Error bars on error bars? Inspired by my recent attendance at an environmental toxicology conference, I have the following question about error bars:
Let's say that I'm drawing samples from some unknown distribution, with finite mean and variance. I want to present the sample mean, and add some error bars. Since I don't know much about the underlying distribution, I just add error bars showing +/- the standard devaition of the samples.
My question is, is there any way I could meaningfully indicate how certain I am of those error bars? Adding error bars to the error bars, so to speak.
As as example, I have drawn 5 samples from some distribution, and I have repeated this 5 times. The sample means, and error bars of +/- the sample standard deviations, are shown below.

We can see that by chance, these sample means and error bars look quite different, and not really mutually compatible. Of course 5 samples isn't very much, but if my samples are obtained via some convoluted experimental procedure (capturing a wild animal and taking a blood sample, for example), it might not be an easy option to get more samples.
Update:
Just to add some notes on how I was thinking:
Coming from a computational physics background myself, I'm used to Monte Carlo methods, and the $1/\sqrt{N}$-error which follows from the central limit theorem. So just like the error in the sample mean has an expected distribution, I thought perhaps it would make sense to ask about the expected error in the sample standard deviation. Of course, the problem is that the distribution of the error in the sample mean is expressed in terms of the (unknown) variance of the underlying distribution, and hence I am left taking the standard deviation of the sample, or something along those lines.
But still, I thought there ought to be some way of indicating that my sample standard deviation is itself quite uncertain, due to the small $N$. But perhaps the only way is simply to list $N$, and be explicit about what the error bars show.

The traditional design of error bars gives an unfortunate impression of some linear distribution of uncertainty, and places a lot of visual emphasis on the the end of the bar, which is where the distribution of the location of your estimate is least likely. Clause Wilke (in his book Fundamentals of Data Visualization, in the chapter Visualizing uncertainty) shows some graphical alternatives to traditional error bars that convey something of the distribution of uncertainty in an estimate:
Image by Claus Wilke, used under an Attribution-NonCommercial-NoDerivatives 4.0 International licence. Original available at https://clauswilke.com/dataviz/visualizing-uncertainty.html
The "graded error bars" in (a) and (b) are formed by plotting the 90%, 95% and 99% CIs simultaneously. Thom Baguley discusses a similar approach he terms "tiered error bars" and provides example R code here: https://seriousstats.wordpress.com/2012/06/21/confidence-intervals-with-tiers/ , although I first saw such an approach being used by Andrew Gelman in his textbook Data Analysis Using Regression and Multilevel/Hierarchical Models.
