Variance I

 

Too Much of Nothing

Too much of nothin' can turn a man into a liar
It can cause some man to sleep on nails
Another man to eat fire
Everybody's doin' somethin', I heard it in a dream
But when it's too much of nothin', it just makes a fella mean
Say hello to Valerie, say hello to Marion
Send them all my salary on the waters of oblivion 

Say hello to Variance too! 

A normal random distribution is defined by its location, often called the Mean/Median/Mode, μ,  and its Variance, σ². The mean/median/mode are not equal, except at infinity. The mean, often called the average, is computed from the total of the observations divided by the number of observations. The variance is the range of the observations but is not as easy to compute. An observation consists of where you are observing a thing,  its x value, and what is the attribute of the thing that you are observing, its f(x). Just because the x-axis is infinite it does NOT mean that the function on that x-axis is also infinite. For example, a wave is a repeating function, f(x)= cos(x). In this case x can be any number, e.g., infinity, ∞, but the value of f(x) can only be between -1 and 1. Mathematically this would be stated that the domain is infinite, but the range is finite and between -1 and 1.  The variance, the range, can be finite even if the input variable, the domain. is infinite.

It is fairly easy to compute the mean of a variable. It is the total of all of the observations divided by the number of the observations. It is a little harder to compute the variance of the observations, but not impossible. If an infinite number of variables is computed then the mean by definition is half of infinity. But it is important to differentiate between infinite variables and infinite domains. Just because the domain is infinite that does not imply that the mean/median/mode is infinite or that the variance is infinite. These are characteristics are of the range, not the domain. The range can be finite even if the domain is infinite, for example, the cosine (x). The mean/median/mode of the cos(x) is sine (x). The variance of the cosine, and the variance of the sine for that matter which is also the mean/median/mode of the cosine, is ½.

The variance is thus a constant. The variance of the range is always a constant, even  if the domain is infinite. Thus it is not inconsistent to say that the variance of infinity is a finite number. The mean of cos(x) is sin(x) whose domain is also infinite, but the variance of that cosine function, and its mean the sine, is, ½,  which is finite.