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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.
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).
In statistical theory, a U-statistic is a class of statistics defined as the average over the application of a given function applied to all tuples of a fixed size. The letter "U" stands for unbiased. [citation needed] In elementary statistics, U-statistics arise naturally in producing minimum-variance unbiased estimators.
The sample mean is a Fisher consistent and unbiased estimate of the population mean, but not all Fisher consistent estimates are unbiased. Suppose we observe a sample from a uniform distribution on (0,θ) and we wish to estimate θ. The sample maximum is Fisher consistent, but downwardly biased.
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 ...
Consider the sample (4, 7, 13, 16) from an infinite population. 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.
If the set is a sample from the whole population, then the unbiased sample variance can be calculated as 1017.538 that is the sum of the squared deviations about the mean of the sample, divided by 11 instead of 12. A function VAR.S in Microsoft Excel gives the unbiased sample variance while VAR.P is for population variance.