TOPICS
Standard Deviation
The standard deviation
of a probability distribution is defined as thesquare root of thevariance
,
 |  |  | (1) |
 |  |  | (2) |
where
is themean,
is the secondraw moment, and
denotes theexpectation value of
. Thevariance
is therefore equal to the secondcentral moment (i.e., moment about themean),
 | (3) |
The square root of thesample variance of a set of
values is the sample standard deviation
 | (4) |
The samplestandard 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,
 | (5) |
The standard deviation
of a list of data is implemented asStandardDeviation[list].
Physical scientists often use the termroot-mean-square as a synonym for standard deviation when they refer to thesquare 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 secondcentral moment. However, a more natural but much less frequently encountered measure of average deviation from themean that is used in descriptive statistics is the so-calledmean 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 aconfidence interval CI is often denoted
, and
 | (6) |
The following table lists theconfidence 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 givenconfidence interval, solve (5) for
, giving
 | (7) |
| CI | range |
| 0.800 |  |
| 0.900 |  |
| 0.950 |  |
| 0.990 |  |
| 0.995 |  |
| 0.999 |  |
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,VarianceExplore this topic in the MathWorld classroomExplore with Wolfram|Alpha
References
Kenney, J. F. and Keeping, E. S. "The Standard Deviation" and "Calculation of the Standard Deviation." §6.5-6.6 inMathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 77-80, 1962.Referenced on Wolfram|Alpha
Standard DeviationCite this as:
Weisstein, Eric W. "Standard Deviation."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/StandardDeviation.html