the variance of the generating probability distribution. In this sense, the concept of population can be extended to continuous random variables with infinite populations. Sample variance Biased sample variance In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a sample of the population. Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution. We take a sample with replacement of n values Y1, ..., Yn from the population, where n < N, and estimate the variance on the basis of this sample. Directly taking the variance of the sample data gives the average of the squared deviations: Here, denotes the sample mean: Since the Yi are selected randomly, both and are random variables. Their expected values can be evaluated by averaging over the ensemble of all possible samples {Yi} of size n from the population. For this gives: Hence gives an estimate of the population variance that is biased by a factor of . For this reason, is referred to as the biased sample variance. Unbiased sample variance Correcting for this bias yields the unbiased sample variance, denoted : Either estimator may be simply referred to as the sample variance when the version can be determined by context. The same proof is also applicable for samples taken from a continuous probability distribution. The use of the term n − 1 is called Bessel's correction, and it is also used in sample covariance and the sample standard deviation (the square root of variance). The square root is a concave function and thus introduces negative bias (by Jensen's inequality), which depends on the distribution, and thus the corrected sample standard deviation (using Bessel's correction) is biased. The unbiased estimation of standard deviation is a technically involved problem, though for the normal distribution using the term n − 1.5 yields an almost unbiased estimator. The unbiased sample variance is a U-statistic for the function ƒ(y1, y2) = (y1 − y2)2/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population. Distribution of the sample variance Being a function of random variables, the sample variance is itself a random variable, and