Dimensional Analysis of Statistical Formulae

Dimensional analysis (also used in Physics) can be used to check statistical formulae.

Define the dimensions
f_i \in F: frequency;
x_i \in X, y_i \in Y: data.

Now define the dimension function D by:

D(f_i) = F;
D(x_i) = X;
D(y_i) = Y;
D(\Sigma \varphi) = F D(\varphi).

Note that, like the complexity notation in computing, these are just notation, and not really formulae in that they can’t be written in reverse; likewise, constant factors are ignored.

Thus, we can check some terms:

D(n) = D(\Sigma 1) = F D(1) = F;
D(\overline{x}) = D(\Sigma x / n) = D(\Sigma x)/D(n) = F X / F = X — this is what we’d expect: for example, if the x_i are velocities (for example), then \overline{x} is also a velocity.

Checking
D(n \Sigma x^2 - (\Sigma x)^2) = F \times F D (x^2) - (F D(x))^2
= F^2 X^2 - F^2 X^2 = F^2 X^2 (*),
we see that the sub-terms are of like dimension and thus may be added or subtracted.

(*) remember: constants factors disappear.

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