Monomial uncertainty propagation (${\sigma }_f^2/f^2\simeq {\sum }_{ij}{a}_i{a}_j\mathrm{Cov}(x_i,x_j)/(x_i{x}_j)$)

For positive measured inputs and $f=C{\prod }_i{x}_i^{a_i}$, logarithmic differentiation gives $df/f={\sum }_i{a}_{i}d{x}_i/x_i$. Applying propagation of uncertainty yields the displayed covariance formula. For independent input errors it reduces to ${\sigma }_f/|f|\simeq [{\sum }_i(a_i{\sigma }_i/x_i{)}^2{]}^{1/2}$. If the input uncertainties are strict bounds rather than standard deviations, a first-order conservative bound is $|\Delta f|/|f|\lesssim {\sum }_i|a_i|\Delta x_i/|x_i|$. These are different statistical interpretations; a product formula alone does not choose between them.

For the same massless spring obeying Hooke's law, the frequency is $\nu =(2\pi {)}^{-1}\sqrt{k/m}$. The displayed mass ratio follows. For small independent frequency errors with common fractional standard uncertainty $ϵ$, monomial uncertainty propagation gives ${\sigma }_R/R\simeq \sqrt{8}ϵ$. If $ϵ$ instead describes individual measurements and each sample mean uses $n$ independent trials, its standard error is smaller by $\sqrt{n}$, giving ${\sigma }_R/R\simeq \sqrt{8/n}ϵ$. Four times as many trials halve this statistical uncertainty; a common calibration error needs its covariance included.