The VARIANCE of random variables measures the spread of the distribution. Variances are always positive because they are the average of squared numbers. The magnitude of the variance determines the average squared distance from the mean of the distribution. We denote the variance of the random variable x as sxx.The variance is mathematically defined as;

• sxx= sum{(xi-mx)(xi-mx)}/n

where sum{} designates summation from i=1 to i=n. The square is written as a product to conform to a general format that will be useful when making comparisons with the covariance.

The STANDARD DEVIATION of a random variable x is simply the square root of the variance

• sx = sqrt(sxx)

It is for this reason that sxx is sometimes written as sx2. Once again, the sxx format is preferred because of the ease of making comparisons with the covariance.

The COVARIANCE between two random variables measures the extent to which they are related. Both the sign and the magnitude of the covariance are interpreted. A positive value means that the variables are directly related, i.e., a positive difference in x is more likely to be associated with a positive difference in y and a negative difference in x is more likely to be associated with a negative difference in y. A negative value means that the variables are inversely related, i.e., a positive difference in x is more likely to be associated with a negative difference in y and a negative difference in x is more likely to be associated with a positive difference in y. A large magnitude indicates the likelihood of these associations are stronger. The mathematical definition of the covariance is

• sxy= sum{(xi-mx)(yi-my)}/n.

For the covariance to be meaningful, the data must be naturally paired.

The CORRELATION COEFFICIENT is invented because of the problem with interpreting the magnitude of sxy as being `large'. How large is large? CORRELATION is defined as a normalized covariance whose magnitude is bounded by -1 and +1. It therefore allows the assessment of a the extent to which the values co-vary. The correlation coefficient is defined as;

• rxy = sxy/(sxsy)

This means that the covariance may be written as sxy=rxysxsy

The mean and variance of a mix, or portfolio of random variables is given by

• mp = sumi{aimi}
• spp = sumi{sumj{aiajsij}}
• sum{ai}=1

These equations are used to determine the mean and standard deviation of returns from a mixture of stochastic returns. They form the basis of portfolio theory. When n=2, the equations reduce to

• kp = a1k1 + a2k2
• spp = a1a1s11+a1a2s12+a2a1s21+a2a2s22
• a1+a2=1

With a little algebra and the commutative property of multiplication and the covariance, the second equation reduces to

• sp = sum{a12s11+2a1a2s12+a22s22}

The plot of kp against sp is a quadratic curve whose shape depends on the correlation between the returns from the two assets 1 and 2 as shown below.