Distribution of Primitive Elements Finite Fields Prime Order

Question

It is well known that the integers modulo a prime $p$ form a finite field and that the multiplicative group of this finite field is cyclic, with $\phi(p-1)$ different possible choices of primitive elements that generate the cyclic group. However, if one calculates these primitive elements for various primes, their distribution seems random. So my question is whether anything is known about this. For example, suppose one works out the average of all primitive elements. One would expect that there is no bias in the size of this average, so that over various primes it should be roughly $p/2$. To quantify this conjecture, one would expect that if you divided the average by $p$ for each prime, then the average of all these values up to a prime $P$ tends to $1/2$ as $P \to \infty $. This seems like one of those statements that seems like it should be true due to lack of "biases" or "conspiracies", but is virtually hopeless to prove. Is anything known about this? ( Note that this example of the averages that should on average be $1/2$ was an example, I wonder in general about distribution: can the primitive elements all be small, all large, can they bunch up, spread out, etc. I hope I got the general gist of my question across. Thanks in advance! )