Is this a reasonable coarse test for normality?

Question

I'm interested in assessing if the marginalized posterior of a parameter obtained through a Bayesian MCMC process is "more or less normal".

I use quotation marks because I'm not trying to asses for normality strictly speaking. I've tried the usual tests and (as stated in Is normality testing 'essentially useless'? and Why would all the tests for normality reject the null hypothesis?) they invariably fail. I need a coarse parameter to tell me if the distribution looks "more or less normal". This is because I usually apply the MCMC to several datasets one after another, and I need a quick way to asses the "degree of non-normality" without having to check each one visually.

What I came up with is a very quick and simple test:

1. Estimate mean, median, and mode for the distribution.
2. Obtain the largest distance between any of these three values.
3. Scale that distance by the range of the distribution to obtain the final $N_t$ parameter, bounded between [0, 1].

The idea is that as the distribution tends to normality then $mean\sim median\sim mode$ thus $N_t\rightarrow0$, and as the distribution becomes "less normal" then $N_t\rightarrow1$.

I've tested it and it seems to give reasonable results, but I'm concerned that I might be missing some fringe cases where this test will fail and get past me.

Is this a reasonable test for coarse normality? Should I instead rely on a more sophisticated test as the one described in What is a good index of the degree of violation of normality and what descriptive labels could be attached to that index??

Answer 1

Here's a counterexample to your approach: mean = median = mode

If you sample from this distribution here is the QQ-plot:

And the Mean-squared displacement from the calibration line is:

set.seed(123)
n <- 1e6
p <- rbinom(n, 2, 0.5)
x <- numeric(n)
X <- cbind(rnorm(n, -2, 0.25), rnorm(n, 0, 0.25), rnorm(n, 2, 0.25))

for(i in 1:n) x[i] <- X[i, p[i]+1]

mean({sort(x) - qnorm(1:n/{n+1})}^2)

gives a stable constant that converges to a fixed, finite value as $n \rightarrow \infty$.

In this case: $\approx$ 0.32