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# Standard Deviation

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

 (1) (2)

where is the mean, is the second raw moment, and denotes the expectation value of . The variance is therefore equal to the second central moment (i.e., moment about the mean),

 (3)

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

 (4)

The sample standard deviation distribution is 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,

 (5)

The standard deviation 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 , and

 (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).

 range CI 0.6826895 0.9544997 0.9973002 0.9999366 0.9999994

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

 (7)
 CI range 0.800 0.900 0.950 0.990 0.995 0.999

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

## References

Kenney, J. F. and Keeping, E. S. "The Standard Deviation" and "Calculation of the Standard Deviation." §6.5-6.6 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 77-80, 1962.

## Referenced on Wolfram|Alpha

Standard Deviation

## Cite this as:

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