Sample mean and covariance

sample meansample covariancesample covariance matrixempirical meanmeansSample mean and sample covariancemean of a random samplemeans of two samplessamplesample average
The sample mean or empirical mean and the sample covariance are statistics computed from a collection (the sample) of data on one or more random variables.wikipedia
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Statistic

sample statisticempiricalmeasure
The sample mean or empirical mean and the sample covariance are statistics computed from a collection (the sample) of data on one or more random variables.

Bias of an estimator

unbiasedunbiased estimatorbias
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a row vector whose j th element (j = 1, ..., K) is one of the random variables.
If the sample mean and uncorrected sample variance are defined as

Estimation of covariance matrices

Covariance estimationShrinkage estimationestimate of the covariance
The maximum likelihood estimate of the covariance
Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix.

Normal distribution

normally distributedGaussian distributionnormal
for the Gaussian distribution case has N in the denominator as well.
The following table gives the quantile z_p such that X will lie in the range with a specified probability p. These values are useful to determine tolerance interval for sample averages and other statistical estimators with normal (or asymptotically normal) distributions:.

Covariance matrix

variance-covariance matrixcovariance matricescovariance
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a row vector whose j th element (j = 1, ..., K) is one of the random variables.
The expected values needed in the covariance formula are estimated using the sample mean, e.g.

Estimator

estimatorsestimateestimates
The sample mean and sample covariance are estimators of the population mean and population covariance, where the term population refers to the set from which the sample was taken.
The central limit theorem implies asymptotic normality of the sample mean \bar X as an estimator of the true mean.

Unbiased estimation of standard deviation

sample standard deviationanti-biasingbias
where is the sample (formally, realizations from a random variable X) and is the sample mean.

Scatter matrix

Given n samples of m-dimensional data, represented as the m-by-n matrix, the sample mean is

Sample (statistics)

samplesamplesstatistical sample
The sample mean or empirical mean and the sample covariance are statistics computed from a collection (the sample) of data on one or more random variables.

Mean

mean valueaveragepopulation mean
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a row vector whose j th element (j = 1, ..., K) is one of the random variables. The sample mean and sample covariance are estimators of the population mean and population covariance, where the term population refers to the set from which the sample was taken. The sample mean is a vector each of whose elements is the sample mean of one of the random variables – that is, each of whose elements is the arithmetic average of the observed values of one of the variables.

Covariance

covariantcovariationcovary
The sample mean and sample covariance are estimators of the population mean and population covariance, where the term population refers to the set from which the sample was taken. The sample covariance matrix is a square matrix whose i, j element is the sample covariance (an estimate of the population covariance) between the sets of observed values of two of the variables and whose i, i element is the sample variance of the observed values of one of the variables.

Vector (mathematics and physics)

vectorvectorsvectorial
The sample mean is a vector each of whose elements is the sample mean of one of the random variables – that is, each of whose elements is the arithmetic average of the observed values of one of the variables.

Average

Rushing averageReceiving averagemean
The sample mean is a vector each of whose elements is the sample mean of one of the random variables – that is, each of whose elements is the arithmetic average of the observed values of one of the variables.

Matrix (mathematics)

matrixmatricesmatrix theory
The sample covariance matrix is a square matrix whose i, j element is the sample covariance (an estimate of the population covariance) between the sets of observed values of two of the variables and whose i, i element is the sample variance of the observed values of one of the variables.

Location parameter

locationlocation modellocation parameters
Due to their ease of calculation and other desirable characteristics, the sample mean and sample covariance are widely used in statistics and applications to numerically represent the location and dispersion, respectively, of a distribution.

Statistical dispersion

dispersionvariabilityspread
Due to their ease of calculation and other desirable characteristics, the sample mean and sample covariance are widely used in statistics and applications to numerically represent the location and dispersion, respectively, of a distribution.

Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
Due to their ease of calculation and other desirable characteristics, the sample mean and sample covariance are widely used in statistics and applications to numerically represent the location and dispersion, respectively, of a distribution.

Definiteness of a matrix

positive definitepositive semidefinitepositive-definite
:Like covariance matrices for random vector, sample covariance matrices are positive semi-definite.

Multivariate random variable

random vectorvectormultivariate
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a row vector whose j th element (j = 1, ..., K) is one of the random variables. :Like covariance matrices for random vector, sample covariance matrices are positive semi-definite.

Bessel's correction

Bessel-correctedBessel corrected variance
The sample covariance matrix has in the denominator rather than due to a variant of Bessel's correction: In short, the sample covariance relies on the difference between each observation and the sample mean, but the sample mean is slightly correlated with each observation since it is defined in terms of all observations.

Random variable

random variablesrandom variationrandom
The sample mean or empirical mean and the sample covariance are statistics computed from a collection (the sample) of data on one or more random variables. This is an example of why in probability and statistics it is essential to distinguish between random variables (upper case letters) and realizations of the random variables (lower case letters).

Realization (probability)

realizationrealizationsobserved data
This is an example of why in probability and statistics it is essential to distinguish between random variables (upper case letters) and realizations of the random variables (lower case letters).

Maximum likelihood estimation

maximum likelihoodmaximum likelihood estimatormaximum likelihood estimate
The maximum likelihood estimate of the covariance

Statistical population

populationsubpopulationsubpopulations
Of course the estimator will likely not be the true value of the population mean since different samples drawn from the same distribution will give different sample means and hence different estimates of the true mean.

Normalizing constant

normalizednormalization constantnormalization
Without loss of generality, assume that the weights are normalized: