Variance

What is variance?

The variance is defined as the square of standard deviation. It is given by σ² = 1 / N ( ∑ fi ( x_{i - )²}

The variance of a random variable is the amount of its statistical dispersion, showing how far the expected value from its values. It is the second central moment and it is also the second cumulant.

Variance of a Random Variable (X):

Its formula is given as, Var( X ) = E [ ( X - μ )² ] Where μ= E(X) and E(X) is the mean of the random variable X. In the lay man's terms it can be said as the average of the square of the distance of each data point from the mean. In other words, it is the mean squared deviation.

Facts about Variance:

• Many distributions, for instance, the Cauchy distribution does not have a variance since the relevant integral diverges.
• In particular cases, if a distribution does not possess expected value, then it will not have variance either. But, the vice versa is not true. That is, there are distributions for which the expected value exists, but variance does not exist.
• The variance will never be negative, since the squares are positive or zero.
• The standard deviation is the square root of the variance.
• The variance does not depend on the mean value μ and it can be proved also. For instance, if the variable is displaced/moved an amount c by taking X+c, the variance of the resulting variable remains untouched.
• But vice versa, if the variable is multiplied by a scaling factor b, the variance is multiplied by b2. More officially, if a and b are real constants and X is a random variable whose variance is given as,

Var (bX + c) = b²var(X)

The often used formula to calculate the variance is as follows:

Var(X) = E(X²) - (E(X))²

Summary:

One of the main reasons for giving preference to the usage of the variance is that the variance of the sum/difference of independent variables is the sum of their variances.