It depends ultimately on the underlying distribution, but for most we can assume an approximately normal distribution. For a normal distribution, the confidence intervals are approximately 68.27% for and 95.45 for . As the mean and standard deviation , this gives the range 40 to 60 as , so we should expect about in this range. Similarly, 30 to 70 is , so we expect about within that range. These formulae are ultimately derived from numerical integration of the probability density function of the standard normal distribution (normal distribution with mean and variance ), which is

.

Thus the fraction of the population within n standard deviations of the mean is approximated by

--Kevin C.