Standard Deviation

In statistics, standard deviation is a simple measure of the variability or dispersion of a data set. A low standard deviation indicates that all of the data points are very close to the same value (the mean), while high standard deviation indicates that the data are “spread out” over a large range of values.
For example, the average height for adult men in the United States is about 70 inches, with a standard deviation of around 3 inches. This means that most men (about 68%, assuming a normal distribution) have a height within 3 inches of the mean (67 inches – 73 inches), while almost all men (about 95%) have a height within 6 inches of the mean (64 inches – 76 inches). If the standard deviation were zero, then all men would be exactly 70 inches high. If the standard deviation were 20 inches, then men would have much more variable heights, with a typical range of about 50 to 90 inches.
In addition to expressing the variability of a population, standard deviation is commonly used to measure confidence in statistical conclusions. For example, the margin of error in polling data is determined by calculating the expected standard deviation in the results if the same poll were to be conducted multiple times. (Typically the reported margin of error is about twice the standard deviation, the radius of a 95% confidence interval.) In science, researchers commonly report the standard deviation of experimental data, and only effects that fall far outside the range of standard deviation are considered statistically significant. Standard deviation is also important in finance, where the standard deviation on the rate of return on an investment is a measure of the risk.
Formulated by Francis Galton in the late 1860s, the standard deviation remains the most common measure of statistical dispersion. A useful property of standard deviation is that, unlike variance, it is expressed in the same units as the data. When only a sample of data from a population is available, the population standard deviation can be estimated by a modified standard deviation of the sample, explained below.