How is it done? I read articles saying the “world’s average temperature this year is ‘x’ degrees, ‘y’ degrees higher than 10/50/100/1000 years ago”.
Do they take a small number of points with reliable temperature records over the years, or do they take a large number of points - many with unreliable (or no) temperature records over the years, and make estimates? In any case, there are bound to be places on Earth with no records (e.g. parts of Siberia or the Amazon).
Are measurements made at ground level? What about oceans (and other bodies of water) and mountains?
There is no single “world’s average temperature”. What there is is a number of scientific models each of which use what those scientists consider the best possible set of measurements. The models use different approaches to deal with varying availability of good data. The ones that produce a global mean, rather than specifically a global surface mean, ocean mean, atmospheric mean, use both land and ocean temperatures.
Preferable any comparison should use points from the same model, but there are overarching analyses joining up models from different time periods with different data availability and scientifically justifying why the can be used for comparison.
Beyond this you will have to read an actual paper on creating such a model. I suggest Berkeley Earth, as they have very good visualizations and were started as a response to criticism of the validity and bias in the previous century’s temperature estimates (the Koch brothers were major funders initially).
And if you can’t make heads or tails of that, here is what they conclude: