# Six sigma in neutrino article?

1. Dec 2, 2011

### zosto

"six sigma" in neutrino article?

in the first paragraph they say "...notwithstanding the assertion that it is a six-sigma effect". What does this mean? I tried to google, but didn't find that very helpful.

By the way, I'm new and wasn't quite sure where to put this.

Thanks :)

Last edited by a moderator: May 5, 2017
2. Dec 2, 2011

### D H

Staff Emeritus
Re: "six sigma" in neutrino article?

It means six standard deviations.

Assume (and this is a big if) that the OPERA people did their statistics right, that they did not make any mistakes in their calculations, that they don't have any unknown errors or biases in their setup, etc. With these assumptions, there results are so solidly far from the speed of light that there is only a tiny, tiny probability those neutrinos were going slower than the speed of light but that some incredibly fluky combination of errors made it look like they were going faster than the speed of light. How tiny? It's a one-sided error that is six standard deviations, or six sigma, from the mean (one-sided because an error that has the neutrinos going even faster than they apparently measured doesn't falsify their results). A one-sided six sigma error is about the same as a one out of a billion chance.

In other words, a six sigma event is rock solid.

3. Dec 2, 2011

### phinds

Re: "six sigma" in neutrino article?

Actually you're making it look about 4,000 times stronger than the actual value of about 4 parts per million (one out of 250,000). Still VERY unlikely, but lets keep the math straight.

4. Dec 2, 2011

### D H

Staff Emeritus
Re: "six sigma" in neutrino article?

$$1 - 0.5(1+\text{erf}(6/\surd 2)) \approx 9.9\times10^{-10}$$

Edit
That's for a one-sided event. If the correct interpretation is as a two-sided event, the probability of an error doubles to about two out of a billion.

Last edited: Dec 2, 2011
5. Dec 2, 2011

### MaxPlank

Re: "six sigma" in neutrino article?

I don't know if you know what a standard deviation is and how errors are treated in physics, so I try to explain it to you in a simpler way...Every result of a measurement (for example x) in physics is given with an error called sigma: σ. This means that the probability that the real value of what you have measured is between x-σ and x+σ is the 68%. So if you imagine to repeat many times the measurement, you expect to obtain 68 times out of 100 a value which is between x-σ and x+σ. You can explain this in an another way: the probability that you obtain a result which is out of the interval x-σ and x+σ due to statistical fluctuations is 32%. Instead if you consider the interval x-2σ and x+2σ the probability that a result is out of the interval is 5%, if you consider 3σ is 0.7% and so on.
In the experiment opera they expect that the difference between the neutrino velocity and the speed of light is zero. So, imagine you perform the experiment obtaining a value of the neutrino velocity equal to v with an error σ. If the difference (v-c) (value measuered and value expected) is 3σ, there's a 0.3% that this value is due to statistical fluctuations. In physics you say that "you discover something new" when this difference is more than 5σ.
So 6σ means that is very unlikely that the result obtained is due to statistical fluctuations. Pay attention that it doesn't mean you're right or that the experiment was performed good: maybe there are systematic errors you haven't taken into account.

6. Dec 2, 2011

### zosto

Re: "six sigma" in neutrino article?

Ah, okay. I don't know anything about statistics. I'm trying to learn...

Why would they expect the neutrinos to travel at the speed of light?

7. Dec 2, 2011

### D H

Staff Emeritus
Re: "six sigma" in neutrino article?

They would expect neutrinos to travel at less than the speed of light. Not equal, not faster.

8. Dec 2, 2011

### zosto

Re: "six sigma" in neutrino article?

so is this not right?

9. Dec 2, 2011

### D H

Staff Emeritus
Re: "six sigma" in neutrino article?

10. Dec 2, 2011

### phinds

Re: "six sigma" in neutrino article?

Hm ... guess I've had it wrong all these years.

EDIT: on the other had, so, apparently, do most of the references on the web since they all seem to say six sigma = 3.4 errors per million events which is what I thought it was.

Perhaps as used in "quality defects" it has a different meaning than what you are referring to.

11. Dec 2, 2011

### D H

Staff Emeritus
Re: "six sigma" in neutrino article?

That is exactly the case. The wiki article on six sigma explains it nicely:

To account for this real-life increase in process variation over time, an empirically-based 1.5 sigma shift is introduced into the calculation. ... [and later]
It must be understood that these figures assume that the process mean will shift by 1.5 sigma toward the side with the critical specification limit.​

So when they say "six sigma" with respect to quality they really mean "4.5 sigma". Using this "1.5 sigma shift",
$$1 - 0.5(1+\text{erf}(4.5/\surd 2)) \approx 3.4\times10^{-6}$$

12. Dec 2, 2011

### phinds

Re: "six sigma" in neutrino article?

Interesting. I was not aware of that. Thanks for the clarification.