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1.5: Multiple Estimates for the Same Quantity

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    24083
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    After making an estimate, it is natural to wonder about how much confidence to place in it. Perhaps we made an embarrassingly large mistake. The best way to know is to estimate the same quantity using another method. As an everyday example, let’s observe how we add a list of numbers.

    12

    15

    +18

    _____

    We often add the numbers first from top to bottom. For 12 + 15 + 18, we calculate, “12 plus 15 is 27; 27 plus 18 is 45.” To check the result, we add the numbers in the reverse order, from bottom to top: “18 plus 15 is 33; 33 plus 12 is 45.” The two totals agree, so each is probably correct: The calculations are unlikely to contain an error of exactly the same amount. This kind of redundancy catches errors.

    In contrast, mindless redundancy offers little protection. If we check the calculation by adding the numbers from top to bottom again, we usually repeat any mistakes. Similarly, rereading written drafts usually means overlooking the same spelling, grammar, or logic faults. Instead, stuff the draft in a drawer for a week, then look at it; or ask a colleague or friend—in both cases, use fresh eyes.

    Reliability, in short, comes from intelligent redundancy.

    This principle helps you make reliable estimates. First, use several methods to estimate the same quantity. Second, make the methods as different from one another as possible—for example, by using unrelated background knowledge. This approach to reliability is another example of divide-and-conquer reasoning: The hard problem of making a reliable estimate becomes several simpler subproblems, one per estimation method.

    You saw an example in Section 1.1, where we estimated the volume of a dollar bill. The first method used divide-and-conquer reasoning based on the width, height, and thickness of the bill. The check was a comparison with a folded-up dollar bill. Both methods agreed on a volume of approximately 1 cubic centimeter—giving us confidence in the estimate.

    For another example of using multiple methods, return to the estimate of the volume of an oil barrel (Section 1.4). We used a roadside asphalt barrel as a proxy for an oil barrel and estimated the volume of the roadside barrel. The result, 60 gallons, seemed plausible, but maybe oil barrels have a completely different size. One way to catch that kind of error is to use a different method for estimating the volume. For example, we might start with the cost of a barrel of oil—about $100 in 2013—and the cost of a gallon of gasoline—about $2.50 before taxes, or 1/40th of the cost of a barrel. If the markup on gasoline is not significant, then a barrel is roughly 40 gallons. Even with a markup, we can still say that a barrel is at least 40 gallons. Because our two estimates, 60 gallons and > 40 gallons, roughly agree, our confidence in both increases. If they had contradicted each other, one or both would be wrong, and we would look for the mistaken assumption, for the incorrect arithmetic, or for a third method.


    This page titled 1.5: Multiple Estimates for the Same Quantity is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Sanjoy Mahajan (MIT OpenCourseWare) .

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