What is variance?

The variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are (Wikipedia 2006). The variance of a real-valued random variable is its second central moment, and it also happens to be its second cumulant (Wikipedia 2006). The variance of a random variable is the square of its standard deviation (Wikipedia 2006).

Variance is defined as

Figure 1

if μ = E(X), where E(X) is the expected value (mean) of the random variable X (Wikipedia 2006). That is, it is the expected value of the square of the deviation of X from its own mean (Wikipedia 2006). In plain language, it can be expressed as "The average of the square of the distance of each data point from the mean", thus it is the mean squared deviation (Wikipedia 2006). The variance of random variable X is typically designated as

Figure 2

Figure 3

This definition of variance can be used for both discrete and continuous random variables (Wikipedia 2006).

Many distributions, such as the Cauchy distribution, do not have a variance because the relevant integral diverges (Wikipedia 2006). In particular, if a distribution does not have expected value, it does not have variance either (Wikipedia 2006). The opposite is not true: there are distributions for which expected value exists, but variance does not (Wikipedia 2006).

Variance will never be negative, provided it is defined, because the squares are positive or zero (Wikipedia 2006). The unit of variance is the square of the unit of observation (Wikipedia 2006). Example: The variance of a set of heights measured in centimeters will be given in square centimeters. This is an inconveinient result, and so the standard deviation is generally used (Wikipedia 2006). The standard deviation is the square root of the variance (Wikipedia 2006).

It can be proven easily from the definition that the variance does not depend on the mean value μ (Wikipedia 2006). That is, if the variable is "displaced" an amount b by taking X+b, the variance of the resulting random variable is left untouched (Wikipedia 2006). By contrast, if the variable is multiplied by a scaling factor a, the variance is multiplied by a2 (Wikipedia 2006). More formally, if a and b are real constants and X is a random variable whose variance is defined,

Figure 4

Another formula for the variance that follows in a straightforward manner from the linearity of expected values and the above definition is:

Figure 5

This is often used to calculate the variance in practice (Wikipedia 2006).

One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of independent random variables is the sum of their variances (Wikipedia 2006).

Exercise: Why is variance sometimes called moments of probability distributions? Answer.

References:Wikipedia. "Variance", Wikimedia Foundation Inc, http://en.wikipedia.org/wiki/Variance, Last accessed 14 February 2006.

Brandon Hodgson