Statistical Engineering and a theory that dates back to 1914

One of the best methods that I have come across which exemplifies the inductive-deductive iterative nature of statistical investigations (see my first post) dates back to 1914 – the so-called “Pi” theorem of E Buckingham; I will illustrate the use of the “Pi” theorem using the well known paper-helicopter experiment, which many people who have taught statistical methods to engineers will be familiar with. If we adopt a completely empirical approach, we might decide to run a response surface experiment to model the flight time of the helicopter as a function of various design parameters; three design parameters might require about 15 runs in the experiment to develop the transfer function. However, if we think for a minute about the physics, we know that the flight time will be a function of the mass of the helicopter, and the area swept out by the rotors, together with the force due to gravity, and the density of air – and all of these quantities are known. The application of the “Pi” theorem, which reduces the dimensionality of the problem, and does not require linearity to ensure dimensional consistency, reveals that the number of experimental runs can be reduced to about three. It is a mystery as to why the “Pi” theorem isn’t referenced in any of the classic texts on response surface methodology and design of experiments; is it because not enough statisticians are interested in engineering?