Monte Carlo method
When you have a curve with data, and apply the Monte Carlo method, with which you obtain a 68% confidence region, can youthen interprete that as the area in which there is a maximal difference of 2 times the standard deviation? Or is something else meant by this? There are no further indications that the data that is measured is distributed with a normal distribution. An example is the gray band at the acceleration spline in this figure.
Thanks in advance,
Re: Monte Carlo method
A distribution is a distribution regardless of whether it was calculated by Monte-Carlo methods, obtained from an empirical sample. or assumed from some analytic distribution.
For distributions like the Normal distribution, the intervals are symmetric but it doesn't always have to be this way. You can for example a chi-square distribution which is skewed and your region (confidence interval) will not have the mean smack bang in the middle but closer to the lower value.
Basically the idea is that for some given alpha value, the chance of getting the true value in a particular interval is 100(1-alpha)%. So if alpha = 0.05 then the chance of the true parameter lying in the interval that was generated is 1-alpha where alpha lies in [0,1] and is typically 0.05.
In other words if we constructed millions of intervals from a sample with alpha 0.05 then we would expect that 95% of the time the true value would be in that interval.
This idea is the basis of fundamental statistical theory.