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Sample mean and covariance

In statistics, the sample mean and sample covariance are key estimators derived from a finite set of observations to approximate the corresponding population parameters. The sample mean, denoted \bar{x}, for a univariate sample of size n consisting of observations x_1, x_2, \dots, x_n, is calculated as the arithmetic average \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i, providing an unbiased estimate of the population mean \mu.^[1] For multivariate data with p variables observed across n samples, this extends to the sample mean vector \bar{\mathbf{x}} = \frac{1}{n} \sum_{i=1}^n \mathbf{x}_i, where each component is the mean of that variable's observations, again serving as an unbiased estimator of the population mean vector.^[2] The sample covariance, which measures the linear relationship and joint variability between variables in the sample, is defined for two variables X_j and X_k as s_{jk} = \frac{1}{n-1} \sum_{i=1}^n (x_{ij} - \bar{x}_j)(x_{ik} - \bar{x}_k), where the division by n-1 ensures it is an unbiased estimator of the population covariance \sigma_{jk}.^[3] In the multivariate case, the sample covariance matrix \mathbf{S} is given by \mathbf{S} = \frac{1}{n-1} \sum_{i=1}^n (\mathbf{x}_i - \bar{\mathbf{x}})(\mathbf{x}_i - \bar{\mathbf{x}})^\top, a symmetric positive semidefinite matrix whose diagonal elements are the sample variances and off-diagonal elements are the sample covariances; it unbiasedly estimates the population covariance matrix and is positive definite when n > p.^[2] These measures are central to descriptive statistics, inference, and multivariate analysis, such as in principal component analysis and hypothesis testing, as they capture central tendency and dependence structure from data.^[3]^[2]

Definitions

Sample mean

The sample mean serves as a fundamental point estimator for the population mean in inferential statistics, providing a measure of central tendency based on a finite set of observations drawn from a larger population.^[4] For a univariate random sample X_1, X_2, \dots, X_n of size n from a population, the sample mean, denoted \bar{X}, is defined as the arithmetic average of these observations:

\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i

This formula represents the total sum of the sample values divided by the number of observations.^[5] To compute the sample mean, first sum all the individual data points in the sample, then divide this total by the sample size n. In descriptive statistics, the sample mean is equivalent to the arithmetic average, serving as the most straightforward summary of a dataset's central location.^[6]

Sample covariance

The sample covariance serves as an estimator for the population covariance, quantifying the linear dependence between two variables from a sample of paired observations. For a dataset consisting of n pairs (X_i, Y_i) where i = 1, \dots, n, the sample covariance s_{XY} is defined as

s_{XY} = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y}),

with \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i and \bar{Y} = \frac{1}{n} \sum_{i=1}^n Y_i denoting the sample means. The choice of n-1 in the denominator ensures that s_{XY} is an unbiased estimator of the corresponding population parameter.^[2]^[7] This statistic measures the directional co-variation between the variables: a positive s_{XY} indicates that deviations from the means tend to occur in the same direction (e.g., both above or both below their means), suggesting a positive linear relationship, whereas a negative value points to opposite directional deviations.^[8] The magnitude reflects the extent of this co-movement, though it is scale-dependent on the units of measurement. For illustration, consider paired data on heights (in inches) and weights (in pounds) from a sample of five individuals: heights = 63, 67, 69, 72, 73; weights = 170, 175, 177, 190, 184. The sample means are approximately \bar{X} = 68.8 and \bar{Y} = 179.2. Computing the sum of the products of centered values yields approximately 115.2, so the sample covariance is $115.2 / (5-1) = 28.8, a positive value consistent with the expected positive association between height and weight.^[9]

Properties

Unbiasedness

The sample mean \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i, where X_1, \dots, X_n are independent and identically distributed random variables with finite expected value \mu, is an unbiased estimator of the population mean \mu. This holds for any distribution with a finite mean, as the expected value satisfies E[\bar{X}] = \mu. The proof relies on the linearity of expectation, which does not require independence or identical distribution beyond the finite mean assumption:

E[\bar{X}] = E\left[ \frac{1}{n} \sum_{i=1}^n X_i \right] = \frac{1}{n} \sum_{i=1}^n E[X_i] = \frac{1}{n} \cdot n \mu = \mu.

^[10]^[11] In contrast, the sample covariance defined using the population mean as \frac{1}{n} \sum_{i=1}^n (X_i - \mu)(X_i - \mu)^T is unbiased for the population covariance matrix \Sigma, assuming finite second moments. However, the standard sample covariance \mathbf{S} = \frac{1}{n} \sum_{i=1}^n (X_i - \bar{X})(X_i - \bar{X})^T, which substitutes the sample mean \bar{X} for \mu, introduces bias because \bar{X} is a random variable dependent on the data. Specifically, under the assumption of independent and identically distributed observations with finite second moments, the expected value is E[\mathbf{S}] = \frac{n-1}{n} \Sigma. To obtain an unbiased estimator, the formula is adjusted to use the denominator n-1:

\mathbf{S} = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})(X_i - \bar{X})^T,

yielding E[\mathbf{S}] = \Sigma. The proof proceeds by decomposing the sum and using the fact that E[(X_i - \bar{X})(X_j - \bar{X})^T] = 0 for i \neq j, while for i = j, E[(X_i - \bar{X})(X_i - \bar{X})^T] = \frac{n-1}{n} \Sigma, leading to the overall factor of \frac{n-1}{n} before correction.^[12]^[13] This denominator adjustment, known as Bessel's correction, ensures unbiasedness for both sample variance (a special case of covariance) and covariance; using n instead results in a downward bias of factor \frac{n-1}{n}. The correction originated with Carl Friedrich Gauss, who applied it in astronomical data analysis as early as 1823, though it is named after Friedrich Bessel for his later independent use in 1818.^[14]^[15]

Consistency

The sample mean \bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i is a consistent estimator of the population mean \mu, converging in probability to \mu as the sample size n \to \infty, provided the random variables X_i are independent and identically distributed (i.i.d.) with finite variance \sigma^2 < \infty. This fundamental result, known as the weak law of large numbers (WLLN), was established using Chebyshev's inequality, which bounds the probability that |\bar{X}_n - \mu| exceeds any \epsilon > 0 by \sigma^2 / (n \epsilon^2), approaching zero as n grows. In the multivariate setting, the sample covariance matrix \mathbf{S}_n = \frac{1}{n} \sum_{i=1}^n (X_i - \bar{X}_n)(X_i - \bar{X}_n)^T (or the unbiased version with n-1) is consistent for the population covariance matrix \Sigma, meaning \mathbf{S}_n \xrightarrow{p} \Sigma as n \to \infty, assuming the i.i.d. vectors X_i \in \mathbb{R}^p have finite fourth moments. This convergence follows from the consistency of the sample mean \bar{X}_n \xrightarrow{p} \mu combined with the continuous mapping theorem applied to the quadratic form defining the covariance, ensuring that deviations in the centered terms vanish in probability. The finite fourth-moment condition is necessary because the variance of elements in \mathbf{S}_n involves expectations of products up to fourth order; without it, consistency may fail for heavy-tailed distributions. Alternatively, the result can be derived via Slutsky's theorem on the decomposition \mathbf{S}_n = \frac{1}{n} \sum_{i=1}^n (X_i - \mu)(X_i - \mu)^T + o_p(1), where the first term converges by a multivariate WLLN and the remainder term accounts for centering. While consistency describes convergence in probability without specifying speed, the central limit theorem refines this by showing that the sample mean scales at rate \sqrt{n}, with \sqrt{n} (\bar{X}_n - \mu) \xrightarrow{d} N(0, \sigma^2) under finite variance, highlighting errors of order O_p(1/\sqrt{n}); analogous asymptotic normality holds for \mathbf{S}_n (with \sqrt{n} scaling for vec(\mathbf{S}_n - \Sigma)), but full details appear in the sampling distributions section. Unbiasedness provides exact finite-sample expectation matching but does not guarantee consistency, though the sample mean and covariance satisfy both properties under the stated conditions. To illustrate empirical convergence, consider simulating n i.i.d. draws from a N(0,1) distribution: for small n=10, sample means vary widely around 0 with standard deviation approximately 0.32; as n increases to 100, the standard deviation drops to about 0.1, and by n=1000, over 95% of simulated means lie within 0.03 of 0, visually tightening toward the population mean in histograms across 1000 replications. A similar pattern holds for the sample variance, approaching 1 more reliably with larger n.

Sampling Distributions

Distribution of the sample mean

The distribution of the sample mean \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i, where X_1, \dots, X_n are independent and identically distributed (IID) random variables drawn from a population, depends on the underlying population distribution. Under the assumption that the population is normally distributed, specifically X_i \sim N(\mu, \sigma^2) for all i, the sample mean follows an exact normal distribution: \bar{X} \sim N(\mu, \sigma^2 / n).^[16] This result arises because the sum of independent normal random variables is itself normal, with mean n\mu and variance n\sigma^2, and dividing by n scales the mean to \mu and the variance to \sigma^2 / n.^[1] A key property of this distribution is its variance, \mathrm{Var}(\bar{X}) = \sigma^2 / n, which holds more generally for any IID population with finite variance \sigma^2 > 0, regardless of the population's shape.^[17] This variance decreases as the sample size n increases, reflecting the averaging effect that reduces sampling variability. For the normal population case, the standard deviation of \bar{X} is thus \sigma / \sqrt{n}, known as the standard error of the mean.^[16] When the population is not normal but the random variables are IID with finite mean \mu and finite variance \sigma^2 > 0, the exact distribution of \bar{X} is generally unknown. However, the central limit theorem provides an asymptotic approximation: as n \to \infty, the standardized sample mean \sqrt{n} (\bar{X} - \mu) / \sigma converges in distribution to a standard normal random variable, N(0, 1).^[18] Equivalently, \sqrt{n} (\bar{X} - \mu) \xrightarrow{d} N(0, \sigma^2).^[19] This theorem justifies using the normal distribution to approximate the sampling distribution of \bar{X} for large n, typically n \geq 30, even for non-normal populations.^[19] These distributional properties enable the construction of confidence intervals for the population mean \mu. If \sigma^2 is known, a (1 - \alpha) \times 100\% confidence interval is given by \bar{x} \pm z_{\alpha/2} \cdot (\sigma / \sqrt{n}), where z_{\alpha/2} is the upper \alpha/2 quantile of the standard normal distribution, exploiting the exact or asymptotic normality of \bar{X}.^[20] When \sigma^2 is unknown, the interval uses the sample standard deviation s and replaces z_{\alpha/2} with the t-distribution quantile t_{n-1, \alpha/2} with n-1 degrees of freedom: \bar{x} \pm t_{n-1, \alpha/2} \cdot (s / \sqrt{n}); this t-based interval is exact under population normality and approximate otherwise via the central limit theorem.^[21]

Distribution of the sample covariance

The distribution of the sample covariance matrix \mathbf{S} is a central topic in multivariate statistical inference, particularly when the observations \mathbf{X}_i, i=1,\dots,n, are independent and identically distributed from a p-variate normal distribution N_p(\boldsymbol{\mu}, \Sigma). In this case, the scaled sample covariance matrix (n-1)\mathbf{S} follows a Wishart distribution with n-1 degrees of freedom and scale matrix \Sigma, denoted as (n-1)\mathbf{S} \sim W_p(\Sigma, n-1). This result, derived from the properties of quadratic forms in normal variables, provides the exact sampling distribution for inference on \Sigma. In the univariate setting, where p=1 and the data X_i \sim N(\mu, \sigma^2), the sample variance s^2 follows a scaled chi-squared distribution: s^2 \sim \frac{\sigma^2 \chi^2_{n-1}}{n-1}. This is a special case of the Wishart distribution, reducing to the gamma family for scalar variances, and underpins exact tests like the F-test for equality of variances. For large sample sizes, regardless of the underlying distribution (assuming finite fourth moments), the sample covariance matrix \mathbf{S} admits an asymptotic normal distribution after appropriate centering and scaling. Specifically, \sqrt{n} (\mathrm{vech}(\mathbf{S} - \Sigma)) \xrightarrow{d} N(0, \mathbf{V}), where \mathrm{vech}(\cdot) stacks the lower triangular elements of the symmetric matrix into a vector, and \mathbf{V} is the asymptotic covariance matrix depending on the fourth-order moments of the distribution. Under multivariate normality, \mathbf{V} simplifies to $2(\Sigma \otimes \Sigma) (I + K_{pp}), where K_{pp} is the commutation matrix, enabling delta-method approximations for functions of \Sigma. A key property under the multivariate normal assumption is the independence between the sample mean vector \bar{\mathbf{X}} and the sample covariance matrix \mathbf{S}. This independence facilitates joint inference, such as in Hotelling's T-squared statistic, by separating location and dispersion parameters.

Extensions

Weighted samples

In scenarios where observations in a dataset have unequal importance or reliability, such as in survey sampling or data with varying precision, the sample mean and covariance can be adapted using weights w_i > 0 assigned to each observation i. Weights can be frequency weights (e.g., multiplicities or inverse inclusion probabilities in sampling designs) or reliability weights (e.g., inverse variances in measurement errors). The weighted sample mean is defined as

\bar{X}_w = \frac{\sum_{i=1}^n w_i X_i}{\sum_{i=1}^n w_i},

which reduces to the standard unweighted sample mean when all weights are equal to 1.^[22] This formulation accounts for factors like inverse variances in measurement errors or sampling probabilities, ensuring that more reliable or representative observations contribute proportionally more to the estimate.^[23] The corresponding weighted sample covariance between two variables X and Y depends on the weight type. For frequency weights, where w_i are positive integers representing observation multiplicities, it is

s_{XY,w} = \frac{\sum_{i=1}^n w_i (X_i - \bar{X}_w)(Y_i - \bar{Y}_w)}{\sum_{i=1}^n w_i - 1},

with the denominator providing an unbiased estimate of the population covariance under the assumption that the effective sample size is \sum w_i. For reliability weights, such as inverse variances, the unbiased estimator requires a different adjustment; a common approximation normalizes the weights so \sum w_i = 1 and uses

s_{XY,w} = \frac{\sum_{i=1}^n w_i (X_i - \bar{X}_w)(Y_i - \bar{Y}_w)}{1 - \sum_{i=1}^n w_i^2},

to account for effective degrees of freedom. These formulations emphasize pairs with higher precision when weights reflect reliability. Under design-based inference frameworks, such as those in probability sampling designs, the weighted sample mean is an unbiased estimator of the population mean, with the weights typically set as the inverse of inclusion probabilities to correct for unequal selection chances.^[24] For example, in stratified sampling, the population is divided into homogeneous strata, and the overall mean is estimated as a weighted average of stratum-specific sample means, with weights proportional to each stratum's population size N_h; this yields \bar{y}_{st} = \sum_{h=1}^H (N_h / N) \bar{y}_h, which is unbiased for the finite population mean.^[25] The weighted covariance inherits similar unbiasedness properties when applied within this design, provided the weights align with the sampling structure and the appropriate formula for the weight type is used.^[26] These weighted adaptations find key applications in handling missing data through inverse probability weighting (IPW), where weights are the reciprocals of estimated probabilities of non-missingness, enabling unbiased inference by effectively upweighting observed cases to represent the full population.^[27] In cluster sampling, weights adjust for the clustered structure by incorporating design effects, such as varying cluster sizes, to produce representative estimates of means and covariances without assuming equal observation importance. Such methods are particularly valuable in complex surveys, where ignoring weights could lead to biased summaries of population relationships.^[28]

Multivariate case

In the multivariate case, observations consist of p-dimensional random vectors \mathbf{X}_i = (X_{i1}, \dots, X_{ip})^T for i = 1, \dots, n, drawn from a population with mean vector \boldsymbol{\mu} and covariance matrix \boldsymbol{\Sigma}. The sample mean vector is defined as \bar{\mathbf{X}} = \frac{1}{n} \sum_{i=1}^n \mathbf{X}_i, which is a p \times 1 vector whose j-th component is the univariate sample mean of the j-th coordinate across all observations.^[2] This estimator provides a point estimate for \boldsymbol{\mu} and generalizes the scalar sample mean to higher dimensions, where the univariate case arises as the special case p=1. The sample covariance matrix \mathbf{S} extends the scalar sample variance to capture both individual and joint variability among the p coordinates. It is a p \times p symmetric matrix given by \mathbf{S} = \frac{1}{n-1} \sum_{i=1}^n (\mathbf{X}_i - \bar{\mathbf{X}})(\mathbf{X}_i - \bar{\mathbf{X}})^T, with diagonal elements s_{jj} representing the sample variances of each coordinate and off-diagonal elements s_{jk} (for j \neq k) denoting the sample covariances between coordinates j and k.^[2] As a Gram matrix of centered data, \mathbf{S} is positive semi-definite, meaning all its eigenvalues are non-negative (\lambda_j \geq 0), which ensures it can serve as a valid covariance structure; it is positive definite if n > p and the data span the full p-dimensional space.^[2] The elements of \mathbf{S} offer key interpretations: the trace \operatorname{tr}(\mathbf{S}) = \sum_{j=1}^p s_{jj} quantifies total sample variability, while the off-diagonals reveal linear dependencies, with positive (negative) values indicating coordinates that tend to increase (decrease) together. The eigenvalues of \mathbf{S} further decompose this variability, where the largest eigenvalues correspond to directions of maximum variance, previewing their role in principal component analysis for dimensionality reduction by projecting data onto these principal axes.^[29] For illustration, consider a bivariate sample from soil calcium and potassium measurements, adapted from a trivariate dataset treated as draws from a bivariate normal distribution, with data points (y_1, y_2): (35, 3.5), (35, 4.9), (40, 30.0), (10, 2.8), (6, 2.7), (20, 2.8), (35, 4.6), (35, 10.9), (35, 8.0), (30, 1.6). The resulting sample mean vector is \bar{\mathbf{y}} = \begin{pmatrix} 28.1 \\ 7.18 \end{pmatrix}, and the sample covariance matrix is \mathbf{S} = \begin{pmatrix} 140.54 & 49.68 \\ 49.68 & 72.25 \end{pmatrix}, where the off-diagonal covariance indicates positive association between the variables.^[30]

Limitations

Degrees of freedom adjustment

The degrees of freedom adjustment in sample covariance estimation involves dividing the sum of cross-products of deviations by n-1 rather than n, where n is the sample size, to account for the one degree of freedom lost when estimating the sample mean from the data itself. This reduction in effective independent observations arises because the sample mean imposes a constraint on the deviations, leaving n-1 degrees of freedom for variance or covariance estimation. In general settings where additional parameters beyond the mean are estimated—such as regression coefficients—the degrees of freedom extend to n - 1 - k, with k representing the number of extra parameters.^[31] Using n in the denominator produces the maximum likelihood estimator of the covariance under normality, but this estimator is biased, systematically underestimating the population covariance because deviations from the sample mean are smaller than those from the true mean. The n-1 adjustment yields an unbiased estimator, ensuring its expected value matches the population covariance exactly.^[15] In the multivariate setting, the sample covariance matrix \mathbf{S} incorporates the same adjustment, such that (n-1) \mathbf{S} follows a Wishart distribution with n-1 degrees of freedom and scale matrix \boldsymbol{\Sigma}, the population covariance matrix. The trace of \mathbf{S}, equivalent to the sum of the diagonal elements (sample variances), then possesses p(n-1) degrees of freedom, where p is the dimensionality of the data.^[32] For a concrete illustration with small n=2, consider univariate observations x_1 and x_2 from a normal distribution with variance \sigma^2. The sample mean is \bar{x} = (x_1 + x_2)/2, and the sum of squared deviations is (x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 = (x_1 - x_2)^2 / 2. Dividing by n-1 = 1 gives the unbiased sample variance s^2 = (x_1 - x_2)^2 / 2, with expected value \sigma^2. Dividing by n=2 instead yields (x_1 - x_2)^2 / 4, with expected value \sigma^2 / 2, confirming the downward bias without adjustment.

Criticisms

The sample mean and sample covariance matrix are highly sensitive to outliers, as even a single extreme observation can substantially distort their values due to the unbounded influence function of these estimators.^[33] This sensitivity arises because the sample mean is pulled toward outliers, and the covariance matrix amplifies this effect by incorporating squared deviations, leading to inflated variances and covariances.^[34] In contrast, robust alternatives like the median and median absolute deviation (MAD) maintain stability, with breakdown points up to 50%, whereas the sample mean and covariance have a breakdown point of zero.^[35] The sampling distributions of the sample mean and covariance rely heavily on the assumption of normality for exact inference, but this assumption often fails in real-world data, resulting in unreliable confidence intervals and hypothesis tests.^[36] Without multivariate normality, the sample covariance does not follow a Wishart distribution, which undermines the validity of standard error estimates and leads to fragile inferences, particularly for small samples.^[37] Degrees of freedom adjustments provide partial mitigation by correcting bias in variance estimates, but they do not address the distributional fragility under non-normality.^[37] In high-dimensional settings where the number of variables p approaches or exceeds the sample size n, the sample covariance matrix \mathbf{S} suffers from the curse of dimensionality, becoming unstable and ill-conditioned unless n \gg p.^[38] This instability manifests as eigenvalue spreading, where the eigenvalues of \mathbf{S} diverge from those of the population covariance, leading to poor performance in downstream tasks like principal component analysis. Historical debates in the early 20th century, particularly following Fisher's work on maximum likelihood estimation, highlighted tensions between unbiasedness and mean squared error (MSE) minimization for the sample mean in small samples.^[39] For instance, in the multivariate case, later developments like the James-Stein estimator showed that biased alternatives can achieve lower MSE than the unbiased sample mean in finite samples.^[40] To address these issues, robust estimators such as the minimum covariance determinant (MCD) have been developed as alternatives, offering high breakdown points while estimating location and scatter.^[41]

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It is assumed that the sample {X1,...,Xn} is IID from a p-variate normal distribution with some unknown mean vector µ and unknown covariance matrix. The ...
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[PDF] A Review and Guide to Covariance Matrix Estimation
Feb 2, 2022 · The sample covariance matrix ST is unbiased and the maximum likelihood estimator under normality. There was a time when it was thought that such ...
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[PDF] arXiv:1203.0967v1 [math.ST] 5 Mar 2012
Mar 5, 2012 · matrix. 1. Page 2. To overcome this curse of dimensionality, several works studied the esti- mation of the population covariance matrix, under ...
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[PDF] Some History of Optimality - Rice Statistics
Asymptotic optimality for estimation goes back to Fisher (1922), as mentioned in Section 2, and is defined as minimum asymptotic variance. For testing ...
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Minimum covariance determinant and extensions - Hubert - 2018
Dec 22, 2017 · The minimum covariance determinant (MCD) method is a highly robust estimator of multivariate location and scatter, for which a fast algorithm is available.DESCRIPTION OF THE MCD... · PROPERTIES · MCD-BASED MULTIVARIATE...