Suppose I have a batch of 11 students with ages distributed as the following:
(Age, Count) or $(\alpha,x_\alpha)$= $(10,4), (11,3), (12,1),(9,3)$. So, $4+3+1+3 = 11$
Now, if I am asked what is the average age of the batch, will I say it is $$\frac{10+11+12+9}{4}= 10.5 = \frac{\sum\limits_{k=1}^N \alpha_k}{N} $$ or $$\frac{10\times4+11\times3+12\times1+9\times3}{4+3+1+3}= 10.18= \frac{\sum\limits_{k=1}^N (\alpha_k \times x_{\alpha_k})}{\sum\limits_{k=1}^N x_{\alpha_k}} $$
My intuition says it should be the second one but I think I'm confusing the definition of a mean. Please tell me where am I going wrong and what are some good sources to learn statistics from. Thanks!
You aren't adding up all the ages in the first one, since you aren't taking the counts into account. The mean age is, indeed, the sum of all the ages, divided by the number of data points. In this case, it would be $$\frac{10+10+10+10+11+11+11+12+9+9+9}{11},$$ which can be rewritten in your second form.
I would personally leave the answer as $10\frac{2}{11}$ or $10.\overline{18},$ but that's a matter of preference.