Error Propagation (Percentage) - sin(x)^2 / x^2

[Strides]

Hey, I'm trying to propagate my percentage errors through some hefty equations and come up on a bit of snag:

I've got a percentage error for x and know how to deal with it for trig functions and powers, however since both errors are from the same source:

y = sin(x)^2 / x^2

Should I just simplify it to: 2*error

 

[Khashishi]

What do you mean by percentile errors? Do you mean you have something like the 25th percentile, 50th percentile (median), and 75th percentile of x?

 

[Strides]

Sorry, I just mean that I've converted the error, from absolute to percentage beforehand. I've just edited my original post to fix the mistake.

 

[Khashishi]

If the errors are small, then you can take the derivative of both sides of the equation. You get something of the form
dy = (...) dx
then just plug your error into dx, and then dy will be your error in y.

 

[Gigel]

As Khashishi said, it is something like: dy = y'(x) dx, where dx, dy are absolute deviations. Relative deviations are dx/x, dy/y:

dy/y(x) = y'(x)/y(x) * x * dx/x

Here x,y,y(x),y'(x) should be computed for x=reference value of x (the one supposedly without errors).

If you take errors dx, dy to be standard deviations, then relative errors would be:

dy/|y(x)| = | y'(x)/y(x) * x | * dx/|x|

and x should be the average value of x. Relative errors can be expressed in % (of the average values). Formulas are approximative and work for a small relative error in x.