Often how the data was collected is not shown/glossed over, so it is tough to know whether the data actually applies to an argument or not.
I would call such a thing a "statistic" not "statistical evidence". That's why checking the source is important - there's a reason researches output a side long paper and not just a three line summary.
Numbers can be twisted around to any purpose the user desires; this was one of the first things I was taught in an "intro to statistics" class, and I think fellow tropers who have been educated in math can back me on that.
The same as above also applies here.
If only the raw data (maybe after some processing through well known mathematical methods) is used for the argument, and not the interpretation of that raw data, then it's no problem.
(Actual example I had some time ago:)
Wrong: "Only 2% of women make false rape accusations"
Less wrong: "2% of the women who make rape accusations were threatened with charges if they wouldn't withdraw the accusations"
The data doesn't actually support the conclusion that got used as an argument. In fact all accusations could be true, and those women were falsely forced to withdraw.
Of course, when people rely on statistical arguments they normally only refer to the first form, so...
After trying to defend statistical evidence, I'll try to defend anecdotal evidence:
There's a math joke that goes: "During a lecture a math student tells the professor: 'Sir, I got a counter-example for your statement' 'Doesn't matter, I have two proofs!' "
If someone makes a statement that wouldn't allow something to happen but you experienced it, it means that the other one was wrong - plain and simple. He can try to pull argument after argument, but a single counter-example is sufficient to disprove an absolute statement.
In an argument, statistical evidence and personal experience serve imho two different purposes and they don't really conflict. Using personal experience to "disprove" statistical evidence is missing the point just as much as using statistical evidence to "disprove" personal experience.