How would you pick a distribution with N atoms to approximate, in some way, the Normal distribution?

Question

My question is not mathematically precise and some of you will probably vote to close it, but nevertheless I feel like there must be precise answer.

Consider Normal distribution $N(0,1)$

If you were told to describe this distribution with only one point in $R$, what point would you choose? For me obvious answer is $0$, mean of distribution, point with highest density.

But, if you were told to describe this distribution with two points in $R$, what points would you choose? I don't know , looking at normal density my intuition tells me that I would choose something like $(-0.5,0.5)$

Is there mathematically precise definition of what I say as "describe distribution"?

How can I choose $N$ points from Normal in $R^d (\mu, \Sigma)$?

Answer 1

$\{-1,1\}$ would have mean $0$ and variance $1$

Answer 2

I would propose describing the standard normal with $d$ points by taking the points $F^{-1}(\frac{k}{d+1})$ for $1\leq k\leq d$. This gives you, in some sense, $d$ "evenly" distributed points acoording to the normal distribution. That is, the probability between two adjacent points is constant.