Let us calculate the probability that a sample from the Gaussian distribution will fall within a single standard deviation of the mean :

Similarly, the probability that the sample is within two standard deviations (within ``'') of the mean is

Hence 68% of the samples will, on average, fall within one , and over 95% of the samples will fall within two of the mean .

The Gaussian distribution will be encountered frequently in this course, not only because it is a fundamental pdf for many physical and mathematical applications, but also because it plays a central role in the estimation of errors with Monte Carlo simulation.