this variable $\Omega(n)$, the number of prime factors of $n$ counting multiplicity, has for large $n$ a normal distribution with mean [*] $1+\log(\log n)$ and standard deviation $[\log(\log n)]^{1/2}$; see, for example, Prime Numbers and Computer Methods for Factorization, page 167 [first edition], page 159 [second edition].

[*] more precisely, this additive constant 1 should be replaced by $1.03465\ldots$ as calculated by Knuth and Trabb-Pardo (appendix A); incidentally, if we don't count multiplicities the normal distribution has mean $0.26+\log(\log n)$ with the same standard deviation $[\log(\log n)]^{1/2}$, so the only difference is a slight displacement of the whole curve.

--- update 2020, in response to query:

the "0.26" number is defined as
$$c_1= \gamma+\sum_{p\;\text{prime}}\biggl(\log(1-1/p)+\frac{1}{p}\biggr)= 0.261497212847643$$
while the "1.03" number is defined as
$$c_2=\gamma+\sum_{p\;\text{prime}}\biggl(\log(1-1/p)+\frac{1}{p-1}\biggr)= 1.034653881897438$$
The number $c_1$ is known as the Meissel-Mertens constant.
Both $c_1$ and $c_2$ are referred to as Hadamard-de la Vallée-Poussin constants (see also this MathWorld entry).