The Ratio: A Possible Clash Between Instinct and Science
Authors: Capen E.C. , Clapp R.V. 
Published: December 2007

The moon is green cheese. Man will never fly. Woman is the lesser sex. 6/2 equals 3. With such long-established "truths" biting the dust, where can a body turn? We can't restore your illusions, but we can improve your ratio calculations. (You say you didn't know they needed improving?) Uncertainty about the denominator of a fraction leads to a calculation bias of dangerous, proportions in some oil patch problems. Luckily there are correction factors to bail us out.

Introduction

Were we to start out by telling you that long division was illegal, you would probably think the editors had lost their usual good sense and gone in for scientific sensationalism. Instead let us soften the blow somewhat and put it this way: In the real world of uncertainty, the arithmetic operation of division, as it is normally carried out, yields answers that tend to be too high (i.e., they are biased) and may lead to erroneous and costly decisions.Less technically - if you divide, you may foul up the works.We shall have several difficulties in presenting this work, skepticism being not the least. And what of our trusted third grade teacher? Are we to make her out a story teller, or worse? And all the other scholars who these many centuries have unashamedly followed the conventional rules of dividing? Would we slander them? Perhaps not. We think many people live in a world of "givens." Given that X = 2 and given that Y = 6, what is Y/X? Under those conditions, we have no quarrel with the answer of 3, which almost everyone teamed as a child. Take away the "givens" and me have a new game to play --- a game that mom accurately describes the real world where little if anything is known for certain. Now we would have to state the problems a bit differently. Say we don't know X but think it is 2. We don't know Y but think it is 6. What should we estimate for Y/X? If we say 3 this time, we could be in for trouble. Not the kind of transitory trouble that comes from having X and Y come out differently from our guess on this one problem, but a more fundamental difficulty. Even if one problem, but a more fundamental difficulty. Even if one argues convincingly that his estimates of X and Y are quite good on the average, we are in the position of concluding that his ratio Y/X is not good on the average.Intuition does not lead to such a conclusion. So we shall have to spend the next several pages trying to tell you why intuition may be unreliable. We need to cover several examples and several ways of whipping the problem. The discussion will start with error analysis problem. The discussion will start with error analysis (reserves or cost estimates) based on data, move to ratios involving data (competitive bidding), hit the question of when - before or after - to do the averaging when summing (mobility ratio), and finally converge on the most common and perhaps most mundane problem---the one using only one piece of data (or no data at all save one's educated guess) and requiring division of one number by another. Example: The much used price-to-earnings ratio as a measure of stock quality comes from dividing an earnings estimate into a known stock price and is thus afflicted with the malady we are describing.The best advice we can give: If you have to divide, watch out.

JPT

P. 483