When looking at the average incomes in a country it is necessary to distinguish between mean income and median income. Mean income tends to be distorted by high earners, so typically median income is used. GDP covers more than individual incomes, including services provided by the state, for example. To look at GDP on an individual level, GDP per capita is used, which seems to suffer from the same problems as mean income. Has anyone devised a way of splitting GDP among individuals to give a 'median' GDP figure for a country.
GDP data are not reported on individual level so that is not directly possible (at least not without going through the expense of collecting such data).
However, output is equal to income (see Mankiw's Macroeconomics pp 47-77). Consequently, if you want to know median output just look at gross median income, they are equal in reality since the value of your income is value of your output. Due to statistical discrepancy, slight statistical methodological differences and measurement error they might be some discrepancy between situation where per-person output would be observable, but aside of that they should be equal.
Also, this statement:
GDP covers more than individual incomes, including services provided by the state, for example.
is not correct.
GDP can be measured by various ways such as income or spending approach. GDP is by definition sum of all individual incomes or sum of all spending (as these have to be equal), so for closed economy (I use closed economy to save time, the point holds for open one as well):
where $i$ is all interest income, $\pi$ all income from profits, $w$ all income from wages, and $r$ all income from rent. As you can see this must be equal to all consumption, investment and government spending. So GDP does not measure more than you can get from individual income statistics (at least from the gross one which are the ones often/typically reported).