# Visualizing distributions of measurements with uncertainty

Suppose I have a vector of measurements, each of which has an associated potential measurement error. E.g. all my measurements are of the form $$2 \pm 0.5$$ or something like that. I can look at the distribution of the measurements (ignoring the standard error) by making a histogram or boxplot or whatever.

But is there a good way to convey through a univariate graphical representation what the uncertainty in the measurements is?

I read this post about error bars for histograms, but it appears that there is no definitive way to do this, if adding error bars to the histogram is even the best way to show potential error collected like this.

Update: I'm assuming that all of the measurements are normally distributed, e.g. $$x_i \sim \mathcal{N}(\text{point measurement}, \ \text{standard error}).$$ I know that if I make this assumption, I can get an approximate distribution for the sum or average or whatever, but I'm really interested in trying to visualize this with uncertainty without having to take any summary statistics.

Here's some example data (formatted for R input).

structure(list(x = c(-0.06, 0.64, -1.71, -1.56, 2.28, 1.03, 0.07,
-0.46, 2.14, 0.59, -2.22, 0.15, 0.62, -0.34, 0.18, -0.87, 0.68,
0.85, 0.25, -0.32, 1.1, -0.21, -0.26, 0.45, -0.68, 0.07, 1.71,
-1.33, -0.73, 0.79, -0.5, 0.4, 0.59, -0.83, 1.42, -0.73, 0.55,
0.58, 0.71, -1.14, 0.56, 0.56, -0.25, -0.48, -0.6, -1.37, -1.56,
-1.27, -0.17, -0.36), se = c(0.13, 0.15, 0.15, 0.15, 0.15, 0.14,
0.14, 0.14, 0.14, 0.13, 0.14, 0.16, 0.14, 0.15, 0.15, 0.12, 0.14,
0.15, 0.13, 0.14, 0.14, 0.14, 0.13, 0.15, 0.13, 0.15, 0.12, 0.13,
0.13, 0.15, 0.14, 0.14, 0.14, 0.14, 0.14, 0.15, 0.13, 0.15, 0.14,
0.13, 0.14, 0.14, 0.14, 0.15, 0.14, 0.14, 0.14, 0.14, 0.14, 0.14
)), class = "data.frame", row.names = c(NA, -50L))
• Is the error distribution the same for all data, symmetrical (eg, normal), & w/ a constant error SD? Can you provide an example dataset for people to work with? Feb 2 at 19:17
• @gung-ReinstateMonica I've updated it and added some example data. I think probably OK to assume a constant error SD, but I'd prefer not to if it isn't necessary. Feb 3 at 1:39
• You can always plot each estimate together with some kind of interval (x +/ se) or some confidence interval. Here the se is so now constant and not large relative to x that I am not clear that one gains anything much visually, although the honesty of quoting uncertainty is always admirable. Feb 3 at 1:58
• (+1) Interesting question! One complicated option could be to use the standard errors to simulate a large number of datasets, generate density plots/histograms of each (with the same bandwidth/interval parameters), and overlay them with very high transparency. The final result may look like a mess, though, and so I'm not sure that this will be very useful.
– mkt
Feb 3 at 8:42
• ... so near constant ... Feb 3 at 15:01