Is it possible to calculate probability density function from a data set? Is it possible to calculate probability density function from a data set of values? I assume this should be some kind of a function fitting exercise. 
 A: No, not quite. It's easy to come up with infinitely many density functions that COULD have lead to a given finite set of observations; it's even possible to come up with infinitely many that make the outcome you observed 'likely'.  So, without making more assumptions, you're pretty much stuck.
That said, we can make some assumptions, and use those assumptions to come up with a potential PDF that seems reasonable.
One common method for approximating PDFs in this fashion is Kernel Density Estimation (KDE).  The idea is that you choose a "bandwidth" $h>0$, and a "Kernel" function $K$ such that (1) $K(x)\geq 0$ for all $x$, and (2) the area under $K$ is $1$, and (3) the mean of $K$ is $0$; then, if your data points are $x_1,\ldots,x_n\in\mathbb{R}$, you define
$$
\hat{f}(x):=\sum_{i=1}^{n}\frac{1}{nh}K\left(\frac{x-x_i}{h}\right).
$$
It is pretty straight-forward to check that this $\hat{f}$ is a density function.
Why is this a reasonable thing to do?  The intuition is that you generally assume $K$ is the density of a random variable that's centered at $0$; then this resulting density function is going to have "high" density at exactly the points that you chose, but also assign some density to the points NEAR those observed data points.  The bandwidth $h$ controls how tightly packed the density is around the observed points; if $h$ is really small, then the density will be really tightly packed around the observed points, whereas large $h$ will spread it out more.
It is very common to use the density of the standard normal as the kernel function.
A: The simplest way is to make a histogram of the data and then normalize it so it has area one. A more sophisticated way is to use a kernel density estimator.
