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19: Deviation from the Mean

  • Page ID
    • Eric Lehman, F. Thomson Leighton, & Alberty R. Meyer
    • Google and Massachusetts Institute of Technology via MIT OpenCourseWare
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    In the previous chapter, we took it for granted that expectation is useful and developed a bunch of techniques for calculating expected values. But why should we care about this value? After all, a random variable may never take a value anywhere near its expectation.

    The most important reason to care about the mean value comes from its connection to estimation by sampling. For example, suppose we want to estimate the average age, income, family size, or other measure of a population. To do this, we determine a random process for selecting people—say, throwing darts at census lists. This process makes the selected person’s age, income, and so on into a random variable whose mean equals the actual average age or income of the population. So, we can select a random sample of people and calculate the average of people in the sample to estimate the true average in the whole population. But when we make an estimate by repeated sampling, we need to know how much confidence we should have that our estimate is OK, and how large a sample is needed to reach a given confidence level. The issue is fundamental to all experimental science. Because of random errors—noise—repeated measurements of the same quantity rarely come out exactly the same. Determining how much confidence to put in experimental measurements is a fundamental and universal scientific issue. Technically, judging sampling or measurement accuracy reduces to finding the probability that an estimate deviates by a given amount from its expected value.

    Another aspect of this issue comes up in engineering. When designing a sea wall, you need to know how strong to make it to withstand tsunamis for, say, at least a century. If you’re assembling a computer network, you might need to know how many component failures it should tolerate to likely operate without maintenance for at least a month. If your business is insurance, you need to know how large a financial reserve to maintain to be nearly certain of paying benefits for, say, the next three decades. Technically, such questions come down to finding the probability of extreme deviations from the mean.

    This issue of deviation from the mean is the focus of this chapter.

    This page titled 19: Deviation from the Mean is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Eric Lehman, F. Thomson Leighton, & Alberty R. Meyer (MIT OpenCourseWare) .

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