Variance II

 

Don’t Fence Me In

I want to ride to the ridge where the west commences
Gaze at the moon till I lose my senses
I can't look at hobbles and I can’t stand fences
Don't fence me in.

But what if the fence is very, very distant?

A fence limits the outcomes of events. In the universe of random choices there are limits. Those limits are very, very large. For example there are estimated to be between 10⁷⁸ to 10⁸² atoms in the universe. Each of these atoms can have as many as 100⁺ electrons. Each of these electrons can be in random, quantum locations. It is such a large number that for all practical purposes the human mind treats this as an infinite number. But the choices still exist and there is a finite, although very, very distant “fence.” 

In a logistics, sech squared, distribution the parameters are the mean, µ, and s, the range of the choices. The variance, σ², in a logistics distribution is s²π²/3, which means that σ=sπ/√3, and s=√3σ/π,  The variance in a logistics distribution also follows the 68/95/99 rule of normal distributions. In fact at µ+3σ there are 99.97% of the outcomes in a normal distribution. Since in a normal distribution, such as the logistics distribution, the median is also µ and the median by definition is 50% of all outcomes, then, with rounding, 2µ=(µ+3σ) or σ= µ/3, which means that s= µ/√3π. It is thus not possible for there to be no fences, a variance of zero, unless the mean is also zero. In a simple binary choice, yes/no, 1/0, the mean, µ, is 0.5 and the variance, σ², is thus .03.  The mean, µ, is by definition n/2, where n is the number of observations in the sample, thus the variance is  n²/36. As noted above the variance in a universe with more than 10¹⁰⁰ , also known as googol, where do you think the name google comes from, choices will be very, very large but it will still exist. Unless there are no choices, s=0, then the variance can not be zero. Just because you can’t see the fence because it is very far away, doesn’t mean that the fence is not there.