Sample mean and covariance
Statistics computed from a sample to estimate population parameters.
The sample mean and sample covariance, also called the empirical mean and empirical covariance, are statistics derived from a sample of data concerning one or more random variables. The sample mean is the average of a set of numbers drawn from a larger population, where "population" refers to the entire set of relevant data, whether observed or not. It is calculated by summing all observed values and dividing by the number of observations. This statistic serves as an estimator for the true population mean; its reliability is assessed using the standard error, which itself depends on the sample variance. For a random sample, the standard error decreases as the sample size grows, and the distribution of the sample mean approaches a normal distribution with increasing sample size. When multiple variables are studied—for instance, sales, profits, and employees of a sample of companies—the sample mean becomes a vector of averages, one per variable. In such multivariate cases, the sample covariance matrix (or variance-covariance matrix) is computed. This K×K matrix, where K is the number of variables, contains estimates of the covariance between each pair of variables, with the diagonal entries representing variances. The sample covariance matrix is positive semi-definite; it is positive definite if and only if the observation vectors have rank K. Both the sample mean and sample covariance matrix are unbiased estimators of their population counterparts. The sample covariance uses N−1 in the denominator (Bessel's correction) because the sample mean is correlated with each observation; if the population mean were known, the denominator would be N. For large samples, the maximum likelihood estimate for a Gaussian distribution, which uses N in the denominator, closely approximates the unbiased estimate.
- field
- Statistics
- known_for
- Estimating population mean and covariance from sample data
- definition_sample_mean
- Sum of values divided by the number of values
- definition_sample_covariance
- K-by-K matrix with entries based on deviations from the sample mean
- use
- Representing location and dispersion of sample distribution and estimating population values
Lore & Background
The sample mean is defined as the average of the values of a variable in a sample, calculated by summing those values and dividing by the number of values. For example, from a sample (1, 4, 1) taken from a population, the sample mean is 2, while the population mean is 1.5. Even if a sample is random, it is rarely perfectly representative, and different samples from the same population yield different sample means. When a statistician considers multiple variables, the sample mean becomes a vector of averages, one for each variable. The sample covariance matrix is a K-by-K matrix where each entry is computed using the sum of products of deviations from the sample means, divided by N-1. This matrix is useful for judging the reliability of sample means as estimators and for estimating the population covariance matrix.
Reader's Guide
The sample mean and sample covariance are widely used in statistics due to their ease of calculation and desirable characteristics. The sample mean serves as an estimator for the population mean, with reliability estimated using the standard error, which is calculated from the sample variance. For random samples, the standard error decreases with sample size, and the distribution of the sample mean approaches the normal distribution as sample size increases. The sample covariance matrix extends these concepts to multiple variables, providing a measure of how variables co-vary. It is a key tool in multivariate statistics, allowing researchers to understand relationships between variables, such as sales, profits, and employees of companies. The sample covariance is also used as an estimate of the population covariance matrix, making it fundamental for inferential statistics and hypothesis testing.
Did You Know?
- The sample mean of (1, 4, 1) is 2, while the population mean of (1,1,3,4,0,2,1,0) is 1.5.
- The sample mean vector contains the average of observations for each variable in a multivariate sample.
- The sample covariance matrix is a K-by-K matrix with entries computed using N-1 in the denominator.
- For a random sample, the standard error of the sample mean falls as the sample size increases.
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