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( Standard deviation)
In probability and statistics, the standard deviation is a measure of the dispersion of a collection of numbers. It can apply to a probability distribution, a random variable, a population or a data set. The standard deviation is usually denoted with the letter s (lowercase sigma). It is defined as the root-mean-square (RMS) deviation of the values from their mean, or as the square root of the variance. Formulated by Galton in the late 1860s,[1] the standard deviation remains the most common measure of statistical dispersion, measuring how widely spread the values in a data set are. If many data points are close to the mean, then the standard deviation is small; if many data points are far from the mean, then the standard deviation is large. If all data values are equal, then the standard deviation is zero. 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. The standard deviation of a (univariate) probability distribution is the same as that of a random variable having that distribution. The standard deviation s of a real-valued random variable X is defined as
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