The physicists who announced the likely discovery of the long-sought Higgsboson particle this week were operating according to an extremely high standard of certainty. As was widely reported, in order to achieve discovery status their experiment had to clear a threshold of "five sigmas" of statistical significance. What precisely the five-sigma mark means, however, wasn't always clearly explained in the coverage of a ground-breaking development that could explain how particles have mass and, by extension, why planets and all other objects exist at all. That is partly because the five-sigma concept is somewhat counterintuitive. It has to do with a one-in3.5-million probability. That is not the probability that the Higgsboson doesn't exist. It is, rather, the inverse: If the particle doesn't exist, one in 3.5 million is the chance an experiment just like the one announced this week would nevertheless come up with a result appearing to confirm it does exist. Agence France-Presse/Getty Images In other words, one in 3.5 millionis the likelihood of Part of the Large Hadron Collider in Geneva that finding a false positive-a fluke produced by random scientists used to discover the Higgs boson particle. statistical fluctuation-that seems as definitive as the findings released by two teams of researchers at the CERNlaboratory in Geneva. The Numbers Guy blog The Particle Proof This is a very high burden of proof as far as science Previously in the Numbers Guy goes. For many medical experiments, researchers need Browser Wars Escalate merely to clear two sigmas. Sigmas don't scale in a Kids Can Be Costly Long After 18 linear way: A two-sigma result can have as much as a Sports Results That Leave Final Score . 5% chance of occurring as a false positive. Three Unclear sigmas, needed to cite evidence-but not discovery-of a new particle in physics, correspond to a one-in-741 chance. Other fields have different benchmarks because less is More on Higgs Boson at stake if a result proves faulty, or, as with pharmaceutical studies, there is more upside to moving ahead quickly with a promising result, statisticians say. "Drugs can be withdrawn, psychological experiments can be refuted, but nobody wants to see the laws of physics proved wrong," says David Spiegelhalter, a statistician at the University of Cambridge. Sigma is the Greek letter used as a symbol for standard Scientists at Europe's CERN research centre deviation-a measure of how far a finding departs from believe they have found one of the basic building blocks of the universe -- a subatomic particle the expected one. The more sigmas attached to a called the Higgs boson. The Director General of result, the more likely it is significant and not due to CERN told an audience of scientists that the discovery is a milestone in mankind's chance. Say a series of experiments on a randomly understanding of nature and has opened up exciting prospects for further revelations in the chosen coin involves flipping it 1,000 times and then field. flipping it 1,000 times again and again. The average number of heads should be 500, but some experiments willyield more and some fewer. A five-sigma finding would be 590 heads. The CERNfinding, then, is the equivalent of getting a lot more heads than expected-which is unlikely to occur by fluke rather than for some systematic reason. The best explanation the CERNphysicists have for this Physicists said they had discovered a new excess signal is the existence of the long-hunted Higgs particle that is consistent with the Higgs boson, a boson. long-sought particle crucial to scientists' current understanding of how the universe is built. WSJ's Gautam Naik explains its significance. Photo: They set such a high bar to rule out two other possible Getty 1m ages explanations: Either their equivalent of the coin is Discovery May Help Tell Universe's Secrets flawed in a way that tacks on extra positive signals; or 7/4/2012 they've run the study enough times and looked for Physicists Near Clue to Universe 7/3/2012 anomalies in so many places in their data-the equivalent of running the coin experiment over and over with different coins-that they've stumbled upon a seemingly unlikely result just by looking too hard for it. Physicists call this the Look Elsewhere Effect, or LEE, and try to account for it. One CERNgroup said its finding's significance falls to between 4.1 and 4.3 sigmas after accounting for LEE. Another factor can pull in the opposite direction from LEE: when another experiment finds the same thing. That bolsters a finding's significance. The Higgsboson was found at the five-sigma level by two CERNexperiments. The five-sigma requirement also helps guard against the equivalent of a faulty coin-some kind of measurement error. That appears to be the explanation for a finding last year that certain particles called neutrinos were traveling faster than the speed of light. It now appears to be the result of a flawed cable. Particle physicists cite many examples of results that cleared three but not five sigmas and weren't replicated by follow-up studies, which helped give rise to the five-sigma rule. University of Pisa physicist Giovanni Punzi, who has collaborated on a parallel hunt for the Higgsboson at the Fermi National Accelerator Laboratory in Chicago, says physicists have debated whether the rule is "exaggeratedly high." It was devised decades ago to provide a margin of safety in case calculations of statistics for an experiment, done without today's computing power, were flawed. But counterarguments prevailed: Modern computing power makes it easier to find false positives; and with so many particles discovered, new ones better clear a high bar. Maybe the rule is "not a bad idea, after all," Dr. Punzi says.

(a) Read about the Higgs boson particle from the WSJ. Using the five sigma cutoff point mentioned in the article and the Excel command NORM.DIST, specify the actual Excel command you use to get a false positive probability of approximately 1 in 3.5 million.

(b) What would the false positive probability be if a 4 sigma standard were used? State your answer in the form 1/X. What is X?