Rapid Calculation Methods
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The following two formulas can represent a running (continuous) standard deviation. A set of two power sums s1 and s2 are computed over a set of N values of x, denoted as x1, ..., xN:
Given the results of these three running summations, the values N, s1, s2 can be used at any time to compute the current value of the running standard deviation:
Where :
Similarly for sample standard deviation,
In a computer implementation, as the three sj sums become large, we need to consider round-off error, arithmetic overflow, and arithmetic underflow. The method below calculates the running sums method with reduced rounding errors. This is a "one pass" algorithm for calculating variance of n samples without the need to store prior data during the calculation. 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.
For k = 1, ..., n:
where A is the mean value.
Sample variance:
Population variance:
Read more about this topic: Sample Standard Deviation
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