Theory of Error

You guys are amazing (which I knew before!). A 54 year old memory of a book I got as a freshman in college, and I even get a photo of it! I'm going to check out Amazon, but the equations will be great until I can do that.

I tried your link.....not $5 any more! Even used copies are $19.95 plus shipping.

My reference on measurement theory was by D.C. Baird. I'm pleased to see that this little book is still in print. It has saved my career more than once when correcting irate experimenters that don't understand the subtle differences of measurement errors and their effect on analysis.

I didn't learn from that book. It was spelled out to me and my cohorts in a book titled "The handling of experimental data" (or some such).

If data is added or subtracted then the component errors are added.

If multiplied or divided than the component percentage (proportional) errors are added.

With due consideration to the units of course.

This is only valid when the probabilistic aspect of errors is NOT considered. This way one obtains extreme values which have low probability to appear but the procedure is simpler.

In fact one can use the definition of a function variation to obtain the equations:

d(F) = Σ (∂F/∂vi) dvi. F= function of variables vi where i is an index from 1 to n and ∂F/∂vi is the parial derivate according to variable vi.

As an alternative to the "academic" approach:

Stick all the numbers and formulas in a spread sheet;
Create several copies of the work area;
Make each independent variable in the copies different from the original by the uncertainty;
Use your intuition/experiment to find out which combinations of positive and negative deviations lead to the biggest deviations in the final results.

Go to the NPL website www.npl.co.uk and search for 'uncertainties' and there are several links that will guide you through the process and how to set up an uncertainty budget.

The reference document (All the information that you need) is the Guide to Uncertainty in Measurement (GUM) that you can download free from BIPM http://www.bipm.org/en/publications/guides/gum.html . This will cover how to propagate uncertainties (your issue) and how to get to the confidence interval (say 95%) for your combined result. The document is fairly clear and shows you all the steps that you will need to go through.

The simplest way to approach uncertainty propagation in a function is to use a Taylor's expansion f(x+delta_x) is approximately f(x)+delta_x*d(f(x))/dx. From this u(f(x))=u(x)*|d(f(x))/dx|. All the terms combine in quadrature (u1^2+u2^2...) if your uncertainties are uncorrelated. If they are correlated it is a more advanced problem.