Concatenation:

I think I figure out the method of calculating the rarities of values that are beyond 3SD(For a variable, when the value is over 3SD, the distribution will not be normal anymore)

Well, for ex, I want to calculate the rarity of 200cm in a country with an average height of 170cm, whose SD is 6cm

3SD is 18cm, and assuming the sample size is sufficient(thus the distribution of the height of country is normally distributed and the average height data is accurate), and since height is normally distributed within 3SD, we can calculate the rarity of 187cm is 0.X%

and we can thus suppose a sufficient sample whose mean is 187cm, but we don't know its SD.

We can calculate its SD by this way:

186cm is within 3SD of the mean of 170cm as well, after knowing the rarities of 186cm and 187cm, and since we know that, the sample whose mean is 187cm subjects to normal distribution which is not violated by the new mean, so the difference of 186 and 187 in addition to the difference of the rarities of them is sufficient to calculate the SD

and the SD is so small so that 200cm is still over 3SD of this hypothesized sample

but never mind, repeat the step

finally you can know the accurate rarity of 200cm which is not influenced by the distortion derived from the value being too far away from the mean

The quantitative definition of coprime numbers and some properties of them:

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