Firstly, 8±3 doesn't mean "it's definitely between 5 and 11". It just means "it's 68% likely that it's in that range", and then it's 95% likely that it's in the range [2:14] and 99.7% likely that it's in the range [-1:17] (really [0:17] in this case, since it can't be negative).

Secondly, why do I say that it's likely to be lower than 8% rather than higher, given that the error bars go both up and down? That's because we're looking at extreme value statistics. Put simply, we're not looking at a random data point, we're looking at the data point with the highest value. That means we're much more likely to see a positive noise fluctuation than a negative one, because a positive noise fluctuation makes a data point more likely to be the highest one while a negative noise fluctuation does the opposite.

Here's a concrete example. Let's say we have two sets of numbers, set A and set B, each with 1000 numbers in them. In set A, each number has a true value of 5 but ±1 in errors. In set B, each number has a true value of 0, but ±10 in errors. So in reality the numbers in set A are much bigger than in set B. But now look at what happens when we take errors into account. In set A, it's unlikely that we will see any numbers higher than around 8, since a +3 error has a likelihood of only 0.15%. Meanwhile, in set B we're almost guaranteed to see numbers higher than 20. So if we're not careful we will incorrectly conclude that B is really bigger than A.

Sorry, that didn't come out as clearly as I had hoped. If you know some simple programming, then I recommend just writing a 5-line program to test it out yourself.