You can find lots of FAQs about sortitional selection on the Common Lot website. Here’s one you’ll find this most recent entry (below) at https://thecommonlot.com/node/56
Question: What is the statistical probability that sortition (random selection) will result in an accurate proportional representation of the population?
For instance, what is the likelihood that sortitionally selecting 500 representatives from the U.S.’s 200 million citizens will result in the exact proportion of men and women as determined by the census? That would be 254 women and 246 men.
Can one speak of a ‘margin of error’ in this calculation?
Answer: (thanks to Yoram Gat, statistician)
Margin of error is not exactly the right term. ‘Margin of error’ is used when using a proportion in a sample to estimate the proportion in the population. In our case the proportion in the population is known (50.8% women; 49.2% men), and we wish to bound the proportion in a sample. I would use something like “random fluctuation”.
To simplify, let us say the proportion is exactly 50-50. So in the case of a sample of 500, you will have at least 239–261 about 70% of the time, at least 227–273 about 95% of the time, and at least 216–284 about 99.5% of the time.
The chance of having a split that is worse than 200/300, by the way, is about 1:100,000.
The chance that either there would be more than 350 men or more than 350 women in the group of 500 is less than 0.2 millionth of a millionth of a millionth (2 x 10^-19)
In making these calculations, the size of the population doesn’t matter unless it is tiny – the statements are as true for a city of 100,000 as they are for a country of hundreds of millions. It is only the size of the sample that matters. As a rule of thumb, on each particular issue the sampling error is about 1 / (2 sqrt(n)), where n is the size of the sample.
This means, further, that if any group makes less than 40% of the population, then the chance that it will form a majority in a group of 500 randomly selected people is less than 3 in a million.