# Normalized confidence measure of point from normal distribution

I have a 1D normal distribution with mean $\mu$ and standard deviation $\sigma$. Given a new number $x$, I want to assign a confidence value to how likely $x$ is to have been drawn from the distribution $\mathcal{N}(\mu,\sigma)$

1 - the cdf gives me $P(X > x)$ but I'm not sure this is what I'm looking for.

Any help is much appreciated

EDIT

Just another thought, the value $x$ will fall on one side of the mean, or on the mean itself. Essentially I want to define a confidence measure that decreases to 0 as the point moves away from the mean, and is 1 at the mean. So what about $0.5 + f(x)$ where $f(x)$ is the pdf of the normal distribution between the bounds $\mu$ and $x$?

Thanks

## 2 Answers

Let's say you get $x = \mu$. While you could say that's the most likely possible value for $x$ if it did come from $N(\mu,\sigma^2)$, you're not in a position to say that the probability that it did come from that normal is anything. To make such a statement would require something like a Bayesian point of view, and then you're dealing with inference conditional on your priors relating to what else it might be coming from.

In the absence of that, you can talk about how relatively likely two values of $x$ would be. That comes from the ratio of how probable they'd be under the model, like so $f(x_1|\mu,\sigma)/f(x_2|\mu,\sigma)$.

• I have used $f(x|\mu,\sigma)/f(\mu|\mu,\sigma)$
– Aly
Apr 12, 2013 at 14:14

For a measure based on distance from the mean, try this:

Use 2P[X > x] for x > mean, 2P[X < x] for x < mean, and 1 for x = mean