A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range1. The standard deviation and mean are often used for symmetric distributions, and for normally distributed variables about 70% of observations will be within one standard deviation of the mean and about 95% will be within two standard deviations(68–95–99.7 rule). For non-normally distributed variables it follows the three-sigma rule.
As shown below we can find that the boxplot is weak in describing symmetric observations.
Generated by this snippet of R code(borrowed from this answer):
normal <- rnorm(10000)
a_vector <- c(-3, -2.65, rep((-2:2)*.674, 5), 2.65, 3)
We can see that the IQR is the same for the two populations
2 but we can see the difference of the two by their means and standard deviations.
mean(normal); var(normal); mean(a_vector); var(a_vector)
We can see from the above case that what median and IQR cannot reflect can be obviously conveyed by the mean and variance.