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Sample mean

by , PhD

Suppose we observe the realizations of a certain number of random variables. Their sample mean is a statistic obtained by calculating the arithmetic average of these realizations. The term is often used in opposition to the term "population mean", which instead refers to the expected value of the random variables in question (provided they all have the same expected value).

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A more precise definition follows.

Definition Let [eq1] be n random variables. Let [eq2] be their realizations. Their sample mean, denoted by $\overline{x}_{n}$, is[eq3]

Note that the term sample mean can also refer to the random variable[eq4]

In other words, before the realizations of the random variables [eq5] become known, their sample mean can be regarded as a random variable.

More details

Under appropriate conditions, the sample mean is an unbiased and consistent estimator of the population mean. The lecture entitled Mean estimation contains an in-depth presentation of the sample mean, including detailed proofs of all its main characteristics and properties.

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