# Finding confidence interval for unimodal function equivalent to and comparable with standard deviation of normal

I'm trying to characterise an arbitrary, unimodal distribution in a way that is a) easily understandable (to a physics audience) and b) comparable with a normal distribution.

My thinking goes like this:

The interval [$$\mu-\sigma$$, $$\mu+\sigma$$] contains ~68% of the probability mass of a normal distribution N($$\mu, \sigma$$), so by quoting $$\mu$$ and $$\sigma$$ I can characterise the interval in which typical values lie.

If for an arbitrary, unimodal distribution I (numerically) find interval bounds $$\sigma_-$$ and $$\sigma_+$$ such that [$$\nu-\sigma_-, \nu$$] and [$$\nu, \nu+\sigma_+$$] each contain ~34% of that distributions probability mass, where $$\nu$$ is its mode, those values are characteristic and comparable.

In my intuition typical values will fall inside the range in both cases and it will represent the same probability mass. This is notably different from the Half Maximum values of both distributions, which I will call $$a_-$$ and $$a_+$$ and define as $$p(a_{\pm})=0.5 p(\nu)$$. The interval [$$a_-, a_+$$] might contain more or less of the probability mass, depending on the shape, and is thus not comparable for normals and non-normals.

My question is whether this logic is flawed and if there are any references to support it. The fact that I haven't found any so far makes me think my idea is either wrong or trivial. If it's wrong I'd like to understand why, if it's trivial a reputable quote would put my mind at ease. Since the goal is a numerical implementation in python, any code or links to libraries is also appreciated, but not necessary.

• My distributions are sufficiently symmetric (un-skewed) that the 34% condition can be fulfilled for $$\sigma_-$$ and $$\sigma_+$$.
• My distributions have zero or negative excess kurtosis such that the tails do not strongly affect the characteristics.