May the cumulative distribution function be used to calculate the inter-quartile range and vise-versa? May the cumulative distribution function be used to estimate the inter-quartile range?
I am drawing their similarities.
The IQR is found via a boxplot which has percentiles. A percentile comes from ranking the data.
A CDF calculates the probability something is above and equal or below and equal to a threshold which is like ranking the range the random variable may assume based on observed frequencies.

It should be possible. Inter-quartile range is given by the difference between third quartile and first quartile, i.e $IQR=Q_3-Q1$
Now according to cumulative distribution function of R.V X,
$P(X\leq Q_1)=0.25$ and $P(X\leq Q_3)=0.75$
Hence, $F(Q_1)=0.25$ and $F(Q_3)=0.75$
$Q_1=F^{-1}(0.25)$ and $Q_3=F^{-1}(0.75)$
But to find the CDF from the IQR, I think you might need some more additional information about the distribution.
