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  • you're talking about uncertainty with such certainty.

  • You're talking about what you don't know, and yet you're able to put numbers on the extent to which you don't know it, which makes me think you do know it.

  • So the symbol is Sigma Andi.

  • It's something you'll just hear thrown away frequently in reports about scientific results, where they're saying basically how significant the results are, how important they are.

  • You hear people saying this is a three signal results.

  • Five signal results are one signal result.

  • We've been hearing a lot about it with regards to the Higgs on the maybe evidence that's that's the Higgs is being seen its CERN at the Large Hadron Collider in the Atlas and CMS detector.

  • I'm sorry, for example, in the context of the Large Hadron Collider where they're inching towards detecting the Higgs Bos on.

  • They keep quoting slightly higher values for this quantity Sigma as a way of indicating that they're getting closer and closer to a detection of it.

  • They won't quite say we found it, but neither will they say we haven't found it.

  • What they talk about is the evidence for it being there, and they use this kind of this symbol signal standard deviation that whenever you make a measurement, were in any kind of science.

  • Astronomy, physics, whatever it is, you make your measurement.

  • But you should also always quote on air a bar on it because you never very, very, really major something precisely.

  • Usually there's some uncertainty with your instrumentation.

  • So, really, the sigma is the size of your Arab, or it's how much uncertainty there is around the measurement you actually made.

  • Imagine you've got a big step, a stadium, a football stadium full of people on.

  • Did you decide to dio measure one's height?

  • Then you plotted.

  • You plot the height, the number of people of a given heights and you just you make distribution of it right so that that distribution will be peaked at some particular height.

  • In other words, the majority of people will.

  • Most people will have this height, and fewer people will have.

  • You know, the lively Bitola or shorter there'll be some maximum where the average is.

  • Okay, so you've got your average.

  • Let's say it's six foot now.

  • You could be us can begin to ask questions about the distribution itself on one way you can do it is, you can say how many people are there within 33% off that average on one side of 33% on the other side.

  • So that's a route 66%.

  • That's one signal.

  • So one sigma is takes into account sort of 66% of the data around the mean so it's the mean, plus 33% plus or minus 33%.

  • If I go to two Sigma, your then saying now I want to think about a rat.

  • I think two signals around 95% of the debtor.

  • So you're including all these people and you can count how many there are that those air in the two Sigma raging three sigma goes to 99 points.

  • 7%.

  • I think it is.

  • So that's taking you up to three segments were almost including everybody now, right, but we're at three signal four Sigma texture to 99.99%.

  • Okay, so that's that's surely that's everything that's not good enough for particle physics.

  • It's the five signal That's the important thing for particle physics will come back to it in a second, but what five Sigma.

  • Means is you're including 99.9999% of all the data, so there's only like one in one million people not included in that 99.99% so that that's what signal is.

  • It's not very difficult.

  • It's it's kind of tells you about the distribution around the mean, and that's all.

  • It does the means, the average and sick.

  • The number of Sigma tells you.

  • Basically, the fraction or the percentage off data which is accompanied is included around that mean scientists are a bit sloppy when they use these things a bit careless.

  • And so the signal that you often hear quoted really should be associated with something where there's a particular kind of error.

  • The most common kind of error is a thing called a gal.

  • See an era eyes, the Gaussian distributions or a particular form of mathematical distribution.

  • They come up all over the place, particularly in uncertainties.

  • It's kind of converse.

  • You met use of it in the converse wearing the science experiment.

  • So in other words, you look for things that are way out in this at the tail of the distribution, so That means a large number of signal is really these things are a measure of how often you're prepared to be wrong in that if you if you publish a result when things get to this sort of 99% significance level, then that means that well, one time in 100 I'm gonna claim a result is true.

  • And actually it's gonna be force.

  • So it's how often you're prepared to be wrong.

  • So one sigma it will be, You know, you'd be wrong with the third of the time and no scientist is prepared to be put up with being wrong 1/3 of the time.

  • You just look stupid in this case, what we're looking at when it's CERN.

  • We're looking at the collision of pro tons on Dhe.

  • From now is collision.

  • Let's concentrate on one of the one of the routes that has led to the possible discovery of the Higgs if it's there and that is the production of photons.

  • So two photons are being produced at the end of the of the of this collision process.

  • Now it I have a model without a Higgs.

  • So let's have, which is the standard model without the Higgs.

  • There are many ways the protons can collide in this model and produce to four tons on.

  • What I can do is I can end up with what it was effectively a distribution off the number of photons I expect as a function of the energy of the system.

  • And now I can ask, Well, if I include the Higgs.

  • What?

  • What happens on Dhe?

  • I can calculate what I expected to happen.

  • But what I can also do is just do the experiment.

  • I could just look for the four tongues, right?

  • I've got these amazing detectors and I could just look for the photons and in particular, what I can ask is at at any given energy, which corresponds, in fact, two to the mass of the Higgs particle, which we're we're trying to test for.

  • I have some average value that I expect there to be, for the number of photon is produced, and then I have one sigma around the average value.

  • Let's make up a number.

  • Let's say we expect to see don't 1000 photons, and then one signal would be including 66% plus and minus.

  • That would be 1000 and whatever that would be and then to signal would be including 95% on I look at the data, and usually, if the debt is lying within that band, then I'm not too excited about it that it's consistent with the background model.

  • You shouldn't always expect to see an average right, because that's why you you have an average.

  • You have some above and some below in the average out.

  • But every now and again, maybe you'll see something which is a three sigma well, three signals bigger than to Cigna, and then you can think that's a bit higher, but you tend not to get too excited.

  • You might think we'll keep an eye on that, and that's where we're at at the moment with the Higgs.

  • In actual fact, the debtor is that this kind of 2.8 signal level, um, what people say Those who are waiting to, you know, before you call up stock home and put your flights two signals, not three signals not good enough.

  • In fact, 46 was not good enough.

  • The 99.99% What they will have to see is evidence of distribute of events, which are at five Sigma away from that average that you would expect the model tow have without a hicks, because when you're up five Sigma, if you just think about it in terms of your underlying model, the model without the Higgs What you're saying is I am seeing events occurring where I would normally have expected them to occur.

  • Once in, everybody gets 1.7 million times.

  • In other words, it's rare, but I'm built and finding lots of them, and that's what they'll do.

  • They'll let their limb to pick up lots, not just one, but quite a few of them on when people are quoting errors on some of these very complicated experiments, like those of the L H C, where the errors they're measuring really on gas and by very long way because things are incredibly concept complicated.

  • That's all sorts of sources of error, systematic errors, uncertainties in the modeling and so on.

  • Almost certainly, the errors are not really gassy in, but they sort of converted into what's the equivalent if the errors workout him?

  • Because that's something that scientists sort of having the back of their heads that if you know if something's significant at a one signal level, you probably don't believe it.

  • If it's a to signal level, you start believing it.

  • By the time it's four or five sigma, you say that's pretty certain.

  • So from what you've told me, tell me if this is an accurate time, and then if you're wanting to confirm something that you already know what you want a low sigma.

  • If you're looking for something new, you good for Ojai Sigma, it's effectively like that.

  • You don't if you hope for it.

  • I mean, you should just be saying, Well, whatever the data gives me, it gives me.

  • But if you if your theory is, you know, bang on, then of course, all the data that's coming in should be perfectly consistent with what your average expectation value is that she should have very few really far out liars.

  • Whereas if your theory is not the full McCoy and is not explaining everything that's going on in that that regime on dhe, there are other important contributors to that process.

  • Then, indeed you you as you get more and more data, then more and more of these new events will emerge that your theory is an accounting for.

  • They'll appear in the outlying regions of your distribution.

  • Large signal.

  • How can you be near near being sure about something when you don't even know if it exists or not?

  • It's actually I think you're looking at it the wrong way around.

  • I think the way you have to look about if you were looking for something like the Higgs Bos on what you're asking is how far my away from so you your starting position should always be the status quo, which is we've never found the Higgs Bos on.

  • So perhaps the status quo is the reason I expose on.

  • And then you start getting more and more evidence that actually may be the reasonings bozo.

  • And so you sort of start getting further and further away from that position of being consistent with their not being a Higgs Bos on So really what you're measuring when you make a measurement with the L.

  • A.

  • C.

  • Is, how far is that measurement away from?

  • They're not actually being a Higgs Bos on a tool.

  • The look elsewhere effect our bread.

  • No.

  • Okay, here we go on Dhe I might trip up on this.

  • So what has happened here?

  • Right.

  • They've got the plots on dhe off the events of the confidence levels versus the pigs mass, the range of hicks masses on.

  • But I mentioned the one at about 100 26 g.

  • V has Bean, the one which has got the most significance.

  • That's called a local significance because you're looking at 126 TV.

  • However, it could be that just random fluctuations, that kind of event could have occurred any of other energy scale that it might have occurred 100 35 gvr 100 23 gvr.

  • So that in itself could be an outline from another graph that yes, I think that might be one way of interpreting it on dso the significance off of assigning that high Sigma Tau.

  • That particular energy scale then becomes diminished on.

  • There's a statistical way of trying to account for that.

  • And the look elsewhere just means you're looking across the band of and the likelihood of this thing appearing as you say as an outlier from a different a different regime.

  • That's unique way of saying it.

  • You should be issued for here.

  • Ready day to day.

  • You're an astronomer, not a particle physicist, as far as I know.

  • Yes, indeed.

  • Do you?

  • Cygnus were signatures.

  • Astronomers are very bad at statistics, I have to say.

  • And so some astronomers use statistics very sloppily.

  • I try not to, but astronomers do you signals.

  • But I'm much more comfortable when you quote something as one of these confidence levels that you've measured something at the 90% confidence level, which basically means that you know, I'm pretty sure I've measured something.

  • But there's a one in 10 chance I'm wrong.

  • The whole of science is best on in some ways narrowing down the uncertainties.

  • What people talk when they come in to do physics at university and no and at school.

  • Or you should be taught at school.

  • It's not the actual value, necessarily.

  • It's the uncertainty in that value.

  • How confident are you?

  • You're talking about uncertainty with such certainty.

  • You're talking about what you don't know.

  • And yet you're able to put numbers on the extent to which you don't know it.

  • Which makes me think you do know it.

  • It's the classic thing, right?

  • There are known unknowns, Aaron there are unknown unknowns.

  • There are known unknowns because we know there are limitations to our experiment.

  • We know that we only recorded this much data and therefore the most precision we could ever get is this much.

  • So there are those things which are sort of we really can't quantify.

  • But then there are the unknown unknowns, which is that it's possible we just set up our experiment wrong.

  • It's possible that something in the experiment we really don't understand.

  • It's possible that some implicit approximation we've made somewhere on the line that actually has a bigger effect than we thought it did.

  • And so those kind of things, though sort of systematic errors where we really don't know what they are, they can always be lurking there.

  • And so even when somebody quotes a result, as you know, a Six Sigma result and therefore hugely significant, if they've missed some systematic error in there, it could still go away.

  • And so indeed, we have to be a little careful that when you quote an error unquote something as being incredibly improbable that you're wrong, there's always the possibility that you've just missed something entirely in the process.

you're talking about uncertainty with such certainty.

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B1 sigma signal average distribution measurement uncertainty

Five Sigma - Sixty Symbols

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    林宜悉 posted on 2020/03/30
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