95 percent are within two standard deviations (μ ± 2σ), and about 99.7 percent lie within three standard deviations (μ ± 3σ). This is known as the 68–95–99.7 rule, or the empirical rule. For various values of z, the percentage of values expected to lie in and outside the symmetric interval, CI = (−zσ, zσ), are as follows: Relationship between standard deviation and mean The mean and the standard deviation of a set of data are descriptive statistics usually reported together. In a certain sense, the standard deviation is a "natural" measure of statistical dispersion if the center of the data is measured about the mean. This is because the standard deviation from the mean is smaller than from any other point. The precise statement is the following: suppose x1, ..., xn are real numbers and define the function: Using calculus or by completing the square, it is possible to show that σ(r) has a unique minimum at the mean: Variability can also be measured by the coefficient of variation, which is the ratio of the standard deviation to the mean. It is a dimensionless number. Standard deviation of the mean Often, we want some information about the precision of the mean we obtained. We can obtain this by determining the standard deviation of the sampled mean. Assuming statistical independence of the values in the sample, the standard deviation of the mean is related to the standard deviation of the distribution by: where N is the number of observations in the sample used to estimate the mean. This can easily be proven with (see basic properties of the variance): (Statistical independence is assumed.) hence Resulting in: In order to estimate the standard deviation of the mean it is necessary to know the standard deviation of the entire population beforehand. However, in most applications this parameter is unknown. For example, if a series of 10 measurements of a previously unknown quantity is performed in a laboratory, it is possible to calculate the resulting sample mean and sample standard deviation, but it is impossible to calculate the standard deviation of the mean. However, one can estimate the standard deviation of the entire population from the sample, and thus obtain an estimate for the standard error of the mean. Rapid calculation methods The following two formulas can represent a running (repeatedly updated) standard deviation. A set of two power sums s1 and s2