# 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: