2010-03-22

Murakamu Haruki's Peter the cat translation to English/German

My friends translated 'Peter the cat' by Murakami, Haruki to English and German to learn German. As far as I know, this article is only available in Japanese. This seems useful for learning Japanese/English/German, I post it here. If there is any problem, please let me know.
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2010-03-17

A 6σ Woman (3)

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!

2010-03-16

A 6σ Woman (2)

Average and variance of salary/month

The story started with variance, I should have told you also another important concept --- average.

If you heard a company A's average salary for a month is 25,500 Euro, you might interested in to join the company. But, this company has only two people, a president and an employee. The president gets 50,000 Euro/month, and the employee gets 1,000 Euro/month. Now you see how average can deceive you. A company B is also two people company, but the president gets 30,000 Euro/month and the employee gets 21,000 Euro/month.

CompanyA
salary of president 50,000 Euro/month
salary of employee 1,000 Euro/month
Average of salary 25,500 Euro/month

CompanyB
salary of president 30,000 Euro/month
salary of employee 21,000 Euro/month
Average of salary 25,500 Euro/month


The averages are the same! If I can choose one of them as an employee, I will choose B. But, if they can provide only average, I can not see the difference. That's bad!

Therefore, statistics considers the squared average of difference from the average. (Oh, well, shall I put an equation here?) This is like putting how much difference from the average on a seesaw. If this is small, then everyone is close to the average like company B. You can consider this value as how much you can trust the average. If the value is large, then, someone is far from the average, means company A condition happens. If you want to know about the detail of this equation, look up ``variance'' or ``standard deviation'' in Google. I just compute the standard deviation of these salary/month here.


Company A
salary of president 50,000 Euro/month
salary of employee 1,000 Euro/month
Average of salary 25,500 Euro/month
Standard deviation 27,300

CompanyB
salary of president 30,000 Euro/month
salary of employee 21,000 Euro/month
Average of salary 25,500 Euro/month
Standard deviation 3,180


The averages are the same for the both companies, but, B is less dispersion, therefore, standard deviation becomes smaller than A. The average didn't give us this information. We use the variance or standard deviation in statistics because of this. We conventionally use σ as deviation. Now you know why we talk about σ.

Now you may see average deceits you. But of course average is important. With standard deviation (or variance) you can see more. This is the reason statistics uses standard deviation.

2010-03-07

A 6σ Woman (1)

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.


Figure 1


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).