Since surface temperature changes are correlated over distances of about 1000 km (it does depend somewhat on the latitude of the stations), it turns that you only need about 60 stations to produce a reasonable surface temperature dataset. [Edit: As Andrew Dessler points out in this comment, this is true for temperature anomalies, but not for absolute temperatures.]
I realise that Nick Stokes has covered this a number of times before. However, it’s probably worth repeating. Also, the main reason I wrote this is because I came across a site that allows you to experiment with this yourself. I have to admit that someone else highlighted this on Twitter, and I can’t remember who it was. If I remember (or someone reminds me) I’ll give credit [Edit: Someone has reminded me. Credit to Marcus N. Hofer]. I’m also not sure of the source of the site [Edit. It’s from Kevin Cowtan and it’s highlighted in this Skeptical Science post.].
I quickly produced the figure above. I used the GHCN adjusted plus ocean data. I initially used all the stations, then 1/10 of the stations, then 1/25, and then 1/80 (only 65 stations). The time-series look very similar (as expected). The mean trend, however, does vary slightly, but the uncertainty (not shown – see Nick Stokes’ posts for a discussion of the uncertainty) also increases. The reason, I think, that the mean trend increases slightly as the number of stations goes down, is that land stations start to dominate more and more over ocean stations, and the land is warming faster than the global average.
To be clear, I’m not suggesting that there aren’t any potential issues with the global surface temperature datasets (see one of Victor’s posts for some discussion of this). I’m mainly just trying to highlight that the sampling is almost certainly not much of an issue; you don’t need lots and lots of stations to produce a reasonable approximation for how global surface tempertures have changed. I also thought others might like to try some other variations, so wanted to highlight the site that allows you to do so (see link below).
Nice comment from Kevin Cowtan suggesting that a somewhat more careful analysis would suggest that you need maybe 130 stations. Doesn’t really change the key point; you don’t need an enormous number of stations if what you’re wanting to estimate how global surface temperatures are changing (temperature anomalies, rather than absolute temperatures). There’s a video explainer, which I’ve posted below.
Tool for producing global surface temperature datasets.
Spectral Approach to Optimal Estimation of the Global Average Temperature (Shen, North and Kim paper suggesting that you only need about 60 stations to produce a global surface temperature time series).
Global trends of measured surface air temperature (Hansen and Lebedeff paper demonstrating that surface temperatures are correlated on scales of about 1000 km).
Why raw temperatures show too little global warming (Victor Venema’s post).
Just 60 stations? (One of Nick Stokes’ original posts about only needing 60 stations).
Global 60 Stations and coverage uncertainty (Nick Stokes’ post about what happens if you cull down to 60 stations).
Are the CRU data “suspect”? An objective assessment (Realclimate post by Kevin Wood and Eric Steig demonstrating that in fact you probably only need about 30 stations).