An important attribute of the standard deviation as a measure of spread is that if the mean and standard deviation of a normal distribution are known, it is possible to compute the percentile rank associated with any given score. The standard deviation has proven to be an extremely useful measure of spread in part because it is mathematically tractable. Many formulas in inferential statistics use the standard deviation.
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Hidden categories: Articles needing additional references from April All articles needing additional references. Namespaces Article Talk. Views Read Edit View history. Financial time series are known to be non-stationary series, whereas the statistical calculations above, such as standard deviation, apply only to stationary series. To apply the above statistical tools to non-stationary series, the series first must be transformed to a stationary series, enabling use of statistical tools that now have a valid basis from which to work.
To gain some geometric insights and clarification, we will start with a population of three values, x 1 , x 2 , x 3. This is the "main diagonal" going through the origin. If our three given values were all equal, then the standard deviation would be zero and P would lie on L. So it is not unreasonable to assume that the standard deviation is related to the distance of P to L.
That is indeed the case. To move orthogonally from L to the point P , one begins at the point:. An observation is rarely more than a few standard deviations away from the mean. Chebyshev's inequality ensures that, for all distributions for which the standard deviation is defined, the amount of data within a number of standard deviations of the mean is at least as much as given in the following table.
The central limit theorem states that the distribution of an average of many independent, identically distributed random variables tends toward the famous bell-shaped normal distribution with a probability density function of. The standard deviation therefore is simply a scaling variable that adjusts how broad the curve will be, though it also appears in the normalizing constant.
If a data distribution is approximately normal, then the proportion of data values within z standard deviations of the mean is defined by:. The proportion that is less than or equal to a number, x , is given by the cumulative distribution function :. This is known as the 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 x 1 , 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. 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:. This can easily be proven with see basic properties of the variance :. 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. The following two formulas can represent a running repeatedly updated standard deviation.
A set of two power sums s 1 and s 2 are computed over a set of N values of x , denoted as x 1 , Given the results of these running summations, the values N , s 1 , s 2 can be used at any time to compute the current value of the running standard deviation:. Where N, as mentioned above, is the size of the set of values or can also be regarded as s 0. In a computer implementation, as the three s j sums become large, we need to consider round-off error , arithmetic overflow , and arithmetic underflow.
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The method below calculates the running sums method with reduced rounding errors. Applying this method to a time series will result in successive values of standard deviation corresponding to n data points as n grows larger with each new sample, rather than a constant-width sliding window calculation. When the values x i are weighted with unequal weights w i , the power sums s 0 , s 1 , s 2 are each computed as:. And the standard deviation equations remain unchanged. The incremental method with reduced rounding errors can also be applied, with some additional complexity.
The above formulas become equal to the simpler formulas given above if weights are taken as equal to one.
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The term standard deviation was first used  in writing by Karl Pearson  in , following his use of it in lectures. This was as a replacement for earlier alternative names for the same idea: for example, Gauss used mean error. From Wikipedia, the free encyclopedia.
For other uses, see Standard deviation disambiguation. See also: Sample variance. Main article: Unbiased estimation of standard deviation. Further information: Prediction interval and Confidence interval. Main article: Chebyshev's inequality.
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Main article: Standard error of the mean. See also: Algorithms for calculating variance. Mathematics portal. Studies in the History of the Statistical Method. Teaching Statistics. The American Statistician. Retrieved 5 February Retrieved 30 May Retrieved 29 October Fundamentals of Probability 2nd ed. New Jersey: Prentice Hall. Retrieved 30 September The Oxford Dictionary of Statistical Terms.
Oxford University Press. Philosophical Transactions of the Royal Society A. Outline Index. Descriptive statistics. Mean arithmetic geometric harmonic Median Mode.
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