<P> This formula is valid only if the eight values with which we began form the complete population . If the values instead were a random sample drawn from some large parent population (for example, they were 8 marks randomly and independently chosen from a class of 2 million), then one often divides by 7 (which is n − 1) instead of 8 (which is n) in the denominator of the last formula . In that case the result of the original formula would be called the sample standard deviation . Dividing by n − 1 rather than by n gives an unbiased estimate of the variance of the larger parent population . This is known as Bessel's correction . </P> <P> If the population of interest is approximately normally distributed, the standard deviation provides information on the proportion of observations above or below certain values . For example, the average height for adult men in the United States is about 70 inches (177.8 cm), with a standard deviation of around 3 inches (7.62 cm). This means that most men (about 68%, assuming a normal distribution) have a height within 3 inches (7.62 cm) of the mean (67--73 inches (170.18--185.42 cm))--one standard deviation--and almost all men (about 95%) have a height within 6 inches (15.24 cm) of the mean (64--76 inches (162.56--193.04 cm))--two standard deviations . If the standard deviation were zero, then all men would be exactly 70 inches (177.8 cm) tall . If the standard deviation were 20 inches (50.8 cm), then men would have much more variable heights, with a typical range of about 50--90 inches (127--228.6 cm). Three standard deviations account for 99.7% of the sample population being studied, assuming the distribution is normal (bell - shaped). (See the 68 - 95 - 99.7 rule, or the empirical rule, for more information .) </P> <P> Let X be a random variable with mean value μ: </P> <Dl> <Dd> E ⁡ (X) = μ . (\ displaystyle \ operatorname (E) (X) = \ mu . \, \!) </Dd> </Dl>

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