Q.For a probability distribution of standard normal variable, state the estimated limits for the middle 50% observations.

Middle $50\%$ of the standard normal: $-0.675\le Z\le0.675$.

For the standard normal variable $Z$ (mean $0$, SD $1$), the central $50\%$ of the area lies between the first quartile $Q_1$ and the third quartile $Q_3$. Since $Q_1=-0.6745$ and $Q_3=+0.6745$ (each cutting off $25\%$ of the area between it and the mean) …

By the symmetry of the normal distribution, $Q_3-\bar X=\bar X-Q_1$, so $Q_3=11+(11-3)=19$.

In a normal distribution the mean equals the median $Q_2$, and the first and third quartiles are equidistant from the mean:

$\bar X-Q_1=Q_3-\bar X.$ …