Standard Deviation - Rapid Calculation Methods

Rapid Calculation Methods

See also: Algorithms for calculating variance

The following two formulas can represent a running (continuous) standard deviation. A set of three power sums s0, s1, s2 are each computed over a set of N values of x, denoted as x1, ..., xN:

Note that s0 raises x to the zero power, and since x0 is always 1, s0 evaluates to N.

Given the results of these three running summations, the values s0, s1, s2 can be used at any time to compute the current value of the running standard deviation:

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 = 0, ..., n:

\begin{align} A_0 &= 0\\ A_k &= A_{k-1}+\frac{x_k-A_{k-1}}{k} \end{align}

where A is the mean value.

\begin{align} Q_0 &= 0\\ Q_k &= Q_{k-1}+\frac{k-1}{k} (x_k-A_{k-1})^2 = Q_{k-1}+ (x_k-A_{k-1})(x_k-A_k) \end{align}

Sample variance:

Standard variance:

Read more about this topic:  Standard Deviation

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