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Your Trail statistics > variance Table of Contents [show]     1 Definition     2 Properties     3 Computing Keywords Standard deviation, bias, expected value, mean Copyright Information This article is licensed under GNU Public License. Contact webmasters for more information on copyright issues. 1 Definition For a random variable X, if the expected value (or mean) is E[X], then the variance of X, or var(X), or usually denoted as σ2, is E[(X-E[X])2]. 2 Properties var(cX) = E[(cX - E[cX])2] = c2var(X) If X and Y are independent, then var(X+Y) = E[(X+Y-E[X+Y])2] = E[(X-E[X])2 + ([Y-E[Y])2] = E[(X-E[X])2 + (Y-E[Y])2 + 2(Y-E[Y])(X-E[X])] = E[(X-E[X])2] + E[(X-E[X])2] + 2E[(Y-E[Y])(X-E[X])] = var(X) + var(Y) + 2E[XY - XE[Y] + YE[X] + E[X]E[Y]] = var(X) + var(Y) Similarly, var(X-Y) = var(X) + var(Y) if X and Y are independent. 3 Computing Suppose the mean value of all items in vector x is x0, then var(x) = σ( x1-x0)2 + x2-x0)2 + + ... + xn-x0)2)/n Note that here we are talking about the whole population. If we only have values from a sample, then S2 the mean value of the squared difference between each sample item and the sample mean could be used as an estimation of the variance, however, keep in mind that this estimation is biased. Therefore, we more often use s2, which is the sum of the squared difference between each sample item and the sample mean divided by the sample size n minus 1. This s2 is an unbiased estimation of variance. [DL]. Standard deviation is the square root of variance. Comments References [DL] http://davidmlane.com/hyperstat/A16252.html Webster Unabridged Va"ri*ance (?), n. [L. variantia.] 1. The quality or state of being variant; change of condition; variation. 2. Difference that produce dispute or controversy; disagreement; dissension; discord; dispute; quarrel. That which is the strength of their amity shall prove the immediate author of their variance. Shak. 3. (Law) A disagreement or difference between two parts of the same legal proceeding, which, to be effectual, ought to agree, -- as between the writ and the declaration, or between the allegation and the proof. Bouvier. A variance, in disagreement; in a state of dissension or controversy; at enmity. "What cause brought him so soon at variance with himself?" Milton. (search time: 56.096 msec) English-Chinese variance ['v/2ri2ns] ²»Ò»ÖÂ,±ä»¯,±äÒì,·ÖÆç,²»ºÍ (search time: 88.858 msec)