Error analysis: simple but confusing matter

[haushofer]

TL;DR Summary

Two rules give different answers for same expression

Dear all,

We learn students to work with significant figures. I guess we all know those rules of thumb. Now imagine I have an expression like L = 2x, where x is a length a the 2 is an amount, i.e. the value of a discrete 'function'. If x=0,72 m (2 significant figures), then L = 1,44 --> 1,4 (2 significant figures); we view the 2 as exact.

But if I write L=x+x, then L = 0,72+0,72=1,44, and I use the rule of thumb that I look at the same amount of decimals. What exactly is the origin of these contradicting accuracies?

Within error analysis, I can define a function z=f(x,y) and calculate the error dz as a function of dx and dy,

$$dz = \sqrt{(\frac{\partial f}{\partial x} dx)^2 + (\frac{\partial f}{\partial y}dy)^2 }$$

If z(x,y) = x+y, then

$$dz = \sqrt{(dx)^2 + (dy)^2}$$

and if z(x,y)=x*y, then

$$dz = |x*y|\sqrt{(\frac{dx}{x})^2 + (\frac{dy}{y})^2}$$

If I take y=x in z(x,y) = x+y, I obtain z=x+x and hence

$$dz = \sqrt{2}|dx|$$

while if I take y=2 (dy=0) in z(x,y) = x*y, I obtain z=2x and hence

$$dz = 2|dx| > \sqrt{2}|dx|$$

Do these two different errors for the same mathematical expression make sense? Does it have it have to do something with different meanings to the expressions z=2x and z=x+x and errors canceling each other out such that the first error is smaller than the second? And is the answer to this question also the answer for my first question about significant numbers and those rules of thumb? Any references to this? Many thanks! :)

 

[Dale]

But if I write L=x+x

The problem with this is that the error in the first x is perfectly correlated with the error in the second x.

Within error analysis, I can define a function z=f(x,y) and calculate the error dz as a function of dx and dy,

You forgot to include the terms for the covariance.

What exactly is the origin of these contradicting accuracies?

In the expressions you used it is important that the errors in the different terms be uncorrelated. In $y=2x$ there is a single term so there is no issue. But in your other equations there are multiple terms and the usual formulas you mentioned only work if the covariance is zero.

 

[etotheipi]

I think @Dale is right, in my notes I have the more general formula, for a function of two variables $z = f(x,y)$,$$s_z^2 = \left( \frac{\partial z}{\partial x} \right)^2 s_x^2 + \left( \frac{\partial z}{\partial y} \right)^2 s_y^2 + 2 \frac{\partial z}{\partial x} \frac{\partial z}{\partial y} s_{xy}$$

 

[haushofer]

Yes, this sounds right. Indeed, the derivation of the error formules assumes no dependence/correlation between the variables.

A nice example of how careful one needs to be with those tules of thumb. Thanks guys!