How to quantify asymmetry of a signal?
Assuming you're looking for symmetry around $x=0$, you can decompose any function into a symmetric and an antisymmetric part:

$$ f(x) = \frac{1}{2}\left(f(x)+f(-x)\right) + \frac{1}{2}\left( f(x) - f(-x) \right)$$

Calling $f_+(x):=\frac{1}{2}\left( f(x) + f(-x) \right)$ and $f_-(x):=\frac{1}{2}\left( f(x) - f(-x) \right)$ you can easily verify that $f_+$ is symmetric while $f_-$ is antisymmetric.

Now you can measure symmetry by taking any suitable norm on your functions. For example the symmetry of $f(x)$ can be measured as 

$$S[f]:=\frac{||f_+||}{||f_+||+||f_-||}$$ 

which ranges from $0$ (fully antisymmetric) to $1$ (fully symmetric). 

A possible useful norm on functions is defined as 

$$||f|| = \sqrt{\int_\mathbb{R} f(x)^2 dx} $$

which can be discretised to 

$$||f|| = \sqrt{ \sum_i f(x_i)^2 \frac{x(i+1)-x(i-1)}{2} }$$