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 |


standard deviation
98.17, 112.3, 102.6, 94.3, 108.1




