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it’s even weirder than this - theoretically the accuracy of a poll depends only on the size of the sample in absolute terms, *not* relative. a 1000-person poll has the same predictive power no matter how large the population it’s drawn from is!

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Tim Urban

@waitbutwhy

· May 2

I used to wonder how polls that surveyed less than a percent of a population could claim to represent the views of the whole population. Running twitter polls has helped me understand. It’s amazing how well the first 300 votes predict the results after 20,000 votes.

11:39 PM · May 2, 2022Twitter for iPhone

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QC (in Berlin 7/29-???)

@QiaochuYuan

May 2

Replying to

@QiaochuYuan

what makes it all work theoretically is the assumption that you’re sampling uniformly randomly which is a very strong assumption that lets you make correspondingly very strong conclusions

QC (in Berlin 7/29-???)

@QiaochuYuan

May 2

the simplest toy model is to think about drawing, say, red or blue balls uniformly at random out of a very large bag, and computing what you can infer about the true proportion of red vs. blue balls by drawing N balls. good bayes theorem exercise

QC (in Berlin 7/29-???)

@QiaochuYuan

May 2

twitter polls obviously can’t claim to be doing this but as a poll percolates through RTs, likes, etc. you can think of its respondents as coming from a sort of random walk on the twitter network, and there are theoretical results about how well these sorts of walks “mix”

Replying to

This is not true. Let population and sample sizes be N and n. Under sampling w/o replacement, with n fixed, the variance of the sample mean increases as N grows. Variance is (p * (1-p) / n) * (1 - (n-1 / N-1)). Approaches variance of estimator w/ replacement when N gets large.

unitroot

@unitroot1

May 3

Easy to see when you consider the n=N case. Then sample mean has zero variance (i.e. "infinite" precision - it's a dirac delta at the population mean).