Abstract: Dimensional Analysis (DA) is a widely used methodology in physics and engineering. The main idea of DA is to extract dimensionless variables based on physical dimensions. Due to its capability in removing dimensional constraints and reducing the number of variables, its overlooked importance in statistics has only been recognized recently. While its properties in physics have been well established, the fundamental statistical theories behind DA remain absent. Such theories are critical in integrating DA into statistical procedures. In this paper, we present a new statistical perspective on DA, which translates the essence of DA into statistical principles. The basis quantities are represented as linear-space bases, while the post-DA variables are formulated as maximal invariant statistics. The proposed statistical properties of DA, the sufficiency and completeness, guarantee the optimality of DA variables. An ocean wave speed example is presented to demonstrate DA methodology. A Meteorology example of planetary boundary layer problem is used to illustrate the proposed statistical properties in a practical context. The proposed representation reveals DA's structural compliance with statistical theories and encourages more appropriate statistical applications.

Key words and phrases: Completeness, dimensional reduction, invariance, sufficiency and transformation.