The tendency is strong that we consider our own personal experience to be representative of what has happened to everyone else. When we hear an explanation for events, we immediately fit it against our own lives, or children, or jobs. If twenty people hear that teenage converts to Christianity tend to be Evangelical at first and then move to mainline churches, nineteen of them will immediately confirm or deny the truth of the statistic on the basis of what happened to them, or their children, or whoever the nearest teenage convert is. I used to think this was more common among women, but I am now convinced this is not so. I even begin to suspect the opposite is true.
"My friend's cousin got the vaccine, and two weeks later he got bitten by a moose. Do the research, people."
I don't quite know what to say to such folks. If you don't already know that the incident that happened near your granddaughter's soccer game is not necessarily representative of what happens in general, I don't think I have a ready explanation to convince you. I readily see why people start there. Sometimes one can actually develop a refutation by noticing that one incident, at least, sharply contradicts that. Whatever else is true about the hairdressers in America, you might be able to assure the assembled crowd "Not always. My hairdresser doesn't own any small dogs at all."
I think we apply this unscientific experiment with N=1 more often when we do not understand the underlying mechanism or have little grasp of the data. It is the opposite of the Bayesian approach, where we first ask how many highschool students there are, then what percentage of highschoool students are basketball players, and what percentage of students in general are taking chemistry this year to begin with before we try and figure out how many basketball players are taking chemistry this year. Looking at whether your kid is a small forward in chem lab is useless. Even estimating from a few years' teams and school enrollments would be more useful.