have a normal distribution. I would like to compare two input probabilities from this population to measure how "similar" they are. Everything is subjective but I wanted to be able to say that $x$ is more "similar" to $y$ than $z$ to $y$, using some sort of an equation against the normal distribution.
For example, if the mean of my population is 10, and my standard deviation is 3. I would like my simple algorithm to say find that two points (19 and 17) are more similar than two other points (9 and 10) simply because it's a lot less likely to get point 17 (since it's more than two sigmas away from the mean), thus getting another random point to be near that first point with the low probability, shows higher similarity than comparing two points that occur with much equally higher probabilities.
Using something like $P(X < p_1) - P(X < p_2)$ is not good enough because I may get $0$ if both points are the same. However, obtaining two points of $9$ and $9$, should score less similarity than two points ($20$ and $20$) since $20$ is a lot less likely to occur than $9$.
I feel like I need to use the difference above but somehow also use the mean and sigma to formulate a similarity "formula".
Is there an existing stat test that captures what I'm trying to do above? If not, does anyone have a suggestion as how I would go about solving the problem above?