Mean age of people

You need EmpiricalDistribution:

data = {14, 15, 16, 22, 24, 25};
weights = {1, 1, 3, 2, 2, 5};

\[ScriptCapitalD] = EmpiricalDistribution[weights -> data];

Expectation[x, x \[Distributed] \[ScriptCapitalD]]
(* 21 *)

values = {14, 15, 16, 22, 24, 25};
weights = {1, 1, 3, 2, 2, 5};

You can also use WeightedData:

Mean @ WeightedData[values, weights]

21

or use Mean with EmpiricalDistribution:

Mean @ EmpiricalDistribution[weights -> values]

21

Both methods also work with symbolic input:

values = Array[Subscript[x, #] &, 5];
weights = Array[Subscript[w, #] &, 5];

Mean @ WeightedData[values, weights] 

TeXForm @ %

$$\frac{w_1 x_1+w_2 x_2+w_3 x_3+w_4 x_4+w_5 x_5}{w_1+w_2+w_3+w_4+w_5}$$

Mean @ EmpiricalDistribution[weights -> values] // Together // TeXForm

$$\frac{w_1 x_1+w_2 x_2+w_3 x_3+w_4 x_4+w_5 x_5}{w_1+w_2+w_3+w_4+w_5}$$

These are all quite fancy solutions so far. For beginners' sake, I'd like to add a somewhat more elementary one that uses somewhat more universal tools:

ages = {14, 15, 16, 22, 24, 25};
counts = {1, 1, 3, 2, 2, 5};

ages.counts/Total[counts]

21

The . (a.k.a. Dot) computes this sum

Sum[ages[[i]] counts[[i]], {i, 1, Length[ages]}]

and Total computes this one:

Sum[counts[[i]], {i, 1, Length[ages]}]

So one can also obtain the result as follows:

Sum[ages[[i]] counts[[i]], {i, 1, Length[ages]}]/Sum[counts[[i]], {i, 1, Length[counts]}]

But Dot and Total are typically faster because they rely on very optimized libraries in the background. And of course, it is also shorter to type.