g., "Mary is twice as able as Joe relative to the difficulty of the test".

William P. Fisher, Jr., Ph.D.

Avatar International Inc.

William Fisher has been selected as an outstanding reviewer for Quality of Life Research Journal's 2007 editorial year.

Choppin, B. H. L. (1985). Lessons for psychometrics from thermometry. Evaluation in Education (now International Journal of Educational Research), 9(1), 9-12.

Roberts, F. S. (1985). Applications of the theory of meaningfulness to psychology. Journal of Mathematical Psychology, 29, 311-32.

Ratios and Meaningfulness in Measurement. W.P. Fisher, Jr. … Rasch Measurement Transactions, 2008, 21:4, 1139

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