Sample Standard Deviation - Rapid Calculation Methods

Rapid Calculation Methods

It has been suggested that this article be merged into Algorithms for calculating variance. (Discuss)

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:

\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:

Population variance:

Read more about this topic:  Sample Standard Deviation

Famous quotes containing the words rapid, calculation and/or methods:

    “The art of watching has become mere skill at rapid apperception and understanding of continuously changing visual images. The younger generation has acquired this cinematic perception to an amazing degree.”
    —Johan Huizinga (1872–1945)

    “Common sense is the measure of the possible; it is composed of experience and prevision; it is calculation appled to life.”
    —Henri-Frédéric Amiel (1821–1881)

    “The methods by which a trade union can alone act, are necessarily destructive; its organization is necessarily tyrannical.”
    —Henry George (1839–1897)