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:

    “More rapid than eagles his coursers they came,
    And he whistled, and shouted, and called them by name:
    “Now, Dasher! now, Dancer! now, Prancer and Vixen!
    On, Comet! on, Cupid! on, Donder and Blitzen!
    To the top of the porch! to the top of the wall!
    Now dash away! dash away! dash away all!””
    —Clement Clarke Moore (1779–1863)

    ““To my thinking” boomed the Professor, begging the question as usual, “the greatest triumph of the human mind was the calculation of Neptune from the observed vagaries of the orbit of Uranus.”
    “And yours,” said the P.B.”
    —Samuel Beckett (1906–1989)

    “There are souls that are incurable and lost to the rest of society. Deprive them of one means of folly, they will invent ten thousand others. They will create subtler, wilder methods, methods that are absolutely DESPERATE. Nature herself is fundamentally antisocial, it is only by a usurpation of powers that the organized body of society opposes the natural inclination of humanity.”
    —Antonin Artaud (1896–1948)