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

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.