## 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.