# Error Calculation

Gold Member
Consider a physical quantity e.g. w,related to some other quantities by $w=f(x,y,z)$.
Imagine an experiment is done for finding the value of w and the measurement errors for x,y and z are known.
I wanna know what is the standard method for calculating the error in w resulting from the errors in x,y and z?
I can think of several ways but don't know which is better!
1-$\Delta w=\frac{\partial f}{\partial x}\Delta x+\frac{\partial f}{\partial y}\Delta y+\frac{\partial f}{\partial z}\Delta z$
2-$\Delta w^2=(\frac{\partial f}{\partial x})^2 \Delta x^2+(\frac{\partial f}{\partial y})^2 \Delta y^2+(\frac{\partial f}{\partial z})^2 \Delta z^2$
and some others...!

Thanks

## Answers and Replies

Simon Bridge
Science Advisor
Homework Helper
The "pythagoras" approach is where x,y,z are independent.

Dale
Mentor
2020 Award
2 is the standard for independent errors.

f95toli
Science Advisor
Gold Member
Why not have a look at the GUM?

http://www.bipm.org/en/publications/guides/gum.html

It is surprisingly readable with quite a few examples. It is also (litteraly) the standard which just about everyone ultimately follows (albeit not always directly), i.e. as long as you folllow the GUM you are pretty safe.

Simon Bridge
Science Advisor
Homework Helper
Maybe the GUM should be made sticky?

Gold Member
GUM is just too long and detailed that you don't know where is the main point!
I couldn't find my answer there!

f95toli
Science Advisor
Gold Member
GUM is just too long and detailed that you don't know where is the main point!
I couldn't find my answer there!

Well, you did ask a very open ended question. Calculating errors "properly" is far from trivial and in some cases the "best way" is a controversial question (just put some people who like Bayesian error estimates in the same room as adherents of "orthodox" frequentist estimates).
Where I work we have a mathematical modelling group which (litteraly) specialises in just this. The GUM is the "basic" document which everyone who needs to do this professionally (e.g. because they do calibration work, quality control or have to certfy equipment) is expected to know.

The most general way of calculating errors (which is frequently used for real data) is to run Monte Carlo simulations, where you've assigned the proper distibution (which usually is the worst case scenario, unless you have very good reason to e.g. assume that the distribution is narrower than this). There is also specialised software you can get that will help you do this.