That's the problem with the whole vote. Both sides had prominent politicians who were talking rubbish.
I have heard a lot of people raise arguments on both sides, pre and post vote, quoting information they believe to be correct when in reality it is a load of rubbish but people believe it because they heard it somewhere and it stuck.
For example a lot of the leave voters I know in Portsmouth wanted out on the basis that we could stop immigration. They completely failed to recognise that immigration is a good thing when done properly and it's unlikely that the numbers will change, albeit their country of origin might.
Despite being a leaver, I tried to explain this point to many of my family and friends (the Mrs side) but they couldn't grasp it and fell back to their Daily Mail arguments because that is what suits their own agenda (no jobs, can't get a council house, etc etc).