Approximating sums of binomials based on the central limit theorem I am looking for approximations of the following function:
$$
Y_n(x) := {1\over 2^n}\sum_{i=0}^{x\cdot n} {n\choose i}
$$
where $n\in\{1,2,\ldots\}$  and $x\in\{0,{1\over n},{2\over n},\ldots,1\}$. Note that $x\cdot n$ is always an integer.
I plotted $Y_n(x)$ for four values of $n$ (25,50,75,100):

It seems very similar to the cumulative distribution function of the normal distribution $N[0.5, 0.5/\sqrt{n}]$:

This makes sense, since by the Berry-Essen theorem (a stronger version of the central limit theorem), the binomial distribution is similar to the normal distribution when $n$ is large.
Is there a way to turn this observation into a quantitative approximation theorem? In particular I am looking for an approximation of the type:
$$ f_n(x) \leq Y_n(x) \leq g_n(x) $$
where $f_n$ and $g_n$ are some functions related to the normal distribution function. 

The Berry-Esseen theorem gives bounds, see here or here.
In essence, as you have p =1/2, the absolute error between the two distribution values is less than  $C/\sqrt N$ with $C < 0.4748$. (Best value as of the year 2012)
