[BruceZ]

We are dealing with the normal distribution, also known as the gaussian distribution (bell-shaped curve). 68% is the area under the curve between +/- 1 standard deviation. 95% is the area between +/-2 standard deviations. The area between +/- 3 standard deviations is actually 99.7%. 99% exactly corresponds to 2.58 standard deviations. You get these numbers from a table of the standard normal distribution, which is the normal distribution with mean = 0 and standard deviation = 1. The definite integral cannot be done in closed form, you must use a table which is derived numerically. The equation for the standard normal distribution is
sqrt(1/2pi)exp^(-x^2/2). It is only possible to do the indefinite integral from minus infinity to +infinity, which of course is 1 since this is the total probability, because Gauss figured out (I think when he was very young) that the integral of exp^(-x^2/2) from minus infinity to +infinity is sqrt(2pi), which is why we divide by this as a normalization constant to get the area to be 1. The trick to doing the indefinite integral is to relate it to an integral in polar co-ordinantes, and then convert it to a double integral which is separable in x and y.

Note that the general formula for any mean u and standard deviation sigma is
sqrt(1/2pi)exp[-(x-u)^2/(2*sigma)]. This is abbreviated N(u,sigma). The standard normal distribution would be N(0,1). When you use the table of the standard normal distribution, you must be aware of what it is giving you. Mine gives the integral from minus infinity to x. This is most useful for doing 1-sided probabilities such as the probability that you will be at least so far ahead after a certain time. For example, the value for 1 standard deviation is .84. This means there is an 84% chance you will be less than 1 standard deviation above the mean, and a 16% chance you will be above this amount (the area under one tail). To get the 2-sided confidence interval that you are familiar with, subtract 2*16% from 1 to subtract off the 2 tails.