to estimate a variance from a residual ( rather than a total ) sum of

A formula for calculating an unbiased estimate of the population variance from a finite sample of n observations is:

For a particularly robust two-pass algorithm for computing the variance, first compute and subtract an estimate of the mean, and then use this algorithm on the residuals.

Based on this sample, the estimated population mean is 10, and the unbiased estimate of population variance is 30.

Algorithm II computes this variance estimate correctly, but Algorithm I returns 29. 333333333333332 instead of 30.

Because the mean and variance of the Cauchy distribution are not defined, attempts to estimate these parameters will not be successful.

But if we use the second experiment, the variance of the estimate given above is σ < sup > 2 </ sub >/ 8.

* The standard deviation of an estimator of θ ( the square root of the variance ), or an estimate of the standard deviation of an estimator of θ, is called the standard error of θ.

In computational statistics, stratified sampling is a method of variance reduction when Monte Carlo methods are used to estimate population statistics from a known population.

In the latter case a sample of data from such a distribution can be used to construct an estimate of its variance: in the simplest cases this estimate can be the sample variance, defined below.

The Gauss – Markov theorem states that the estimate of the mean having minimum variance is given by:

The Allan variance is intended to estimate stability due to noise processes and not that of systematic errors or imperfections such as frequency drift or temperature effects.

where is the number of frequency samples used in variance, is the time between each frequency sample and is the time-length of each frequency estimate.

where s < sup > 2 </ sup > is the sample variance of our estimate, σ < sup > 2 </ sup > is the true variance value, d. f.

where the true residual variance σ < sup > 2 </ sup > is replaced by an estimate based on the minimised value of the sum of squares objective function S. The denominator, n-m, is the statistical degrees of freedom ; see effective degrees of freedom for generalizations.

The statistical analyses required to estimate the genetic and environmental components of variance depend on the sample characteristics.

Like all behavior genetic research, the classic twin study begins from assessing the variance of a behavior ( called a phenotype by geneticists ) in a large group, and attempts to estimate how much of this is due to genetic effects ( heritability ), and how much appears to be due to shared or unique environmental effects-events that affect each twin in a different way, or events that occur to one twin but not another.

The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate.

In regression analysis, the term mean squared error is sometimes used to refer to the estimate of error variance: residual sum of squares divided by the number of degrees of freedom.

For a quantity measured multiple independent times with variance, the best estimate of the signal is obtained

) Then X < sub > i </ sub > is the value ( or realization ) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μ < sub > i </ sub > and variance σ < sub > i </ sub >< sup > 2 </ sup > for all times i. Then the definition of the autocorrelation between times s and t is

More generally, if X < sub > 1 </ sub >, ..., X < sub > n </ sub > are independent random variables, not necessarily identically distributed, but all having the same variance, then

A simple example is one in which X has a normal distribution with expected value 0 and variance 1, and Y = X if | X | > c and Y = − X if | X | < c, where c > 0.

The log-normal distribution is the maximum entropy probability distribution for a random variate X for which the mean and variance of is fixed.

Let X be a random variable with finite expected value μ and finite non-zero variance σ < sup > 2 </ sup >.

Let ρ be the correlation coefficient between X < sub > 1 </ sub > and X < sub > 2 </ sub > and let σ < sub > i </ sub >< sup > 2 </ sup > be the variance of X < sub > i </ sub >.

An assumption of finite variance Var ( X < sub > 1 </ sub >)

If we assume that P < sub > X </ sub >( x ) is Gaussian with variance σ < sup > 2 </ sup >, and if we assume that successive samples of the signal X are stochastically independent ( or, if you like, the source is memoryless, or the signal is uncorrelated ), we find the following analytical expression for the rate – distortion function:

Here Var ( X ) is the variance of X, defined as:

In probability theory, Maxwell's theorem, named in honor of James Clerk Maxwell, states that if the probability distribution of a vector-valued random variable X = ( X < sub > 1 </ sub >, ..., X < sub > n </ sub > )< sup > T </ sup > is the same as the distribution of GX for every n × n orthogonal matrix G and the components are independent, then the components X < sub > 1 </ sub >, ..., X < sub > n </ sub > are normally distributed with expected value 0, all have the same variance, and all are independent.

In probability theory, the law of total variance or variance decomposition formula, also known by the acronym EVVE ( or Eve's law for short ), states that if X and Y are random variables on the same probability space, and the variance of Y is finite, then

Here, < sub > n </ sub > denotes the sample mean of the first n samples ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >), s < sup > 2 </ sup >< sub > n </ sub > their sample variance, and σ < sup > 2 </ sup >< sub > n </ sub > their population variance.

Next consider the sample ( 10 < sup > 8 </ sup > + 4, 10 < sup > 8 </ sup > + 7, 10 < sup > 8 </ sup > + 13, 10 < sup > 8 </ sup > + 16 ), which gives rise to the same estimated variance as the first sample.

where A is the arithmetic mean, H is the harmonic mean, M is the maximum of the interval and s < sup > 2 </ sup > is the variance of the set.

The mean of the sample ( m ) is also distributed normally with variance s < sup > 2 </ sup >

s < sup > 2 </ sup > is the variance of the reciprocals of the data

where σ < sup > 2 </ sup > is the variance.

This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.

When U and V are two independent normally distributed random variables with expected value 0 and variance 1, then the ratio U / V has the standard Cauchy distribution.

Both the mean and the variance may be infinite ( if it includes at least one term of the form 1 / 0 ).

Power law distributions have a well defined mean only if the exponent exceeds 1 and have a finite variance only when the exponent exceeds two.

While the first one may be seen as the variance of the sample considered as a population, the second one is the unbiased estimator of the population variance, meaning that its expected value E is equal to the true variance of the sampled random variable ; the use of the term n − 1 is called Bessel's correction.

The unbiased sample variance is a U-statistic for the function ƒ ( x < sub > 1 </ sub >, x < sub > 2 </ sub >)

The degrees of freedom of the weighted, unbiased sample variance vary accordingly from N − 1 down to 0.

For example, consider the weighted mean of the point 0 with high variance in the second component and 1 with high variance in the first component.

If taking the time-series and skipping past n − 1 samples a new ( shorter ) time-series would occur with τ < sub > 0 </ sub > as the time between the adjacent samples, for which the Allan variance could be calculated with the simple estimators.

This correctly estimates the variance, due to the fact that ( 1 ) the average of normally distributed random variables is also normally distributed ; ( 2 ) the predictive distribution of a normally distributed data point with unknown mean and variance, using conjugate or uninformative priors, has a Student's t-distribution.

Note that knowing that alters the variance, though the new variance does not depend on the specific value of a ; perhaps more surprisingly, the mean is shifted by ; compare this with the situation of not knowing the value of a, in which case x < sub > 1 </ sub > would have distribution

For example, if the average of n uncorrelated random variables Y < sub > i </ sub >, i = 1, ..., n, all having the same finite mean and variance, is given by

The overall shape of the probability density function of the t-distribution resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider.

As the number of degrees of freedom grows, the t-distribution approaches the normal distribution with mean 0 and variance 1.