How to get a Gaussian curve fitting a given range of values?

Question

I was trying to find a way to make a gaussian function out of a range of values:

$1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12\ 13\ 14\ 15\ 16$

I want the mean to be the most probable value, $8$ and the other values get smaller probabilities in such a way that it looks like a bell curve.

So if the range is from $1 - 100$, I need $50$ to be the most likely value.

Answer 1

Assume the mean $\mu$ to be the median ($50$, in your example). In a normal distribution, around $99.7%$ of values are within $3\sigma$ ($3$ times the standard deviation), of the mean, so we want the range to be $6\sigma$ (so in your example, $6\sigma = 100 \Rightarrow \sigma = 50/3$).

For example, if the values are $0, 1, \ldots, 100$, we have $\mu = 50$, and $\sigma = 50/3$. So $X \sim N\left(50, \left(\dfrac{50}{3}\right)^2\right)$. The pdf is:

$f(x) = \dfrac{3}{50\sqrt{2\pi}}\exp\left[-\dfrac{1}{2}\left(\dfrac{3(x - 50)}{50}\right)^2\right]$