Variance

Variance measures the dispersion of a set of data points around their mean value. It is a quantitative expression of the variability or spread in a dataset. In finance it is the base measure of risk for a single asset or a portfolio. Mathematically, the variance (${\sigma }^2$, or $\text{Var}<!--\; FUNCTION\; APPLICATION\; -->(X)$) of a random variable $X$ is defined as:

where $E$ is the expectation operator and $\mu =E[X]$ is the mean of $X$. In probability theory, the expected value $E$ is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of the possible values a random variable can take, weighted by the probability of those outcomes.

Expanding the square gives an equivalent form worth knowing because it is the one spreadsheets and one-pass algorithms actually compute — mean of the squares minus square of the mean, no second pass over the data required:

If $X$ is given as a set of samples $(x_1,x_2,\dots x_n)$, then mean (arithmetic average) is

In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a sample of the population. This is generally referred to as sample variance or empirical variance. Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution. Correcting for this bias yields the unbiased sample variance, denoted $S^2$.

In practical terms, a higher variance indicates that the data points are more spread out from the mean.

Standard deviation $\sigma$ is a square root of variance.