The answer provided by Jason R in the comments --
"dB values" are really just a way of representing values on a logarithmic scale. Arithmetic-averaging values on a logarithmic scale is equivalent to geometrically averaging the original values (in linear scale). This probably isn't what you want. You probably would like a measure of total power in each band, which you would get by just summing the power spectrum bins across each band. You could divide by the number of bins, but if the bands aren't equal-width, that will add some bias to the result.
does a good job of identifying the problem but I'm not sure it is a correct solution (or maybe I'm just reading it wrong). If a summation is to occur, it should occur on the raw magnitudes of the bins (before conversion to dB).
The issue is this
3dB + 3dB != 6dB
to do a summation on the decibel scale is more complex than simply adding the values together. It can be done but if so you should do some reading on the dB scale and the nature of adding relative power. Doing the addition when still in a linear domain is easiest.