of this effect, since it equals the income level that is exceeded by half the population, regardless of by how much. Finally, a third candidate for the measure of central value is the mode, which is the most likely value of the distribution or the outcome with the highest probability. However, the expected value is by far the most widely used measure of central or average tendency. We now turn to the characterization of the risk implied by the nature of the probability distribution of returns. In general, it is impossible to quantify risk by a single number. The idea is to describe the likelihood and magnitudes of "surprises" (deviations from the mean) with as small a set of statistics as is needed for accuracy. The easiest way to accomplish this is to answer a set of questions in order of their informational value and to stop at the point where additional questions would not affect our notion of the risk-return trade-off. The first question is, "What is a typical deviation from the expected value?" A natural answer would be, "The expected deviation from the expected value is ." Unfortu- nately, this answer is not helpful because it is necessarily zero: Positive deviations from the mean are offset exactly by negative deviations. There are two ways of getting around this problem. The first is to use the expected ab- solute value of the deviation which turns all deviations into positive values. This is known as MAD (mean absolute deviation), which is given by   n Pr(s) Absolute value[r (s) E(r)] s 1   The second is to use the expected squared deviation from the mean, which also must be positive, and which is simply the variance of the probability distribution:   n 2 Pr(s)[r(s) E(r)]2 s 1   Note that the unit of measurement of the variance is "percent squared." To return to our orig- inal units, we compute the standard deviation as the square root of the variance, which is II. Portfolio Theory 6. Risk and Risk Aversion The McGraw−Hill Companies, 2001           CHAPTER 6 Risk and Risk Aversion 173     Figure 6A.1 Skewed probability distributions for rates of return on a portfolio.     A B