To print asterisks based on inputs
Shell programs should start with a shebang, so that when executed, the OS can select the correct interpreter:

#!/bin/bash


With Bash, we can read prompted input using read -p:

read -p "Enter the mininum number of stars: " min


We should check that we have a valid positive integer before using $min.

The same comments also apply to $max.  For both variables, consider accepting arguments instead of prompting for interactive input.

Prefer printf to echo -n - that's a good practice for portable scripting.

Most of those for loops can be eliminated with suitable printf formatting.  We can produce n stars with printf '%.${n}s' '' | tr ' ' '*', and we can put that into a space that's m wide, with printf '%${m}s' "$(printf '%.$ns' '' | tr ' ' '*')".  If we did keep the loops, we don't need $ for variable expansion within the arithmetic context (( )).

It seems strange to use arithmetic evaluation but also to invoke expr to increment $sum - it's more consistent to simply write ((++sum)).

The echo command with a single empty argument is equivalent to echo with no arguments, so no need to write "" there.