Is this what you are looking for:
mysql> SELECT country, AVG(lat), AVG(lng) FROM c2014 GROUP BY country;
+---------+---------------------+---------------------+
| country | AVG(lat) | AVG(lng) |
+---------+---------------------+---------------------+
| ad | 42.527902188508406 | 1.537318801102431 |
| ae | 24.891538320635465 | 55.51244805853463 |
| af | 34.04674051673764 | 67.26681304757301 |
| ag | 17.085518998526485 | -61.79143601297681 |
| ai | 18.21587371826172 | -63.039283025832404 |
| al | 41.088975844950475 | 20.014610561816568 |
| am | 40.26937918217537 | 44.84238364011771 |
| an | 13.779007457889145 | -67.10738386983765 |
| ao | -10.394024888552527 | 16.401272589394896 |
| ar | -32.13739248870794 | -63.40704697093584 |
| at | 47.62893757137828 | 14.375785999308414 |
| au | -32.58224675171816 | 144.07049698825728 |
| aw | 12.509566132918648 | -69.99144844386889 |
| az | 40.179647407439965 | 47.49281764155795 |
| ba | 44.13574533529882 | 17.93174898133396 |
| bb | 13.1662623028257 | -59.554101581004126 |
...
| yt | -12.817898351638043 | 45.14693607267786 |
| za | -29.213355151321988 | 27.271365789847575 |
| zm | -13.74802438257624 | 27.937999186534476 |
| zr | -3.150024081275917 | 23.1336162007831 |
| zw | -19.047764062703678 | 29.515317293888238 |
+---------+---------------------+---------------------+
234 rows in set (1.22 sec)
This pseudo-centroid is easy to compute, and it "reasonable" for most countries. But it can be quite far from any sort of "geographic" centroid. Algeria is easy to understand:
The pseudo-centroid for Algeria (country code DZ), for example, (lat: 35.6, lng: 3.24) is about 75 miles (120 km) from the Mediterranian, yet the country is about 1200 mi / 2000 km from north to south. The issue is that most cities are very near the coast, not scattered around the Sahara. So, the pseudo-centroid found from city lat-lngs is skewed far to the north.
A similar problem happens with cities that have annexed a lot of green space. Or counties with all their cites on a river; etc.
I looked for, but failed to find, a country where the pseudo-centroid was clearly in another country. For the curved country of Viet Nam, it almost lands in Laos. Several countries (eg, New Zealand) have it land off the coast; but I decided that does not really count.