But to say that your results are three standard deviations from the mean, you need to know what the standard deviation for tests like this is. I have no idea what that would be, but my intuition suggests that 13.4% is probably within three of them.

We know exactly what standard deviation is. In fact, we know everything about this distribution (it's what is called a normal distribution) because this is the classic high school binomial distribution problem. But in fact, even if it was not so simple, a lot of other scenarios (almost all that anyone cares about) converge to the same thing. Read up on the central limit theorem if you want to know why.

By the way, here are my results so far: 74 tried, 15 successes.