Standard Deviation and Variance

Standard deviationis a widely used measure of variability or diversity used instatisticsandprobability theory. It shows how much variation or "dispersion" exists from the average (mean, or expected value). A low standard deviation indicates that the data points tend to be very close to themean, whereas high standard deviation indicates that the data points are spread out over a large range of values.

The standard deviation of astatistical population, data set, orprobability distributionis thesquare rootof itsvariance. It isalgebraicallysimpler though practically lessrobustthan theaverage absolute deviation.^[1][2]A useful property of standard deviation is that, unlike variance, it is expressed in the same units as the data.

In addition to expressing the variability of a population, standard deviation is commonly used to measure confidence in statistical conclusions. For example, themargin of errorinpollingdata is determined by calculating the expected standard deviation in the results if the same poll were to be conducted multiple times. The reported margin of error is typically about twice the standard deviation ­– the radius of a 95 percentconfidence interval. Inscience, researchers commonly report the standard deviation of experimental data, and only effects that fall far outside the range of standard deviation are consideredstatistically significant– normal random error or variation in the measurements is in this way distinguished from causal variation. Standard deviation is also important infinance, where the standard deviation on therate of returnon aninvestmentis a measure of thevolatilityof the investment.

When only asampleof data from a population is available, the population standard deviation can be estimated by a modified quantity called the sample standard deviation,explained below.

Contents

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• 1 Basic examples
• 2 Definition of population values
  • 2.1 Discrete random variable
  • 2.2 Continuous random variable
• 3 Estimation
  • 3.1 With standard deviation of the sample
  • 3.2 With sample standard deviation
  • 3.3 Other estimators
  • 3.4 Confidence interval of a sampled standard deviation
• 4 Identities and mathematical properties
• 5 Interpretation and application
  • 5.1 Application examples
    • 5.1.1 Climate
    • 5.1.2 Sports
    • 5.1.3 Finance
  • 5.2 Geometric interpretation
  • 5.3 Chebyshev's inequality
  • 5.4 Rules for normally distributed data
• 6 Relationship between standard deviation and mean
• 7 Rapid calculation methods
  • 7.1 Weighted calculation
• 8 Combining standard deviations
  • 8.1 Population-based statistics
  • 8.2 Sample-based statistics
• 9 History
• 10 See also
• 11 References
• 12 External links

[edit]Basic examples

Consider apopulationconsisting of the following eight values:

These eight data points have the mean (average) of 5:

To calculate the population standard deviation, first compute the difference of each data point from the mean, andsquarethe result of each:

Next compute the average of these values, and take the…