What is a dimensionless indicator?

Question

What is a dimensionless indicator? I assume it refers to the fact that it is consistent in the way that it doesn't vary disproportionately at the variation of its factors.

Answer 1

I'm an engineering guy, so my first thought was "dimensionless number" but I wanted to be sure. I looked around and found your term here, and here that support a variation on it.

A dimensionless number or dimensionless variable is something used extensively in physics and engineering because they sometimes can be amazingly informative.

Here are some dimensionless numbers (ref):

• Archimedes Number: $ Ar = \frac {g \rho L^3}{\mu^2}$ (ratio of gravitational force to viscous force)
• Biot Number: $ Bi = \frac {h L} {k}$ (ratio of internal thermal resistance to boundary thermal resistance)
• Cauchy Number: $ \frac {\rho {\nu}^2} {E}$ (ratio of inertial force to modulus)

... et cetera.

Not every dimensional number is applicable to every model or situation.

The Buckingham Pi theorem is a way to make candidate dimensionless numbers given your phenomena. It doesn't say which is useful, but it gives a finite set.

Here are some relevant links on the subject:

• http://www.eng.wayne.edu/legacy/forms/4/Buckinghamforlect1.pdf
• http://ocw.mit.edu/courses/mechanical-engineering/2-25-advanced-fluid-mechanics-fall-2005/readings/07_pi_theorem.pdf
• http://www.jhu.edu/virtlab/course-info/ei/notes/dimension_notes.pdf

EDIT

Some "sneakier" indicators:

• Signal to noise ratio: $ SNR = \frac {\mu} {\sigma}$ where $ \mu$ is expected signal and $ \sigma$ is standard deviation in signal.
• Reynolds number: $ Re = \frac {\nu L \rho} {\mu}$ ratio of inertial and viscous forces.
• Gini Coefficient (link)

Answer 2

In additional to @EngrStudent's excellent answer, dimensional analysis is directly applicable to statistical science. There is a splendid exposition aimed at statistical readerships in

D. J. Finney. 1977.
Dimensions of statistics. Journal of the Royal Statistical Society. Series C (Applied Statistics), 26: 285-289. (Finney 1917$-$2018)

Thinking about dimensions and units helps avoid misconceptions and increases understanding. Perhaps the most familiar example is appreciating that standard deviations have the same units as the original variables (and a related point that variance is often less easy to think about whenever it doesn't have the same units).

A slightly more subtle example is realising that probability density of a single variable has units that are the reciprocal of the units of the original variable. It is common to encounter puzzlement that probability densities can exceed 1, which is a consequence of not appreciating that they have units and are not probabilities.