From where does the uncertainty formula come from? [duplicate]

Question

The uncertainty formula is the one used in the laboratories to find the uncertainty of a variable. Say $X$ is a function of $Y$ and $Z$ such that $X=X(Y,Z)$ then it's uncertainty can be found with: $$\Delta X=\sqrt{(\frac{\partial X}{\partial Y}\Delta Y)^2+(\frac{\partial X}{\partial Z}\Delta Z)^2}$$ where does it come from? I know that for a small differential, for the same example we have $$dX=\frac{\partial X}{\partial Y}\Delta Y+\frac{\partial X}{\partial Z}d Z$$ but I don't know how to relate them or if they are even related. Where dors the uncertainty equation come from?

Answer 1

If you assume that the errors are small so that a linear approximation to the function $X=X(Y,Z)$ is appropriate for values $Y=Y_0+\eta$ and $Z=Z_0+\zeta$ where $\eta$ and $\zeta$ are independent random variates of zero expectation representing the errors, then you can approximate $X$ by the linear terms in its Taylor expansion $$X = X(Y_0+\eta, Z_0+\zeta) \approx X(Y_0,Z_0)+ \frac{\partial X}{\partial Y}\eta +\frac{\partial X}{\partial Z}\zeta \tag{1}$$ The error in $X$ is then the difference $X-X_0$ where $X_0=X(Y_0,Z_0)$, so that $$X-X_0=\frac{\partial X}{\partial Y}\eta +\frac{\partial X}{\partial Z}\zeta \tag{2}$$. Since the the variance is additive for independent random variates we can write that $$\mathbf V[X-X_0]=\mathbf V[\frac{\partial X}{\partial Y}\eta +\frac{\partial X}{\partial Z}\zeta]\\ =\left(\frac{\partial X}{\partial Y}\right)^2\mathbf V[\eta] +\left(\frac{\partial X}{\partial Z}\right)^2\mathbf V[\zeta] \tag{3}$$ The standard deviation (dispersion) of random variate $A$ is defined by $\Delta A =\sqrt{\mathbf V[A]}$ and it gives the formula you were asking for.