I've heard Neil deGrasse Tyson giving a short list of the most common elements in the universe on a video, which went on something like:
As I understand astronomers use spectroscopy to determine an object's chemical composition. My question is how can they determine the most common elements for certain, when most of what we see is a just a fraction of the stars, nebulas and an even smaller fraction of planets out there?
Neil de Grasse Tyson is extrapolating.
Chemical abundances can and have been measured in a huge numbers of stars in our own Galaxy and considerably fewer in local galaxies. The chemical abundances of glowing gas clouds can also be measured and this technique has a much greater reach. In addition, there are ways in which the summed light from all the stars in a galaxy can be used to give crude information about their average chemistry, and again, this can be used out to large distances.
From these measurements we have a pretty good idea of the chemistry of our local part of the universe. One can then construct an inventory to make a table like the one in your question. This is dominated by stars and gas - planets cannot be measured but are a negligible fraction of mass.
Now there is a complication. The chemistry of the universe changes with time, because hydrogen and helium are gradually being turned into heavier elements inside stars, and then much of the products are distributed into the interstellar and intergalactic medium when stars die. Thus to get an "up to date" inventory one should exclude the older stars and perhaps focus more on the gas, which gives an idea of current chemistry in the interstellar medium.
Having done all this, you get the table in your question - which applies to the local universe.
It is then a fundamental assumption in cosmology that the universe is homogeneous on large scales. There is thus no reason or evidence to suppose that things are different elsewhere. Indeed, given that we now pretty much understand why the chemical league table looks the way it is - a simple consequence of the physics of star formation, nuclear fusion and mass loss in stars - then it is difficult to imagine any scenario in which it could be very different anywhere else.
They have theoretically and experimentally determined the possible nuclear reactions between the elements. Then they calculated how fast these reactions occur. Then they calculated the abundances for elements right from the beginning: at the beginning there were only hydrogen nucleus (only protons). Then they calculated it onwards using the calculated reactions. And here we have it! A table of the abundances of the elements.
