The Flaw of Averages

1. The Flaw of Averages Although understood by mathematicians & statisticians for centuries, the Flaw of Averages was named by Sam Savage, currently a Consulting Professor at Stanford University. To paraphrase Sam, An estimate based on the assumption that average conditions will occur will almost always be wrong (with there usually being no way to determine ahead of time whether the estimate will be optimistic or pessimistic). • For this portion of the session, the learning objectives are: • Learn why managers should confront and communicate uncertainties instead of ignoring them. • Learn how a manager can get into trouble by not understanding “The Flaw of Averages”.

2. Statistician Drowns in River with Average Depth of 3 Feet!

3. Statistician is Comfortable on Average! A statistician has is his head in a _________ and his feet in a ________. When asked how he feels, he says, “On average, I feel comfortable.”

4. Example 1 -- Optimistic This week’s demand for an item is estimated to be one of 6 equally-likely values: 5000, 7000, 9000, 11,000, 13,000, or 15,000 units. The item’s unit selling price is $9 and unit production cost is $6. As CFO, you have been informed that this week’s production quantity will be 10,000 and have been asked to provide a SINGLE NUMBER as the estimate for this week’s contribution to profit and overhead; that is, you have been asked to estimate the average contribution to profit and overhead. What is your estimate? Units Sold = Revenue from Sales = Cost of Production = Contribution to Profit & Overhead =

5. Example 1 (continued) An estimate of ($9-$6)(10,000) = $30,000 is incorrect because it is based on the Flaw of Averages (FofA). To understand the flaw, consider the following table (keeping in mind that the production quantity is fixed at 10,000): Whereas FofA estimates average Contribution as $30,000, the actual average is: FofA results in a very OPTIMISTIC estimate, overestimating the actual average by The optimism results because FofA counts “upside” even though it will never occur. Instead of $30,000 being the AVERAGE Contribution, it is the MAXIMUM!

6. Example 2 -- Pessimistic Your firm has purchased for $1 a (European) call option on a stock with a exercise date of one week from today and an exercise price of $18; that is, for $1, you have purchased the guaranteed right to sell the stock one week from today at $18. The price per share of the stock one week from today is estimated to be one of 6 equally-likely values: $15, $17, $19, $21, $23, or $25. As CFO, you have been asked to provide a SINGLE NUMBER as the estimate for option’s value; that is, you have been asked to estimate the option’s average value. What is your estimate? Stock’s Price (one week from today) = Option’s Payoff = Option’s Cost = Option’s Net Value =

7. Example 2 (continued) An estimate of ($20-$18)-$1 = $1 is incorrect because it is based on the Flaw of Averages (FofA). To understand the flaw, consider the following table (keeping in mind that the option’s exercise price is $18): Whereas FofA estimates the option’s average value as $1, the actual average is: FofA results in a very PESSIMISTIC estimate, underestimating the actual average by The pessimism results because FofA counts “downside” even though it will never occur.

8. Review of Terminology and Notation • A random variableX is a value about which we are uncertain. • The following are equivalent ways of referring to the same thing: • The mean of X, denoted by µX. • The expected value of X, denoted by E[X]. • The (weighted)average value of X, denoted by .

9. Example 3 -- with Two Random Variables Jeff and Jackie are analysts for J&J Industries. They have been asked to forecast the average daily sales (S) for a product with both uncertain daily production (P) and uncertain daily demand (D). The tables below summarize this uncertainty, where it assumed that P & D are independent random variables. • Jeff and Jackie note that: • If, for example, P=100 & D=112, then S=100. • If, for example, P=100 & D=96, then S=96. • In general, for a given P & D, S = min(P,D). So, Jeff’s and Jackie’s goal is to forecast the mean of min(P,D).

10. To forecast the mean of min(P,D), Jeff notes that Mean of P = 100 and Mean of D = 104 and concludes that However, Jackie disapproves of Jeff’s approach. To forecast the mean of min(P,D), Jackie uses the following tabular approach:

11. FLAW OF AVERAGES In general, if X1, X2, …, Xn are random variables, and f(X1, X2, …, Xn) is a function of these n random variables, then In words instead of symbols, the “Flaw of Averages” states that the mean of a function of random variables does not equal the function of the means of the random variables. does NOT equal Whose forecast of sales is correct -- Jeff’s forecast of 100 or Jackie’s forecast of 95? Jackie is correct, since she computed the mean of min(P,D) from “first principles”, whereas Jeff is incorrect, because he incorrectly assumed that Mean of min(P,D) = min(Mean of P, Mean of D) Jeff’s error illustrates the so-called

12. In performing an analysis, instead of directly confronting uncertainty, many managers “hide” from uncertainty by using the mean of a random variable as a proxy for the random variable’s probability distribution. Because of the Flaw of Averages, “hiding” from uncertainty almost always results in incorrect results. In our “baby” example, Jeff’s “hiding” from uncertainty resulted in an incorrect forecast of 100, whereas Jackie’s confronting uncertainty resulted in a correct forecast of 95. So, even in our “baby” example, the forecast error for Jeff was As we will see, Simulation enables us to confront and communicate uncertainty instead of “hiding” from it. In particular, the software Crystal Ball enables simulation within an Excel spreadsheet.