A 6 sigma event every 2 months!

“So much or so much sigma, that is a mathematical impossibility.” I have seen this sentence many times on LinkedIn or similar places.

The supposed meaning of this sentence is that the probability of an event corresponding to x — with x being 3 or 6 or similar — standard deviations has a probability so small that it can be consider for all practical matter as impossible.

These comments are based on a misunderstanding of what “sigma” represents together with a confusion between “normal distribution” and “any distribution”.

I write this post with a discrete distribution notation but it can be extended easily — with a little bit of extra notations — to continuous distribution.

Suppose that my distribution has n points with values a_i and probabilities p_i = 1/n (i = 1, … , n). The mean is

m = sum_i=1ⁿ a_i p_i = 1/n sum_i=1ⁿ a_i

To simplify further my notations, I suppose that the mean is 0.

The standard deviation is the square root of variance, with variance

V = 1/n sum_i=1ⁿ (a_i)²

Let me take a very simple distribution with n>2 and where all the values are 0, except the first 2 that are b and -b. The mean is indeed 0. The variance is

V = 1/n * 2 * b²

and the volatility (sigma) is

s = sqrt(2/n) * b

What is the probability of an event with value larger or equal (in absolute value) to x>0 standard deviation? We have

|a_i| ≥ x * s = x * √(2/n) * b

if and only if

i = 1 or 2 and b ≥ x * √(2/n) * b.

The latter is equivalent to

n ≥ 2 * x^2.

Take x = 3. As soon as n≥18, there are results above 3 standard deviations. The probability of that happening is 2 * 1/n. For the lowest number of points, n=18, the probability is 1/9, more that 10%.

What that shows is that for at least one distribution, the probability of a 3 sigma event is more than 10%, i.e. one day every two weeks, if we consider only business days.

We can do the same with different x. The second column report the number of point and the third the probability that the distribution above (denoted B) is in absolute value above x.

x	n	P (|B| ≥ x)	P (|N| ≥ x)
3	18	11.1111111%	0.2699796%
4	32	6.2500000%	0.0063342%
5	50	4.0000000%	0.0000573%
6	72	2.7777778%	0.0000002%

This is compared in the last column with the probability of the normal distribution (denoted N) is in absolute value above x.

A 6 sigma event, it is only a one in 40 events, i.e. once every 2 months for daily measures. Nothing impossible there!