It's 38% for one dice roll because each result in the probability space(the set of integers from 1 to 100) is equally likely, but when you add in more dice rolls, you start to form a bell curve and then stuff like the law of large numbers starts to kick in and values closer to the mean are more likely, and values far away form the mean are less likely—this is the same principle that says flipping a million
coins, we should expect around 500 thousand heads and tails, and getting less than 400 thousand heads would be very very unlikely. Derpmind posted a good visual representation of this with the probability space of 2d6
here, and I made my own
here.
Another way to frame the problem is that we want to find the chance that we roll a
152 or less on a 4d100. The number of ways to add up to 152 (and the numbers less than it) is 20,291,150. The entire probablity space of 4d100 is
100,000,000 ways to add up numbers. Dividing, we get 0.2029115, or roughly 20.3% to get an average of 38 or lower.