where . Similarly, the second term on the right-hand side becomes where and . Thus the total variance is given by A similar formula is applied in analysis of variance, where the corresponding formula is here refers to the Mean of the Squares. In linear regression analysis the corresponding formula is This can also be derived from the additivity of variances, since the total (observed) score is the sum of the predicted score and the error score, where the latter two are uncorrelated. Similar decompositions are possible for the sum of squared deviations (sum of squares, ): Calculation from the CDF The population variance for a non-negative random variable can be expressed in terms of the cumulative distribution function F using This expression can be used to calculate the variance in situations where the CDF, but not the density, can be conveniently expressed. Characteristic property The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. . Conversely, if a continuous function satisfies for all random variables X, then it is necessarily of the form , where . This also holds in the multidimensional case. Units of measurement Unlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared. For this reason, describing data sets via their standard deviation or root mean square deviation is often preferred over using the variance. In the dice example the standard deviation is , slightly larger than the expected absolute deviation of 1.5. The standard deviation and the expected absolute deviation can both be used as an indicator of the "spread" of a distribution. The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers arising from measurement anomalies or an unduly heavy-tailed distribution. Propagation Addition and multiplication by a constant Variance is invariant with respect to changes in a location parameter. That is, if a constant is added to all values of the variable, the variance is unchanged: If all values are scaled by a constant, the variance is