# Analyzing math equations

This could be to a certain extent a philosophical question and I might know the answer already but I am interested in what others think.

My question is roughly the following. Let's say I am reading some new math material and let that be Statistics for my example but it could apply in any branch of math.

I come across population or sample mean and the definition is something like: $$\mu = \frac{1}{n}\sum_{i=1}^n x_i$$

For some reason this is intuitive enough, the average is the sum of the values divided by their number.

Now you get to the equation for variance: $$S^2 = \frac{1}{n} \displaystyle\sum_{i} (x_i - \bar{x})^2$$

Now according to my understanding, the variance is a measure of data spread from the mean. So translates to subtraction in the equation. All good, then comes the squaring ^2.

Looking around online, many theories come about why there is a squaring operation in there. To get rid of negative values, to allow for analysis of a continuous function and so on and so forth.

I find my self in this situation very often, where I see an equation and I don't understand how it came about. It seems as if the inventor or author just had an intuition that he didn't document or maybe it's trivial and its up to the reader to deduce the knowledge.

What is it though? Why isn't such information captured? Is it just about practicing? Or we don't even care, we just study useful properties or if things make sense? How do mathematicians or scientists arrive at such conclusions? hunch? trial and error?

• Good and thorough question. If I have some time, I will try to answer it, very humbly. – Jean Marie May 2 at 15:26
• You could begin by reading this – Jean Marie May 2 at 15:29
• You could also have a look to this. – Jean Marie May 2 at 15:30
• Basically, in order to avoid that two points "very far" from the mean, one in positive and one in negative, will cancel each other conveying the wrong information that tehre is no dispersion at all. – Mauro ALLEGRANZA May 2 at 15:33
• @JeanMarie - Do not worry :-) We are here (also) to exchange points of view. My point is that, IMO, there is nothing "philosophical" regarding variance. – Mauro ALLEGRANZA May 2 at 17:26