Originally, I would like to continue to talk about the determinant, but, for a while, I would like to write about sigma story. And I will come back to the determinant again.
Starting with a question
Have you ever met a 6σ person?
What is 6σ?
When I lived in Saarbruecken, I saw a huge woman working in the Karstadt. I believe it's not true, but a rumor said, she stacked in a register seat when she was working. But I cannot say that is not a true... It is hard to find such a person in Japan. The distribution of Japanese's height or weight seems narrower than Germany's one. I feel the distribution of USA people is wider than those two countries. It means, large people are larger than others in USA, I think. 'Average' is an important concept in statistics, and this 'distribution' is the second important concept in statistics. The distribution, i.e., how the people have different weight, or how the people are close to their average weight.
In mathematics, this is called 'variance (σ^2)' or its square root 'standard deviation (σ).' Variance or standard deviation can show how many people are far away from the average. Using mathematical words, standard deviation of US people's weight distribution would be larger than Japanese one.
Figure 1 shows a normal distribution or Gauss distribution. The upper Figure is σ = 1 case, the bottom is σ = 2 case. The upper figure has smaller σ than the bottom one, we could call it Japanese type and the bottom one is US type. However, there are Political correctnessist (I just made this word up) -- who believe if people don't say the problem acculately, then, the problem will be gone -- would complain me. So, let's stick to the traditional name, the Gauss distribution.
Roughly speaking, The X axis of figure indicates samples, for instance, weight of people, the Y axis indicates how many people have that weight at X. The highest point at the center is average in Gauss distribution. This means, average is most common. If the average is 60kg, many of the people has 60kg weight most probably. Actually, we need to integrate this distribution, but, I try to give you a rough idea.
Standard deviation describes how the fat/thin people are distributed in the society, but, if the distribution can be approximated by the Gauss distribution, we could say, how a person is special by a single number. We can use the how many σ-special-degree someone is.
Because Gauss distribution is completely described by an average and a standard deviation, we can describe someone's location by using the average and $σ$. Here the average was chosen as zero, then only standard deviation is needed to say how someone is special or unusual. Gauss distribution is very powerful, but, there are many distributions that don't follow the Gauss distribution.
Next time, I would like to talk about that where is one. For Japanese students, this is also known as Hensachi (deviation).
19 hours ago