Sample mean and covariance

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.