How Many Sigma?

by Dr. Tahir Yaqoob on February 6, 2012

Quite often, a scientific conclusion has to be assigned a probability of being true, and this is often expressed in jargon as a certain number of “sigma.” We will need to refer to this terminology often, so the purpose of this article is to explain what the jargon means.

First of all, there is an implicit assumption that the pertinent quantity that is being assigned a probability is being drawn for a very specific distribution, namely the normal distribution (or bell curve). Another term for it is Gaussian distribution. Now in reality, it will often be impossible to know whether this assumption is true or not, but it may be the best that can be done. What determines whether something follows such a distribution or not is beyond the scope of this article, but maybe we’ll come back to it another time.

Whereas probabilities are calculated from areas between two points on the horizontal axis, the “number of sigma” is simply the horizontal distance from zero that encloses an area corresponding to the required probability. The units are such that “1 sigma” corresponds to a probability of 0.683 (to 3 decimal places). In fact, “1 sigma” is precisely the 1 standard deviation from the mean, where standard deviation has the usual meaning in statistics. This is illustrated below for 1 sigma, 2 sigma, and 3 sigma.

3 sigma area of normal curve

Now, a scientific conclusion is not considered to be worthy of merit unless it has a confidence level of 3 sigma (translation: to avoid getting laughed at you had better make sure your result is at least 3 sigma). Generally it must be at least 5 sigma for any serious claim of a “discovery.” For the record, I’ve given below the probabilities to 10 or more decimal places, corresponding to 1, 2, 3, 4, and 5 sigma. You can see from the awkwardness of the numbers that it is handy to be able to refer to probabilities in terms of the number of sigma.

Probability (P) 1 – P
1 sigma 0.6826895475 (0.31731045)
2 sigma 0.9544997215 (0.045500279)
3 sigma 0.9973001480 (0.0026998520)
4 sigma 0.99993669986724854 (0.0000633001327515)
5 sigma 0.99999940395355225 (0.0000005960464478)

 

To get more of a feel for these probabilities, let’s consider the toss of a coin and ask for a given number of tosses, what is the probability of not obtaining all heads or all tails? We can then express these in terms of the number of sigma. For a small number of tosses the distribution is not actually a bell curve, but we can do the exercise anyway, just to get a feel for it. Below are the results, just picking out the number of tosses closest to one of 1, 2, 3, 4, or 5 sigma probabilities.

Number of coin tosses Sigma
3 1.15
6 2.15
10 3.10
15 4.00
22 5.06

So, to ensure that you don’t get all heads or all tails at a confidence level of 5 sigma you have to toss a coin 22 times. By the way this also means that if you want to have kids that don’t all have the same gender, at a confidence level of 5 sigma, you have to have 22 kids!

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