Variance

In probability theory and statistics, variance is a measure of dispersion, meaning it is a measure of how far a set of numbers are spread out from their average value. It is defined as the expected value of the squared deviation from the mean of a random variable. The standard deviation is the square root of the variance. Technically, it is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by ⁠ σ 2 {\displaystyle \sigma ^{2}} ⁠, ⁠ s 2 {\displaystyle s^{2}} ⁠, ⁠ Var ⁡ ( X ) {\displaystyle \operatorname {Var} (X)} ⁠, ⁠ V ( X ) {\displaystyle V(X)} ⁠, or ⁠ V ( X ) {\displaystyle \mathbb {V} (X)} ⁠. An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished. Another disadvantage is that the variance is not finite for many distributions. There are two distinct concepts that are both called "variance". One, as discussed above, is part of a theoretical probability distribution and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real-world system. If all possible observations of the system are present, then the calculated variance is called the population variance. Normally, however,...