Statistics and Error Analysis in Physics |

**In General: Data, Fluctuations and Errors**
> s.a. particle statistics [spin-statistics]; probability in physics.

* __Statistical uncertainties__:
They vanish in general for *N*_{obs}
→ ∞, except for certain systems said to possess non-averaging
properties, as in random media.

* __Epistemic uncertainty__:
A kind of uncertainty whose complete probabilistic description is not
available, largely due to incomplete knowledge.

@ __Books__: Hacking 90;
Roe 92;
Epps 13;
Willink 13;
> s.a. statistics.

@ __General references__: Herbut a1512 [ensemble theory and experiment].

@ __Related topics__: Lévy a0804 [use of the median vs the mean in physics];
Ishikawa a1207 [quantum-linguistic formulation];
Chen et al JCP(13)
[epistemic uncertainty, flexible numerical approach for its quantification];
Vivo EJP(15)-a1507 [aspects of Extreme Value Statistics];
> s.a. Benford's Law.

**Experimental Errors** > s.a. physics teaching.

* __Types__: They can
be statistical/random or systematic; Errors in reading measuring
instruments can be either type.

* __Combining uncertainties__:
There is no universally accepted prescription for combining statistical
and systematic errors into one number, so they are usually given separately;
In terms of probabilities, the only way to deal with issues like this one
is to abandon the frequentist view in favor of 'degrees of belief'.

* __Variance__:

* __Confidence interval__:

* __Error propagation__: The rule

*σ*_{u}
= [ ∑_{i}
(∂*u*/∂*x*_{i})^{2}
*σ*_{i}^{2}
]^{1/2}

applies to variances of random, uncorrelated variables, not to confidence intervals.

@ __Error analysis__: Taylor 97;
Silverman et al AJP(04)aug [error propagation];
Berendsen 11;
Nikiforov A&AT-a1306
[algorithm for the exclusion of "blunders"].

**Data Analysis, Inference** > s.a. Paradoxes.

* __Curve fitting__:
This is a minimization problem, in which one minimized an error function;
For non-linear curve fitting (non-linear regression) the most widely used
algorithm is the Levenberg-Marquardt method, an iterative one based on
computing the gradient of the error as a function of the parameters in the
fit; As a rule of thumb, if the fit involves *n* parameter values,
one should have at the very least 3*n* data points for the fit to
be meaningful.

* __Statistical significance__:
In particle physics the gold standard for reporting a discovery is obtaining
an experimental result that is 5 standard deviations (5 σ) away from a
theoretical prediction, corresponding to a one-in-3.5-million chance of an
observation being a fluke (3.3 σ corresponds to a one in 1,000 chance,
3.3 σ to a one in 40,000 chance).

@ __General references__:
Bevan 13 [II].

@ __Bayesian__:
Lemm 03;
Lee 04;
James 06;
Sivia & Skilling 06 [II].

@ __Curve fitting__: Sorkin pr(80);
Sorkin IJTP(83)ap/05 [Occam's razor and goodness of fit];
Turney BJPS(90) [balancing stability and accuracy];
Gould ap/03 [linear fits];
Transtrum et al PRL(10) [non-linear fitting process];
Banerji CP(11) [least-squares method];
> for a different, but related concept see Spline.

@ __Related topics__: Maltoni & Schwetz PRD(03)hp [compatibility of data sets];
Pilla et al PRL(05)phy [signal in noisy background];
Łuksza et al PRL(10) [statistical significance of structures in random data];
Cubitt et al PRL(12) ["extracting dynamical equations from experimental data is NP hard"];
Murugan & Robertson a1904 [topological data analysis, introduction].

**Specific Areas and Topics** > s.a. correlations;
random processes; stochastic processes.

@ __In quantum mechanics__: Rylov qp/01;
Rajeev MPLA(03).

@ __In astrophysics / cosmology__: Szapudi ap/00-proc [variances of correlations];
Hill ap/01-proc [Bayesian statistics in neutrino detection];
Feigelson & Babu ap/04-conf;
Verde a0712-ln,
LNP(10)-a0911;
Feigelson a0903-en [rev];
Heavens a0906;
Madore AJ(10)-1004;
Feigelson & Babu a1205-ch [rev];
Feigelson & Babu 12
[r CP(14)];
> s.a. observational cosmology.

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 15 apr 2021