Random sampling and the central limit theorem

Let's try to understand these two important statistical concepts using an example. Suppose one wants to find the average age of one state of India, lets say Tamil Nadu. Now, the safest and brute-force way of doing this will be to gather age information from each citizen of Tamil Nadu and calculate the average for all these ages. But, going to each citizen and asking their age or asking them to tell their age by some method will take a lot of infrastructure and time. It is such a humongous task that census, which attempts to do just that, happens once a decade and what will happen if you decided to do so in a non-census year?

The statisticians face such issues all the time. The answer lies in random sampling. Random sampling means that you take a group of 1000 individuals (or 10000, depending on your capacity, obviously the more the merrier) and calculate the average for this group. You call this A1. Getting to this is easier as 1000 or 10000 is...