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 wellstudied and wellunderstood, function.
However, consistent with widespread inconsistent and ambiguous terminology, the square root of the biascorrected 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 rootmeansquare 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 socalled 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 