Statistical proof

.

The term "statistical proof" has several uses, both technically and colloquially.

* Data analysis and statistics — When mathematical statistics is used to provide a mathematical proof of a proposition regarding the probability of data under specified assumptions and hypotheses, this is a “statistical proof”. Examples of the assumptions are that the data are independent and have an underlying distribution that is a normal distribution, or the assumptions of a nonparametric analysis. In a typical statistical test, the ratio of probabilities of an event occurring under each of two hypotheses (the likelihood ratio) is evaluated, and if the likelihood ratio is "sufficiently large", it is asserted that there is “statistical” proof of one hypothesis. Since the assumptions of mathematical statistics themselves require further justification, and the justification is often uncertain or merely “probable” in an un-quantified way, the resulting concluded probability statement is no longer certain, as in the medical, natural, and social sciences. In that case, the term "statistical proof" is still sometimes used to refer to the outcome of a statistical test. ["Optimism is no longer merely the product of gut feelings but is supported by statistical proof both from the providers themselves and, much more importantly, from independent authorities like industry analyst Laing & Buisson" [http://www.hi-mag.com/healthinsurance/article.do?articleid=20001721061&adname=his_features "Health Insurance and Protection"] ] [""I don’t have any statistical proof of this, but I firmly believe that cheating and plagiarism have increased in recent years," Brown, who is the history department chair, said." [http://bupipedream.com/current/index.php/articles/view/8622 "Pipe Dream"] , Birmingham University Student Newspaper] ["Based on clear statistical proof about which combination of design works most successfully in converting sales and delivering customers, Maxymiser enables marketers to refine their email content in real-time to capitalize on strong performing features and remove the less successful elements." [http://www.e-consultancy.com/news-blog/366275/maxymiser-launches-email-optimisation-software.html "Maxymiser launches email optimization software"] , Maxymiser press release] In these "statistical proofs", with assumptions that are only "probably" true, not certain, this usage of the term "proof" no longer refers to a mathematical proof, even if extensive mathematical tools and methods are employed.

* Bayesian analysis — A statistical proof in Bayesian analysis is a mathematical proof primarily employing Bayes' theorem. The conclusion is a statement regarding a measure of quantified subjective belief or degree of certainty.

* Physics — The term refers to analyzed data obtained in a physics experiment, such as resulting from a collision in a particle accelerator [“ the Large Hadron Collider could start to indicate exotic new things like the Higgs boson within months - though it could take a year or two to have the statistical proof needed to confirm its presence” [http://www.channel4.com/news/articles/science_technology/qa+the+large+hadron+collider/2449067] ] , which may result in “statistical proof” of the existence of a predicted new particle, or in a data in a cosmological “experiment” (observational study), regarding fractal distribution of galactic clusters.Fact|date=September 2008

* Exploratory data analysis — “Statistical proof” may refer to a scatter plot or product of non-mathematically rigorous exploratory data analysis. (See picture at right of data points so numerous and linear that they appear to create a line.)

* Colloquial uses — Statistical proof may colloquially refer to raw data where the data makes the conclusion “obvious”. [“Southern Cal’s defensive line dominated Ohio State last weekend. Here’s statistical proof” [http://nflblogs.profootballweekly.com/NFLdraft/2008/09/freeman_falters.html] ] Such colloquial use of “statistical proof is often interchangeable with a colloquial use of “mathematical proof”, even though there is no formal mathematical proof at all. Fact|date=September 2008 For example, the large number of stars together with their clustering into galaxies so tightly that galaxies appear from a distance to be points, is statistical proof of the existence of a gravitational force binding the stars together.

* Pure mathematics — In various branches of pure mathematics, such as cryptography, number theory, random number generation, mathematical statistics, chaotic series, and others, the expression “statistical proof” may be used in a well defined manner, or colloquially. [“a statistical proof can just mean ‘most’ cases, which may mean that some cases which fit under the guarantee are not really secure” [http://www.mail-archive.com/[e-mail]/msg03572.html] ] [“in number theory and commutative algebra... in particular the statistical proof of the lemma.” [http://www.jstor.org/pss/2686395] ] [“Whether constant "π" (i.e., pi) is normal is a confusing problem without any strict theoretical demonstration except for some statistical proof”” (Derogatory use.) [http://www.springerlink.com/content/nj34v59p71m11125/] ] [“these observations suggest a statistical proof of Goldbach's conjecture with very quickly vanishing probability of failure for large E” [http://people.web.psi.ch/gassmann/eneseminare/abstracts/Goldbach1.pdf] ]

Notes