Asymptotic distribution of the interquartile range
The asymptotic distribution of the interquartile range for the normal distribution is shown here. Let $f$ be the density, $F$ the CDF and the population quantile function be $F^{-1}(p)$ of a random variable. Further, let $F^{-1}(p) = \xi_{p}$. Then, the following holds asymptotically:
$$
\sqrt{n}\left(\mathrm{IQR} - \left(\xi_{\frac{3}{4}}-\xi_{\frac{1}{4}}\right)\right)\xrightarrow{d} \mathrm{N}\left(0, \frac{1}{16}\left[\frac{3}{f^{2}(\xi_{\frac{3}{4}})}+\frac{3}{f^{2}(\xi_{\frac{1}{4}})}-\frac{2}{f(\xi_{\frac{1}{4}})f(\xi_{\frac{3}{4}})}\right]\right)
$$
For iid observations of a normal distribution $\mathrm{N}(\mu, \sigma^{2})$, this result simplifies to:
$$
\sqrt{n}\left(\mathrm{IQR} - 1.349\sigma\right)\xrightarrow{d} \mathrm{N}\left(0, 2.476\sigma^{2}\right)
$$
So asymptotically, the standard deviation is $1.573\sqrt{\frac{\sigma^{2}}{n}}$.
In summary, the IQR of a normal distribution $\mathrm{N}(\mu, \sigma^{2})$ is asymptotically normally distributed with mean $F^{-1}_{\mu, \sigma^2}(3/4)-F^{-1}_{\mu, \sigma^2}(1/4)$ (i.e. the population IQR) and variance $2.47569\sigma^2/n$.
Let's check that with a small simulation:
# Parameters
mu <- 100
sigma <- 15
n <- 5000
Asymptotic variance of IQR
varfac <- (1/2)exp(2(qnorm(1/2/2, lower.tail = FALSE)/sqrt(2))^2)*pi
Population IQR
true_iqr <- qnorm(3/4, mu, sigma) - qnorm(1/4, mu, sigma)
Simulation
set.seed(142857)
res <- replicate(1e5, {
IQR(rnorm(n, mu, sigma))
})
Mean and variance of simulated IQRs
mean(res)
[1] 20.2288
var(res)
[1] 0.1117288
varfac*sigma^2/n
[1] 0.111406
For $n=5000$, the agreement is excellent.
Relative efficiency
For the standard deviation, we have according to the delta method
$$
\sqrt{n}(s_n-\sigma)\xrightarrow{d} \operatorname{N}\left(0, \frac{\mu_4-\sigma^4}{4\sigma^2}\right)
$$
where $\mu_4$ is the 4th central moment. For a normal distribution, this simplifies to $\operatorname{N}\left(0, 1/2\sigma^2\right)$. A consistent estimate of $\sigma$ of a normal distribution is $\operatorname{IQR}/1.349$ which has an asymptotic variance of $2.476/1.349^2=1.361$. Hence, the asymptotic efficiency of the interquartile range relative to the standard deviation is the ratio of their asymptotic variances, namely $(1/2)/1.361=0.367$.
