Let's visualise how a someone is special in the Figure1. In Figure 1, the area filled by blue region is x <= +- 1σ people. Again, in the Gauss distribution, the center is the average. People are apart from the average, then number of such special people rapidly decreases. The red area of Figure 2 shows the people who are not in the area of x <= +-1σ. Figure 1: Inside of sigma 1 region.

Figure 2: Outside of 1sigma region.

Gauss choose the parameter to make the total area of the curve to one. So, we can just compute any area of this graph, then we know the degree of speciality from the area. If you are in the area 0.7, you are in the range of 70% people. Great job, Gauss!

I cannot compute this area by hand, therefore I asked a clever program. That answered me that the blue area of Figure 1 is 0.6827. This means, 68% people is in 1σ (more accurately +-1σ). By the way, some statistics defines the ``normal person'' living in between +-2σ (95.4%) or +-3σ (99.7%). Unfortunately, there is no definition about ``normal person'' in mathematics. Someone must decide who are the normal, this is a political decision. For example, some government always changes this σ depends on what they measure. It looks not so good. This is quite arbitrary, for example, when some market research tries to find ''how many customers are satisfied?'', but excludes ''unusual'' people. Then they could set the normal people as in +-0.5 σ. This means if they can satisfied 38% customers, then all the customers are satisfied since 62% customers are abnormal. The mathematics is valid according to this political decision. Oh, well...

Where can we find my friend as the 6σ person? She is in the red area in Figure 3. We cannot see!

Figure 3: Here is sigma 6.

The probability of 6σ person is the following. (I use matlab/octave here.)

ans =

1.9732e-009

This means, 1 over 200 million. Maybe there is only one in Japan or none since population of Japan is 120 million. The number my friend told us is 0.0001 * 0.03 = 0.00003 (=3e-6), this is 3 over million, therefore, 6σ seems a bit too high. I computed to find something this order, 4.5σ seems somehow this range. We could call her as a 4.5σ person.

By the way, how many company managers in Japan? I don't know a million is a realistic number or not. If there are around 300,000, the number 0.0001 * 0.03 = 0.00003 makes her unique. Coincidentally, I work in the same company of her brother. What a fate!

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