How is Google+ population estimated? I keep reading that people are estimating the Google+ population based on statistical estimates:

My model is simple. I start with US Census Bureau data about surname popularity in the U.S., and compare it to the number of Google+ users with each surname. I split the U.S. users from the non-U.S. users. By using a sample of 100-200 surnames, I am able to accurately estimate the total percentage of the U.S. population that has signed up for Google+. Then I use that number and a calculated ratio of U.S. to non-U.S. users to generate my worldwide estimates. My ratio is 1 US user for every 2.12 non-U.S. users. That ratio was calculated on July 4th through a laborious effort, and I haven't updated it since. That is definitely a weakness in my model that I hope to address soon. The ratio will likely change over time.

How is this possible? I don't see how a fixed sample size tells you what percentage of the U.S. population is participating. Let's take 2 cases:


*

*case 1: there are 10,000 Google+ users

*case 2: there are 1,000,000 Google+ users


Why would the samples be statistically different?
 A: This exercise will be pretty useless unless the sample of the surnames is statistically sound, i.e., a random sample with known probabilities of selection. Otherwise, you are estimating the number of female drivers by first picking a color (say yellow), counting the fraction of female drivers in the yellow cars, and then obtaining the estimate of the population total as the (total # of cars) * (fraction of women drivers based on the red cars). If you did not pick up your color at random (and preferably repeated this selection a bunch of times to ensure better coverage of different types of cars), God only knows how good your estimate might be.
Getting a good sample of US surnames is far from a trivial task. The studied distributions of surnames are very odd, to say the least. Most of the surnames will be unique, with just a handful of people having this last name (mine is an example). On the other hand, a few surnames (Smith, Johnson, Williams) may cover as much as 1% of the population).
The problem of weird distributions is frequently encountered in establishment surveys where you have monstrous corporations like Microsoft and Adobe, and millions shops-on-the-corner with two-three local geeks. One practice in working with the distributions like that is to perform probability proportional to size sampling: you take the whole list, but you will sample the surnames (or companies) with greater probabilities if they represent a greater share of the total. You then control for unequal probabilities of selection with weights. Another approach is to use cut-off sampling: you sample all the surnames with frequency greater than (businesses with sales greater than) 0.1% of the total, and then wave hands about the potential non-sampling error for the remaining surnames.
A: Two assumptions are made: (1) the rate of US citizens to all people is the same within the Google+ population as in the global population, and (2) for US citizens, the rate of people with any surname to all US citizens is (on average) the same within the Google+ population as in the global population.
So: you take, say, 200 surnames, and count how many US Google+ subscribers there are with these surnames ($USG_s$). Given assumption (2) (say the rate is $r_s$, found by dividing the number of US citizens with these surnames by the total number of US citizens), an estimate for the total number of US subscribers is found like this:
$USG\sim USG_s/r_s$
Then, using assumption (1) you can use the same 'trick' to find an estimate of the total number of Google+ users.
Simply put: if there are less people that are Google+ subscribers, there will be less US citizens that are Google+ subscribers (assumption (1)). By this and assumption (2), there will also be less US citizens with a given surname that are Google subscribers.
