Standard Deviation - Identities and Mathematical Properties

Identities and Mathematical Properties

The standard deviation is invariant under changes in location, and scales directly with the scale of the random variable. Thus, for a constant c and random variables X and Y:

The standard deviation of the sum of two random variables can be related to their individual standard deviations and the covariance between them:

where and stand for variance and covariance, respectively.

The calculation of the sum of squared deviations can be related to moments calculated directly from the data. The standard deviation of the sample can be computed as:

\operatorname{stdev}(X) = \sqrt{E} = \sqrt{E - (E)^2}.

The sample standard deviation can be computed as:

\operatorname{stdev}(X) = \sqrt{\frac{N}{N-1}} \sqrt{E}.

For a finite population with equal probabilities at all points, we have

\sqrt{\frac{1}{N}\sum_{i=1}^N(x_i-\overline{x})^2} = \sqrt{\frac{1}{N} \left(\sum_{i=1}^N x_i^2\right) - \overline{x}^2} = \sqrt{\frac{1}{N} \sum_{i=1}^N x_i^2 - \left(\frac{1}{N} \sum_{i=1}^{N} x_i\right)^2}.

This means that the standard deviation is equal to the square root of (the average of the squares less the square of the average). See computational formula for the variance for proof, and for an analogous result for the sample standard deviation.

Read more about this topic:  Standard Deviation

Famous quotes containing the words mathematical and/or properties:

    “The most distinct and beautiful statement of any truth must take at last the mathematical form.”
    —Henry David Thoreau (1817–1862)

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)