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Variance
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===={{visible anchor|Unbiased sample variance}}==== Correcting for this bias yields the ''unbiased sample variance'', denoted <math>S^2</math>: <math display="block">S^2 = \frac{n}{n - 1} \tilde{S}_Y^2 = \frac{n}{n - 1} \left[ \frac{1}{n} \sum_{i=1}^n \left(Y_i - \overline{Y}\right)^2 \right] = \frac{1}{n - 1} \sum_{i=1}^n \left(Y_i - \overline{Y} \right)^2</math> 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 {{math|''n'' β 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 {{math|''n'' β 1.5}} yields an almost unbiased estimator. The unbiased sample variance is a [[U-statistic]] for the function {{math|1=''f''(''y''<sub>1</sub>, ''y''<sub>2</sub>) = (''y''<sub>1</sub> β ''y''<sub>2</sub>)<sup>2</sup>/2}}, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population. ===== Example ===== For a set of numbers {10, 15, 30, 45, 57, 52, 63, 72, 81, 93, 102, 105}, if this set is the whole data population for some measurement, then variance is the population variance 932.743 as the sum of the squared deviations about the mean of this set, divided by 12 as the number of the set members. 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.
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