What is the formula utilized to calculate the age of the Universe?
In terms of a formula, one that is used is the inverse of the Hubble Constant, so the formula is
There isn't a standardized formula to estimate the age of the Universe. There are, however, 2 methods that are generally looked to as ways to estimate the age of the Universe.
Please note that we are talking about estimates - there is currently no way known of determining exactly how old the Universe is - and in fact, as scientists study the Universe in its younger days, the composition of it changes so significantly (consider a Universe that only contains tightly compressed plasma - that was the state of the Universe 13 billion years ago) that at some point even the concept of time gets tossed out the window.
So to the 2 ways scientists do their calculations:
look at the oldest structures in the Universe and know that the Universe is at least as old as them and maybe older. Globular clusters are among the oldest structures we can see and age estimates put them anywhere between 11 and 18 billion years old, so there is a floor of an age at 11 billion years (it can't be any younger but can be older).
look at the expansion of the Universe, described by Hubble's Constant, then simply take the rate of the expansion and reverse it to see how long it takes to get the estimated size of the Universe down to the size of a baseball. That equation looks like this:
Looks so simple, doesn't it? The solving is in the first link below. Anyway, this calculation gives solutions in the 10-20 billion year range.
The current accepted age range is 13.5 and 14 billion years.