 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 Nobs → ∞, 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

σu = [ ∑i (∂u/∂xi)2 σi2 ]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 3n 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.