Question: 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):

[![enter image description here](https://i.stack.imgur.com/jzvzh.png)](https://i.stack.imgur.com/jzvzh.png)

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

[![enter image description here](https://i.stack.imgur.com/j46SZ.png)](https://i.stack.imgur.com/j46SZ.png)

This makes sense, since by the [Berry-Essen theorem](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_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.

Answer: 
The Berry-Esseen theorem gives bounds, see [here](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem) or [here](https://www.johndcook.com/blog/normal_approx_to_binomial/).

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)
*Only a note.* 

The question can be considered as a problem of interpolation.

E.g. one can define $\enspace\displaystyle Y\_n(x):=\frac{1}{2^n}\sum\limits\_{v=0}^\infty {\binom{xn+1}{v+1}}\sum\limits\_{j=0}^v (-1)^{v-j}{\binom{v}{j}}{\binom{n}{j}}$ 

which includes your definition *without* $\,x=0\,$ for $\,n<\infty\,$.

You can check for $\,0<x\leq 1\,$ if $\,n\in \{25;50;75;100\}\,$ and compare the results. 

Does it help you to find an approximation ?