the mean (67–73 inches)one standard deviationand almost all men (about 95%) have a height within 6 inches of the mean (64–76 inches)two standard deviations. If the standard deviation were zero, then all men would be exactly 70 inches tall. If the standard deviation were 20 inches, then men would have much more variable heights, with a typical range of about 50–90 inches. Three standard deviations account for 99.73% of the sample population being studied, assuming the distribution is normal or bell-shaped (see the 68–95–99.7 rule, or the empirical rule, for more information). Definition of population values Let μ be the expected value (the average) of random variable X with density f(x): The standard deviation σ of X is defined as which can be shown to equal Using words, the standard deviation is the square root of the variance of X. The standard deviation of a probability distribution is the same as that of a random variable having that distribution. Not all random variables have a standard deviation. If the distribution has fat tails going out to infinity, the standard deviation might not exist, because the integral might not converge. The normal distribution has tails going out to infinity, but its mean and standard deviation do exist, because the tails diminish quickly enough. The Pareto distribution with parameter has a mean, but not a standard deviation (loosely speaking, the standard deviation is infinite). The Cauchy distribution has neither a mean nor a standard deviation. Discrete random variable In the case where X takes random values from a finite data set x1, x2, …, xN, with each value having the same probability, the standard deviation is or, by using summation notation, If, instead of having equal probabilities, the values have different probabilities, let x1 have probability p1, x2 have probability p2, …, xN have probability pN. In this case, the standard deviation will be Continuous random variable The standard deviation of a continuous real-valued random variable X with probability density function p(x) is and where the integrals are definite integrals taken for x ranging over the set of possible values of the random variable X. In the case of a parametric family of distributions, the standard deviation can be expressed in terms of the parameters. For example, in the case of the log-normal distribution with parameters μ and σ2, the standard deviation is Estimation One can find the standard deviation of an