# Standard Deviation in Particle Physics

I'm familiar with sigma, and how its usually calculated and used, but would like to know how it's applied to particle physics. I recall reading that the discovery of the Higgs would only be credible unless it meets a 5 sigma criterion. What are the particle physics factors/numbers that would go into a sigma formula for a credible Higgs discovery?

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Good question! I think that the 5-sigma criterion is "cultural," rather than being based on a rigorous statistical analysis, but I'm not a particle physicist, and I could certainly be wrong. I think the reason it's so high is just an acknowledgment that it's hard to be sure you've accounted for all possible backgrounds and systematics. – Ted Bunn Apr 16 '11 at 18:08
There is a good reason for the 5 sigma criterion: in the halycon days of particle zoo many "particles" were "discovered" which were just statistical noise. Big particle physics experiments have thousands or ten-of-thousands of channels. If you plot a bunch of histograms totaling 10,000 bins, the odds that you'll see a bump that looks significant at the 2--3 sigma level approach 50/50 even if the input is noise. – dmckee Apr 16 '11 at 18:18

Errors in particle physics are of two kinds. Statistical, and systematic. Statistical is the usual standard deviation of gausian distributions, sqrt(n)/N for 1 sigma. It is the systematics that take a lot of effort, and often are not taken well into account.

Systematic errors come from

1) the background to the signal expected. The background is calculated theoretically and entered into a monte carlo program that simulates the production of the data. There are also errors in the parameters the theory has used that enter as systematics.

2)Effects from the limitations of the measuring apparatus, which is also inserted into the monte carlo simulation of events

3) from the method of analysis, i.e. the cuts made in order to isolate the signal in the data and in the simulated events.

Ideally, the monte carlo simulation will have many more events than the data and so the statistical error from the MC can be ignored. One should estimate the errors from uncertainty in theory and from defects of detectors by varying the parameters in the MC program to the 1 sigma level of the important parameters and cuts and observing the change in the distributions.

Of course there are so many parameters and cuts that this procedure is not adhered to strictly, as happened with the new 3 sigma announcement of the "fifth force" by CDF, where the errors are just statistical from the events. That is why for discovery one asks 5 sigma. It is very hard to have an effect of 5 sigma due to the systematics enumerated above, whereas 3 sigma announcements have often disappeared . The ALEPH Higgs for example was a 3 sigma one that disappeared when the other 3 LEP experiments looked. I have known a 4 sigma resonance that was the effect of not estimating the systematics of the cuts. 5 sigma is for playing it safe.

I should add here that the different systematics in different experiments are the main reason why at least two expensive experimental set ups are approved and built in the collider setups: independent confirmation in parallel .

Edit: I want to correct this entry due to the discussions on statistical significance for the very much in the physics news OPERA neutrino speed measurement. In blog discussions I learned that the current way of dealing with systematic errors is of assuming randomness and adding them in quadrature. One interested in the details of how particle physics is treating errors should go to the CERN Yellow Report 99-03, July 1999, that has an html version.

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I don't have your depth of experience, but I too have seen a ~2.8 sigma bump just disappear as more data came in. – dmckee Apr 16 '11 at 19:27
Anna: I think I understand potential errors. Can you be more specific as to what items go into the sigma formula? – Michael Luciuk Apr 16 '11 at 19:41
@Michael Luciuk sqrt(N)/N +syst1 + syst2 + syst3 ..... The systematic errors as estimated by analyzing MC events and effects of cuts. – anna v Apr 17 '11 at 4:40
Thank you Anna. – Michael Luciuk Apr 17 '11 at 12:27
correction : that should be in the anser to @Michael Luciuk Sqrt(N/N2 +syst12 + syst22 + syst33+...) ,"syst" properly normalized . – anna v Sep 27 '11 at 11:47