In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. The variance of a real-valued random variable is its second central moment, and it also happens to be its second cumulant. The variance of a random variable is the square of its standard deviation. Variance - Definition. If μ = E(X) is the expected value (mean) of the random variable X, then the variance is ...

Including:

Variance - Definition Variance - Properties Variance - Population variance and sample variance
Variance - An unbiased estimator
Variance - Generalizations Variance - History Variance - Moment of inertia

Read more here: » Variance: Encyclopedia - Variance

See also:

Variance, Variance - Definition, Variance - Properties, Variance - Population variance and sample variance, Variance - An unbiased estimator, Variance - Generalizations, Variance - History, Variance - Moment of inertia

Read more here: » Variance: Encyclopedia II - Variance - Definition

If the variance is defined, we can conclude that it is never negative because the squares are positive or zero. The unit of variance is the square of the unit of observation. For example, the variance of a set of heights measured in centimeters will be given in square centimeters. This fact is inconvenient and has motivated many statisticians to instead use the square root of the variance, known as the standard ...

See also:

Variance, Variance - Definition, Variance - Properties, Variance - Population variance and sample variance, Variance - An unbiased estimator, Variance - Generalizations, Variance - History, Variance - Moment of inertia

Read more here: » Variance: Encyclopedia II - Variance - Properties

Agnostic atheism is the philosophy that encompasses both atheism and agnosticism. Due to definitional variance, an agnostic atheist does not believe in God or gods and by extension holds true one or more of these statements: The existence and nonexistence of deities is currently unknown and maybe absolutely unknowable. Knowledge of the existence and nonexistence of deities is irrelevant or unimportant. Abstention from claims of knowledge of the e ...

Read more here: » Agnostic atheism: Encyclopedia - Agnostic atheism

Effective population size may be defined in two ways, variance effective size and inbreeding effective size. These are closely linked, and derived from F-statistics. Effective population size - Variance effective size. In an idealized population, the variance in allele frequency (p) is given by: then this gives: Effe ...

See also:

Effective population size, Effective population size - Definitions, Effective population size - Variance effective size, Effective population size - Inbreeding effective size, Effective population size - Examples, Effective population size - Variations in population size, Effective population size - Dioeciousness, Effective population size - Non-Fisherian 1:1 sex-ratios, Effective population size - Unequal contributions to the next generation, Effective population size - Overlapping generations and age-structured populations, Effective population size - Measurement in wild

Read more here: » Effective population size: Encyclopedia II - Effective population size - Definitions

Effective population size - Variations in population size. Population size varies over time. Suppose there are t non-overlapping generations, then effective population size is given by the harmonic mean of the population sizes: For example, say the population size was N = 10, 100, 50, 80, 20, 500 for six generations (t = 6). Then the effective population size is the harmonic mean of these, giving: ...

See also:

Effective population size, Effective population size - Definitions, Effective population size - Variance effective size, Effective population size - Inbreeding effective size, Effective population size - Examples, Effective population size - Variations in population size, Effective population size - Dioeciousness, Effective population size - Non-Fisherian 1:1 sex-ratios, Effective population size - Unequal contributions to the next generation, Effective population size - Overlapping generations and age-structured populations, Effective population size - Measurement in wild

Read more here: » Effective population size: Encyclopedia II - Effective population size - Examples

The above definition of variance in terms of differences contained by the data does not involve the arithmetic mean. It seems plausible to assume that the information contained in the above matrix could have been also obtained from a matrix of all possible differences between the data elements and their mean, which can be obtained as Squaring the elements of the above matrix results in a matrix with n columns of squared deviation scores x with column sums (5.00) divided by n (4) equal to the variance computed by divid ...

See also:

True variance, True variance - Computation of the true and unbiased variance, True variance - Changing true variance to unbiased variance and vice versa, True variance - Degrees of freedom, True variance - Degrees of freedom: Monte Carlo simulation, True variance - True variance and all possible differences between values of a variable, True variance - Matrices of differences, True variance - Differences between data elements and their mean, True variance - Variance and Information, True variance - Retrospect, True variance - Conventional language of computation

Read more here: » True variance: Encyclopedia II - True variance - Differences between data elements and their mean

A slightly faster (significantly for running standard deviation) way to compute the population standard deviation is given by the formula (but this can exacerbate round-off error) Similarly for sample standard deviation Or from running sums: ...

See also:

Standard deviation, Standard deviation - Definition and shortcut calculation of standard deviation, Standard deviation - Examples, Standard deviation - Interpretation and application, Standard deviation - Geometric interpretation, Standard deviation - Rules for normally distributed data, Standard deviation - Relationship between standard deviation and mean, Standard deviation - Rapid calculation methods, Standard deviation - An axiomatic approach, Standard deviation - Common predefined functions

Read more here: » Standard deviation: Encyclopedia II - Standard deviation - Rapid calculation methods

Let be a vector of real numbers. We write meaning that is estimated by the mean value , and the standard deviation is is a real number, and is a signless real number, meaning that and are considered equivalent. The case n = 2 is per definition Note the special case The case justifies the use of the sign A few rules apply. If then Sym ...

See also:

Standard deviation, Standard deviation - Definition and shortcut calculation of standard deviation, Standard deviation - Examples, Standard deviation - Interpretation and application, Standard deviation - Geometric interpretation, Standard deviation - Rules for normally distributed data, Standard deviation - Relationship between standard deviation and mean, Standard deviation - Rapid calculation methods, Standard deviation - An axiomatic approach, Standard deviation - Common predefined functions

Read more here: » Standard deviation: Encyclopedia II - Standard deviation - An axiomatic approach

Suppose we are given a population x1, ..., xN of values (which are real numbers). The arithmetic mean of this population is defined as . (see sigma notation) and the standard deviation of this population is defined as . The standard deviation of a random variable X is defined as . Note that not all random variables have a standard deviation, since these expected values need not exist. If the random variable X< ...

See also:

Standard deviation, Standard deviation - Definition and shortcut calculation of standard deviation, Standard deviation - Examples, Standard deviation - Interpretation and application, Standard deviation - Geometric interpretation, Standard deviation - Rules for normally distributed data, Standard deviation - Relationship between standard deviation and mean, Standard deviation - Rapid calculation methods, Standard deviation - An axiomatic approach, Standard deviation - Common predefined functions

Read more here: » Standard deviation: Encyclopedia II - Standard deviation - Definition and shortcut calculation of standard deviation

We will show how to calculate the standard deviation of a population. Our example will use the ages of four young children: { 5, 6, 8, 9 }. Step 1. Calculate the mean/average . . We have N = 4 because there are four data points:       Replacing N with 4 ...

See also:

Standard deviation, Standard deviation - Definition and shortcut calculation of standard deviation, Standard deviation - Examples, Standard deviation - Interpretation and application, Standard deviation - Geometric interpretation, Standard deviation - Rules for normally distributed data, Standard deviation - Relationship between standard deviation and mean, Standard deviation - Rapid calculation methods, Standard deviation - An axiomatic approach, Standard deviation - Common predefined functions

Read more here: » Standard deviation: Encyclopedia II - Standard deviation - Examples

The standard deviation is a measure of the degree of dispersion of the data from the mean value. Stated another way, the standard deviation is simply the "average" or "expected" variation around an average (i.e., square all individual deviations around the average, add these up, divide by N, then take the square root. You then have the root of the mean squared deviation (RMS, in a very simple sense the average or expected variation around the average). In fact the standard deviation is sometimes called the expected deviation, though this may be confusing as the expec ...

See also:

Standard deviation, Standard deviation - Definition and shortcut calculation of standard deviation, Standard deviation - Examples, Standard deviation - Interpretation and application, Standard deviation - Geometric interpretation, Standard deviation - Rules for normally distributed data, Standard deviation - Relationship between standard deviation and mean, Standard deviation - Rapid calculation methods, Standard deviation - An axiomatic approach, Standard deviation - Common predefined functions

Read more here: » Standard deviation: Encyclopedia II - Standard deviation - Interpretation and application

The mean and the standard deviation of a set of data are usually reported together. In a certain sense, the standard deviation is a "natural" measure of statistical dispersion if the center of the data is measured about the mean. This is because the standard deviation from the mean is smaller than from any other point. The precise statement is the following: suppose x1, ..., xn are real numbers and define the function Using calculus, it is not di ...

See also:

Standard deviation, Standard deviation - Definition and shortcut calculation of standard deviation, Standard deviation - Examples, Standard deviation - Interpretation and application, Standard deviation - Geometric interpretation, Standard deviation - Rules for normally distributed data, Standard deviation - Relationship between standard deviation and mean, Standard deviation - Rapid calculation methods, Standard deviation - An axiomatic approach, Standard deviation - Common predefined functions

Read more here: » Standard deviation: Encyclopedia II - Standard deviation - Relationship between standard deviation and mean

Consider a sample of size N drawn from a normal distribution of mean μ and unit variance (i.e., ). The sample mean, , of the sample , defined as has variance . This is equal to the reciprocal of the Fisher information from the sample (this is clear from the definition) and thus, by the Cramér-Rao inequality, the sample mean is effici ...

See also:

Efficiency statistics, Efficiency statistics - Efficient estimator, Efficiency statistics - Asymptotic efficiency, Efficiency statistics - Examples, Efficiency statistics - Relative efficiency

Read more here: » Efficiency statistics: Encyclopedia II - Efficiency statistics - Examples

For a sample of n values the sample kurtosis is where m4 is the fourth sample moment about the mean, m2 is the second sample moment about the mean (that is, the sample variance), xi is the ith value, and is the sample mean. ...

See also:

Kurtosis, Kurtosis - Definition of kurtosis, Kurtosis - Terminology and examples, Kurtosis - Sample kurtosis, Kurtosis - Estimators of population kurtosis

Read more here: » Kurtosis: Encyclopedia II - Kurtosis - Sample kurtosis

The risk free asset is the (hypothetical) asset which pays a risk free rate - it is usually proxied by an investment in short-dated Government bonds. The risk free asset has zero variance in returns (hence is risk free); it is also uncorrelated with any other asset (by definition: since its variance is zero). As a result, when it is combined with any other asset, or portfolio of assets, the change in return and also in risk is linear. Because both risk and return change linearly as the risk free asset is introduced into a portf ...

See also:

Modern portfolio theory, Modern portfolio theory - Risk and reward, Modern portfolio theory - Mean and variance, Modern portfolio theory - Diversification, Modern portfolio theory - The efficient frontier, Modern portfolio theory - The risk free asset, Modern portfolio theory - Portfolio leverage, Modern portfolio theory - The market portfolio, Modern portfolio theory - Capital Market Line, Modern portfolio theory - Asset pricing, Modern portfolio theory - Systematic risk and specific risk, Modern portfolio theory - Capital Asset Pricing Model, Modern portfolio theory - Securities Market Line

Read more here: » Modern portfolio theory: Encyclopedia II - Modern portfolio theory - The risk free asset

An informative prior expresses specific, definite information about a variable. An example is a prior distribution for the temperature at noon tomorrow. A reasonable approach is to make the prior a normal distribution with expected value equal to today's noontime temperature, with variance equal to the day-to-day variance of atmospheric temperature. This example has a property in common with many priors, namely, that the posterior from one problem (today's temperature) becomes the prior for another problem (tomorrow's temperatu ...

See also:

Prior probability, Prior probability - Prior probability distribution, Prior probability - Informative priors, Prior probability - Uninformative priors, Prior probability - Improper priors

Read more here: » Prior probability: Encyclopedia II - Prior probability - Informative priors

The power spectral density, PSD, describes how the power (or variance) of a time series is distributed with frequency. If f(t) is a signal, the spectral density Φ(ω) of the signal is the square of the magnitude of the continuous Fourier transform of the signal. where ω is the angular frequency (2π times the cyclic frequency) and F(ω) is the conti ...

See also:

Spectral density, Spectral density - Definition, Spectral density - Properties, Spectral density - Related concepts, Spectral density - Applications, Spectral density - Colorimetry

Read more here: » Spectral density: Encyclopedia II - Spectral density - Definition

Given a sub-set of samples from a population, the sample kurtosis above is a biased estimator of the population kurtosis. The usual estimator of the population kurtosis (used in SAS, SPSS, and Excel but not by MINITAB or BMDP) is G2, defined as follows: where k4 is the unique symmetric unbiased estimator of the fourth cumulant, k2 is the unbiased estimator of the population variance, m4 is the fourth sample moment about the mean, m2 is the sample ...

See also:

Kurtosis, Kurtosis - Definition of kurtosis, Kurtosis - Terminology and examples, Kurtosis - Sample kurtosis, Kurtosis - Estimators of population kurtosis

Read more here: » Kurtosis: Encyclopedia II - Kurtosis - Estimators of population kurtosis

Characteristic classes are in an essential way phenomena of cohomology theory — they are contravariant constructions, in the way that a section is a kind of function on a space, and to lead to a contradiction from the existence of a section we do need that variance. In fact cohomology theory grew up after homology and homotopy theory, which are both covariant theories based on mapping into a space; and characteristic class theory in its infancy in the 1930s (as part of obstruction theory) was one major reason wh ...

See also:

Characteristic class, Characteristic class - Definition, Characteristic class - Motivation

Read more here: » Characteristic class: Encyclopedia II - Characteristic class - Motivation