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The sample mean ¯ (the arithmetic mean of a sample of values drawn from the population) makes a good estimator of the population mean, as its expected value is equal to the population mean (that is, it is an unbiased estimator). The sample mean is a random variable, not a constant, since its calculated value will randomly differ depending on ...
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. [1] For example, the sample mean is a commonly used estimator of the population mean. There are point and interval ...
which is an unbiased estimator of the variance of the mean in terms of the observed sample variance and known quantities. If the autocorrelations are identically zero, this expression reduces to the well-known result for the variance of the mean for independent data. The effect of the expectation operator in these expressions is that the ...
The sample mean, on the other hand, is an unbiased [5] estimator of the population mean μ. [3] Note that the usual definition of sample variance is = = (¯), and this is an unbiased estimator of the population variance.
We estimate the parameter θ using the sample mean of all observations: = = . This estimator has mean θ and variance of σ 2 / n, which is equal to the reciprocal of the Fisher information from the sample. Thus, the sample mean is a finite-sample efficient estimator for the mean of the normal distribution.
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It arose sequentially in two main published papers. The earlier version of the estimator was developed in 1956, [1] when Charles Stein reached a relatively shocking conclusion that while the then-usual estimate of the mean, the sample mean, is admissible when , it is inadmissible when .
To estimate μ based on the first n observations, one can use the sample mean: T n = (X 1 + ... + X n)/n. This defines a sequence of estimators, indexed by the sample size n. From the properties of the normal distribution, we know the sampling distribution of this statistic: T n is itself normally distributed, with mean μ and variance σ 2 /n.