Comparing two probabilities from the same normal distribution

Question

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?

Answer 1

From what I understand, you are trying to combine in a single metric two things: a) the distance between two possible values a random variable can take and b) the probability that the random variable will take values "around there". It does seem that the way to do it, is to measure the probability mass that is enclosed by these two points, and then declare as more "similar" (in your combined sense) the pair that encloses the smaller probability mass. You discuss this yourself, but you write that it doesn't cover the case where the value is exactly the same. But this can be overcome by simply writing this metric as a piecewise expression, say

$$M = P(X<p_1) - P(X<p_2),\; p_1\neq p_2$$ $$M = P(X<p_1), \; p_1=p_2$$

or compactly using the indicator function

$$ M = P(X<p_1) - P(X<p_2)I_{\{p_1\neq p_2\}},\;\; p_1\ge p_2$$

After all, if the two points have zero distance between them, it is only "fair" to take into account only the probability that must be accumulated in order for the random variable to "reach there". The higher $M$ for a pair of points compared to another pair of points, the less "similar" is the first pair, in your sense.

ADDENDUM : Checking the conceptual consistency of the measure Since by the OP's comments to my answer it seems that the OP believes that the answer may be useful, I note that some simple "conceptual consistency" checks must be run, by trying various combinations of pairs of points and see whether this approach gives a logical result. I give one example:

The way it is defined above, it means that the pair of points $(x_1,x_1)$ will always be considered "less similar" than any pair of $(x_1, x_i)$ , $x_1 > x_i$. Is this reasonable regarding the purpose at hand?