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The standard deviation is invariant under changes in location,

and scales directly with the scaleof the random variable.

Thus, for a constant cand random variables Xand Y:

The standard deviation of the sum of two random variables

can be related to their individual standard deviations and thecovariance between them:

Identities and Properties

http://en.wikipedia.org/wiki/Location_parameterhttp://en.wikipedia.org/wiki/Scale_parameterhttp://en.wikipedia.org/wiki/Scale_parameterhttp://en.wikipedia.org/wiki/Location_parameter

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The calculation of the sum of squared

deviations can be related

to moments calculated directly from the data.

The standard deviation of the sample can be

computed as:

The sample standard deviation can be

computed as:

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For a finite population with equal probabilities

at all points, we have

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Interpretation and Application

A large standard deviation indicates that the datapoints are far from the mean and a smallstandard deviation indicates that they areclustered closely around the mean.

For example, each of the three populations {0, 0,14, 14}, {0, 6, 8, 14} and {6, 6, 8, 8} has a mean of7. Their standard deviations are 7, 5, and 1,respectively. The third population has a much

smaller standard deviation than the other twobecause its values are all close to 7. It will havethe same units as the data points themselves.

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Application

Climate

As a simple example, consider the average daily maximumtemperatures for two cities, one inland and one on the coast. It ishelpful to understand that the range of daily maximumtemperatures for cities near the coast is smaller than for cities

inland. The standard deviation of the daily maximum temperaturefor the coastal city will be less than that of the inland city as, on anyparticular day, the actual maximum temperature is more likely to befarther from the average maximum temperature for the inland citythan for the coastal one.

Particle physics

Particle physicsuses a standard of "5 sigma" for the declaration of adiscovery.[3]At five-sigma there is only one chance in nearly twomillion that a random fluctuation would yield the result.

http://en.wikipedia.org/wiki/Particle_physicshttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Particle_physics

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Relation b/w Standard deviation and

mean

The mean and the standard deviation of a set ofdata are descriptive statisticsusually reportedtogether. In a certain sense, the standard

deviation is a "natural" measure of statisticaldispersionif the center of the data is measuredabout the mean. This is because the standarddeviation from the mean is smaller than from any

other point. The precise statement is thefollowing: suppose x1, ..., xnare real numbers anddefine the function:

http://en.wikipedia.org/wiki/Descriptive_statisticshttp://en.wikipedia.org/wiki/Statistical_dispersionhttp://en.wikipedia.org/wiki/Statistical_dispersionhttp://en.wikipedia.org/wiki/Statistical_dispersionhttp://en.wikipedia.org/wiki/Statistical_dispersionhttp://en.wikipedia.org/wiki/Descriptive_statistics

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Standard deviation of mean

Often, we want some information about the precision of the

mean we obtained. We can obtain this by determining the

standard deviation of the sampled mean.

where Nis the number of observations in the sample used toestimate the mean. This can easily be proven with

Hence Resulting in