Standard Deviation (original) (raw)

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The standard deviation sigma of a probability distribution is defined as the square root of the variance sigma^2,

where mu=x^_=<x> is the mean, mu_2^'=<x^2> is the second raw moment, and <x> denotes the expectation value of x. The variance sigma^2 is therefore equal to the second central moment (i.e., moment about the mean),

sigma^2=mu_2. (3)

The square root of the sample variance of a set of N values is the sample standard deviation

s_N=sqrt(1/Nsum_(i=1)^N(x_i-x^_)^2). (4)

The sample standard deviation distributionis a slightly complicated, though well-studied and well-understood, function.

However, consistent with widespread inconsistent and ambiguous terminology, the square root of the bias-corrected variance is sometimes also known as the standard deviation,

s_(N-1)=sqrt(1/(N-1)sum_(i=1)^N(x_i-x^_)^2). (5)

The standard deviation s_(N-1) of a list of data is implemented as StandardDeviation[_list_].

Physical scientists often use the term root-mean-square as a synonym for standard deviation when they refer to the square root of the mean squared deviation of a quantity from a given baseline.

The standard deviation arises naturally in mathematical statistics through its definition in terms of the second central moment. However, a more natural but much less frequently encountered measure of average deviation from the mean that is used in descriptive statistics is the so-called mean deviation.

Standard deviation can be defined for any distribution with finite first two moments, but it is most common to assume that the underlying distribution is normal. Under this assumption, the variate value producing a confidence interval CI is often denoted x_(CI), and

x_(CI)=sqrt(2)erf^(-1)(CI). (6)

The following table lists the confidence intervals corresponding to the first few multiples of the standard deviation (again assuming the data is normally distributed).

To find the standard deviation range corresponding to a given confidence interval, solve (5) for n, giving

n=sqrt(2)erf^(-1)(CI). (7)

See also

Central Moment, Confidence Interval, Mean, Mean Deviation,Moment, Normal Distribution,Root-Mean-Square, Standard Deviation Distribution, Sample Variance, Sample Variance Distribution, Standard Error, Variance Explore this topic in the MathWorld classroom

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References

Kenney, J. F. and Keeping, E. S. "The Standard Deviation" and "Calculation of the Standard Deviation." ยง6.5-6.6 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 77-80, 1962.

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Standard Deviation

Cite this as:

Weisstein, Eric W. "Standard Deviation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StandardDeviation.html

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