On the Catalan referendum turnout

Yes-Yes: 1,861,753
Yes-No: 232,182
Yes-?: 22,466
No: 104,772
Blank: 12,986
Others: 71,131
Total votes cast overall: 2,305,290
Registered voters: 5,413,868
Total turnout: 2,305,290/5,413,868 = 42.583%

Compare with the voter turnout in the 2012 Catalonian election, which had a voter turnout of:
3,635,170/5,413,868 = 67.146%

Now, referenda definitively aren't exclusive, and the voices of those who merely chose to make themselves represented are taken to be a proportional representation of the population as a whole (as determining the opinions of those who don't make themselves heard are impossible to gauge, and calling it illegitimate on the basis of those who chose not to vote undermines the entire electoral system), barring an electoral/referendum boycott - which there wasn't. Low turnout numbers may be explained by the referendum being a non-binding referendum.

However, for the sake of argument, let's project the results needed among those who, in actuality, didn't vote. I'll take the population as a whole and the voter turnout for Spain as a whole (2011 general election, 68.940%).

If we take every single person that could have voted - there were 3,108,578 people who could have voted, but didn't. If we remove the invalid votes and the votes that aren't complete, we can extrapolate a valid/explicitness rate of 95.377%, meaning that among the 3,108,578 that we're taking into account, 2,964,856 would have cast valid and explicitly yes or no votes. For "No" and "Yes-No" combined to win, it would require an extra 1,524,800 votes in either response, requiring 51.429% of the non-cast votes. For just "No" to win, it would require an extra 1,756,982 votes, requiring 59.260% of the non-cast votes.

This alone implies quite a sizeable and unlikely gap between the results of those who voted and those who didn't required, taking in the virtually impossible situation of a 100% voter turnout as input. If we examine a more likely scenario, the referendum having the same voter turnout as the Spanish general election in 2011, 3,732,330 of those 5,413,868 would have voted providing us with an additional 1,427,070 - 1,361,091 of which would be valid and explicit. This more realistic projection renders both "No", and "No" and "Yes-No" combined impossible - the latter requiring 1,524,800 additional votes (112.028% of the remaining vote) and the former requiring 1,756,982 additional votes. (129.086% of the remaining vote)

For a combined "Yes-No" and "No" result to even be able to scrape through with a total of 1,861,754 votes between them, the voter turnout would have to be at least (taking into account invalid and inexplicit votes) 3,900,588, or a 72.048% turnout. For a "No" result to be able to scrape through with 1,861,754 votes, the voter turnout would have to be at least 4,143,504, or 76.535%.

While the abnormally low voter turnout may have raised questions of legitimacy in a referendum were the results closer, the absolute overwhelming victory for "Yes-Yes" means that, even if additional votes were added, all for opposing options with no additional votes for "Yes-Yes", it would still win even under abnormally high voter turnouts.

However, votes of any kind, be they referenda, election, etc. rely on the desire of all covered people being projected using the votes cast by those that decided to cast votes. While the opinions of those that choose not to make their voices heard can't be taken into consideration (for obvious reasons), they can be presumed to roughly correspond proportionally to the opinions of those that did choose to make their voices heard. Not only is it unlikely that the public of Catalonia as a whole would have such a massively diverging opinion from the voting public to delegitimise the referendum, but the sway could go either way. For all we know, 100% of those that didn't vote in the referendum could have supported a "Yes" position.



On a side note, I love working with statistics. It gives me the same kind of pleasure as filling out a Sudoku or Futoshiki grid, or doing a crossword :D