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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.
Suppose the population is (0,0,0,1,2,9), which has a population mean of 2 and a population variance of /. A sample of n = 1 is drawn, and it turns out to be = The best estimate of the population mean is ¯ = / = / =
In statistics and in particular statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated value of the standard deviation (a measure of statistical dispersion) of a population of values, in such a way that the expected value of the calculation equals the true value.
The arithmetic mean of a population, or population mean, is often denoted μ. [2] 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).
Based on this sample, the estimated population mean is 10, and the unbiased estimate of population variance is 30. Both the naïve algorithm and two-pass algorithm compute these values correctly. Next consider the sample ( 10 8 + 4 , 10 8 + 7 , 10 8 + 13 , 10 8 + 16 ), which gives rise to the same estimated variance as the first sample.
Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator: it underestimates the variance by a factor of (n − 1) / n; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by n-1 instead of n, is called ...
For example, a single observation is itself an unbiased estimate of the mean and a pair of observations can be used to derive an unbiased estimate of the variance. The U-statistic based on this estimator is defined as the average (across all combinatorial selections of the given size from the full set of observations) of the basic estimator ...
However, the sample standard deviation is not unbiased for the population standard deviation – see unbiased estimation of standard deviation. Further, for other distributions the sample mean and sample variance are not in general MVUEs – for a uniform distribution with unknown upper and lower bounds, the mid-range is the MVUE for the ...