In general, considering the number of coprimes to $n$ in an arbitrary interval of length $h$, the expectance is given by 
$$h \frac{\phi(n)}{n},$$
where $\phi(n)$ is the Euler totient function. If $n$ is odd, the variance is given by
\begin{align*}
G(n,h) 
	= \prod_{ \substack{ {p \mid n} } } \left( 1-\frac{2}{p} \right)	 
	\sum_{ \substack{ {d \mid n} } }  \frac{\mu^2(d)}{\rho(d)}		
	d \left\{\frac{h}{d}\right\} \left( 1 - \left\{\frac{h}{d}\right\} \right).	
\end{align*}
If $n$ is even the variance is given by 
\begin{align*}
G(n,h) 
	= \prod_{ \substack{ {p \mid n'} } } \left( 1-\frac{2}{p} \right)	 
	\sum_{ \substack{ {d \mid n'} } }  \frac{\mu^2(d)}{\rho(d)}		
	d \left\{\frac{h}{2d}\right\} \left( 1 - \left\{\frac{h}{2d}\right\} \right).	
\end{align*}
Here $n'$ is the largest odd divisor of $n$, $\mu(n)$ is the Möbius function, $\rho(n) := \prod_{p \mid n}(p-2)$ for any squarefree integer $n$, and {x} is the fractional part of x. The derivation of these expressions can be found in the article [On the mean square distribution of primitive roots of unity][1] by Hausman and Shapiro.

Note that through the inequality $\{x\} ( 1 - \{x\} ) \leq x$, $G(n,h)$ satisfies the upper bound 	
\begin{align*}
G(n,h) \leq h \frac{\phi(n)}{n}.
\end{align*}
This bound holds for even and odd $n$. 

For an actual example of the supremum and infimum of the number of coprimes to $p_k\#$ in an interval of length $h$, consider the following plot for $k=5$ (red and blue, mean subtracted) plotted together with the standard deviations $\pm\sqrt{G(p_k\#,h)}$ (black):

![coprime bounds][2]


  [1]: http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160260407/pdf
  [2]: https://i.sstatic.net/20698.jpg