I'm sorry to ask such a basic maths question here, but the reason I'm asking is that just maybe academics apply a different process for averaging out statistics that I don't know about. I need to rely upon the averages in the percent tables on this page of government statistics and none of them seem right to me.
The far right percentage table totals 607/44 = 13.79%. And yet the table states 15%. The other percentage tables are similarly wrong. Or is it just me that is wrong?
The answer to the lede question is "No, the statistics are not wrong."
Generally speaking, one cannot simply "average the averages" or average the percentages, as has been done in the question. As an extreme example, suppose that you have two bags holding marbles (probabalists and statisticians like marbles, right?): the first bag has $10000$ marbles, $1$ of which is blue, and $9999$ of which are red; the other bag has one blue marble.
What percentage of the marbles are blue?
If the percentages are averaged, the result is \begin{align} \frac{\frac{\text{blue marbles in the first bag}}{\text{total marbles in the first bag}} + \frac{\text{blue marbles in the second bag}}{\text{total marbles in the second bag}}}{\text{total number of bags}} &= \frac{\frac{1}{10000} + \frac{1}{1}}{2} \\ %&= \frac{10001}{2\cdot 10000} \\ &= 0.50005 \\ &\approx 50\%. \end{align} This does not give the percentage of marbles which are blue. That said, this computation does have some meaning: it gives the chance of drawing a blue marble in an experiment where (1) a bag is chosen at random, and then (2) a marble is drawn from the chosen bag at random. This kind of computation can be useful in some contexts, but it seems unlikely to be meaningful in the dataset presented in the question.
If the goal is to understand the percentage of marbles which are blue, it is necessary to consider the entire population of marbles as a whole. Hence the percentage of marbles which are blue is the total number of blue marbles divided by the total number of marbles, i.e. \begin{align} \text{percent of marbles which are blue} &= \frac{\text{total number of blue marbles}}{\text{total number of marbles}} \\ &= \frac{2}{10001} \\ &= 0.00019998 \\ &< 1\%. \end{align} This computation gives the chance of drawing a blue marble in an experiment where all of the marbles are mixed into one bag, then a marble is drawn at random.
The second computation is much more in line with what is generally expected for geographical data: information is given for each of many smaller regions or divisions, and totals are given to describe the larger geographical area. The total is supposed to answer the question "If a random person is chosen from this region, what is the probability that they have characteristic [x]?"