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.


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:


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)
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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.

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