Quote Originally Posted by wufwugy View Post
Statistics question: let's say a poll gives somebody a 48% approval rating with a 3 point margin of error. Does this mean that the actual result is anywhere between 51% and 45% at the same probability per each value?
There's a likelihood function that is symmetrical and normal-distribution-shaped for normal samples. So if the sampling data are normally distributed, the likelihood is highest around the reported value (48% in this case) and tapers off above and below it in a normal distribution shape. So, e.g., 48% is the most likely value of the approval rating, 47% and 49% are less likely but equal to each other, and so on as you fall further from the mean.

It's not quite appropriate to apply that to percentage data though, and the likelihood distribution should be a bit asymmetrical, and a bit log-linearish as well, but with values close to 50% the difference between the actual shape of the likelihood distribution and one that is symmetrical and normal ought to be negligible. Certainly values closer to the 48% will be more likely than values further from 48% regardless.

The 95% confidence interval used as the standard 'margin of error' is a bit non-intuitive because it relates to sampling, not population means. It implies that 95% of the times you sampled the data, you would get a value of the mean +/- the margin of error. Doesn't necessarily follow that the true value is in there 95% of the time.