# Rigorous error analysis theory

Now that I finished my pregraduate laboratory courses I've been thinking a lot about the error theory we used for our measurements. I tried to find a mathematically rigorous book on error analysis but it was pointless. I was wondering if any of you guys knew literature on this subject. The topics I am searching for go along the lines of:

1. considering the result of a measurement as a probability measure;
2. generalizing the notion of "best value" or the value with highest probability to measures that do not have a density (radon-nikodym derivative) associated to them, such as the Dirac measure;
3. a concrete definition of uncertainty which coincides with the standard deviation in the case of a gaussian distribution but isn't as cumbersome as the 68% confidence interval,
4. given a set of measurement results $\mu_1,\dots,\mu_n$ and a function of those measurements $f$, how to find the result of the measurement $f$ (notice that solving this question generalizes the problem of error propagation).

If any of you guys can give a quick explanation of any of these topics it would also be greatly appreciated.

• Other than something without a probability distribution, you can find the answers to all of it in standard statistics or random processes books. Google them – Bob Bee Dec 17 '16 at 20:47
• I was looking for books that made an emphasis on physics. Still, some mathematics literature is fine. Any particular suggestions? – Iván Mauricio Burbano Dec 17 '16 at 20:51
• I can't give you a specific one other than Papoulis, which is too theoretical and on random processes. Others I'd just google, such as 'statistical error analysis books for physics', quite a few come up and you can see what might work for you. Others can perhaps give you their preferred one. – Bob Bee Dec 17 '16 at 21:08
• J B Scarborough. Numerical mathematical analysis – inquirer Jan 27 '18 at 7:55

I have had to refer to some books and resources in my current research where some rigourous error analysis is required. Here are two that I have used alot:

Introduction to Statistics and Data Analysis for Physicists (Bohm and Zech, 2010) - this reference is completely available online (all 412 pages!). In the preface, the authors state in the preface that the reference is written

with the focus on modern applications in nuclear and particle physics

The book is purportedly aimed at Masters and PhD level students, compiled from lecture notes from the University of Siegen in Germany. The reference has less of an emphasis on the mathematical foundations, but of an appeal to the intuition of the reader. Examples and formulas are derived in a stepwise fashion with considerable explanation.

A quick search (control-F) of 'Dirac' shows a couple of examples related to that topic.

Measurements and their Uncertainties (Haese and Hughes, 2010) - this reference is also completely available online. This reference is considerably more general than the first reference (and shorter at 153 pages!).

The authors state in the preface that the book is from the restructure of a practical first year physics course at Durham University - but is accessible to all levels of physics experience, with an emphasis on being able to be used while in the lab and for scientific-computing applications (where I used it) - specifically:

The scope of this book is to cover all the necessary groundwork for laboratory sessions in a first- and second-year undergraduate physics laboratory and to contain enough material to be useful for final-year projects, graduate students and practising professional scientists and engineers.