Using Sigma Instead of p for Higgs Boson
Date: Summer 2012
Can you help me with something? I assume this is a physics thing, and as I am not a physics person, I am just not speaking the language. All the news about the Higgs Boson is going on about the "sigma level" assurance for the Higgs Boson. Statisticians do not usually refer to a "sigma level" but a p-value (probability under the null). Is that a physics thing?
My follow-up question is what distribution are they using, and where are they getting it from (e.g. normal, and there is a reason to think normal is appropriate)? I really do not know much about this, but I cannot begin to think how they are determining their significance level... or the distribution under the null hypothesis (which is, presumably, that there is no Higgs Boson).
This is an excellent question. Let us look at the follow-up question first. P(a) is under a normal distribution, which assumes tails that are equal.
The Boson energy is in a range we refer to as a fat tail. The distribution is not normal. There is a lot of "noise" from already known events occurring in the energy range of interest.
Take a look at the graph(if it does not come out properly, please use the reference): You will notice a fat tail of the weighted events/1.67GeV vs mass(GeV). Please recall that because E=mc^2, mass may be thought of as electron volts(eV) and is cancelled out in the slope calculation.
Deviations from this fat tail, or the slope are merited as how far out from the standard deviation (1 sigma) they occur. 1 sigma means there is no change, 1.5 to 2 sigma is noise, 3 sigma is a promise, 4 sigma is a result and 5 sigma is a discovery. You should be aware that these numbers and the interpretation varies with the researchers. Each increasing sigma has an increasing confidence limit that the occurrence was not random. Sigma is not limited to physics. It is also used for market predictions, environmental assays and predictions for blood chemistry analytes.
Attaining this 5 sigma deviation from the fat tail required many collisions in order to see the decay results of the boson. Those collisions were hidden in a "sea" of data and required 2 years of data collection in order for that little "bump" at 125(GeV) to become statistically significant.
Bottom line is that there is now sufficient data to make a claim that there is something at 125(GeV) that causes a change in decay events. The reason to suspect the Higgs boson is that the mass is great enough to match mathematical predictions.
There is plenty more to do.
Hope this helps!
Peter E. Hughes, Ph. D. Milford, NH
When analyzing results, the physicists are looking at energy values
registered on a detector (when a particle hits the detector, it
"donates" its energy to the detector, which is registered as the
detector's signal). However, there are random thermal and other energy
fluctuations (background noise) at the same time as the particles
hitting. So the signal from the detector is "noise plus intentional
signal". So whenever to register a 'bump' up in the detector, the
question is, "Was that a particle, or was that a random
The five-sigma comment is a statement of the probability of the signal
observed being a real particle, versus just a random thermal
fluctuation, five standard deviations from the normal distribution.
The normal distribution describes the odds of a random event causing a
deviation from the expected value. You gather data to assess the
variability of the system (calculate its standard deviation, to fit
the system to the normal distribution curve). Then, when a value falls
outside the distribution, you can assess the odds that it was random,
or not random. The more data you generate showing the signal, the
higher the likelihood it is not a random occurrence (eventually, all
random events will regress to the normal distribution if the system
follows a normal distribution). At one sigma (one standard deviation),
the odds are 1/3 that a deviation is random. In other words, the
deviation from expected is small enough that it occurs pretty
frequently. However, at larger sigma values, the odds shrink -- at 3
sigmas, the odds are under 1% that a deviation is just random. And at
Five sigmas, the odds are over 1 in 1 million.
A fair coin could land on 5 heads in a row, but it's unlikely...). In
this case, they've gotten the rough equivalent of 21 heads in a row,
which is a standard (ha!) threshold for asserting that the signal is
real, and not random. They are saying that it's not just a thermal
fluctuation, but a real particle. Given the energy level (125 GeV)
fits into the possible/expected energy for the Higgs Boson, they're
saying that's what it is.
Hope this helps,
You are really getting into some “heavy duty” statistics, which is really more mathematics than physics. There are various statistical “tests” of significance of a variable (or observation). In general, the “tests” are designed to ask/answer the question, “What is the probability that a variable (or observation) can be “explained” or is the result of a random variation?” Or put the other way, “What is the probability that a variable (or observation) is NOT due to a random change in the observation?”
Just how you go about answering those questions is often not easy to determine, especially when the changes are small, or the data points few. In those cases it often takes thousands (or even more) measurements under carefully controlled conditions. And sometimes the “answer” is, “We cannot tell!” Both the physics and the mathematics get pretty sophisticated.
In the case of the Higgs boson specifically, the “weight of the evidence” seems to favor that the observation is “significant”. But notice here there is some qualification (“the weight of the evidence”) favors the existence of the Higgs. In comparison, recall that a couple of months ago there was a claim that certain sub-atomic particles traveled faster than the speed of light. After a lot of further experimentation and computation the “Yes” answer turned into a “No” answer. The observation was found to be due to a loose connection.
My PhD mentor had a favorite expression that I have always kept in mind, “Measuring nothing accurately is very difficult!!”
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Update: November 2011