Normal Distributions, and Order from Diversity via the Central Limit Theorem
As the head of a team of scientists, you are interested in compiling data on the heights of adult American men (by “American” we mean those who live within the borders of the United States). To do this, you would first want to choose a large sample of this population according to scientific principles ensuring that the sample is random. We will have more to say about randomness in the next several chapters, but essentially what you want is a sample representative of the entire population under study. So, for example, since you are interested in the population of adult American men your sample will not consist exclusively of men living in the Bronx in New York, nor will it consist solely of men residing on the west coast, or of those who earn more than $100,000 a year, or, in general, any category whatsoever restricting the sample from being representative of the entire population. One way you could go about choosing a random sample is to assign a unique number to each adult male in the population, and then select a certain portion of these numbers using a random device such that each number has the same probability of being selected. For example, you could write the numbers on cards, toss all the cards into a (very large) hat, mix well, and then select numbers with replacement (that is, after each selection you put the selected card back into the hat, mix again and select another card, etc.). Replacement of the cards is necessary to ensure that each card always has the same probability of being selected. If you happen to select a card that you had already chosen previously, it corresponds to an adult male chosen before, so we just ignore this observation, replace the card, mix, and keep repeating until a previously unchosen card is selected. In practice, of course, this procedure would be impossible to carry through—for one thing you would be hard pressed to find a hat big enough to hold the cards! It is also very hard to mix a lot of things so that a reasonable approximation to a uniform distribution model is achieved—see Section 13.6 for an interesting example. Statisticians have more sophisticated ways of finding a random sample; we’ll take another look at this in Chapter 14. For now let’s suppose you have this sample. What you are going to do is measure these individuals as accurately as you can and record these measurements. You will have a bunch of very large books in which each person’s height is written down, and when the head of the Department of Demographics comes to your office and asks “What can you tell me about the height of the American adult male?” you will proudly haul out the books and say, “Take a look—it’s all in there.”
KeywordsCentral Limit Theorem Standard Normal Distribution Standard Normal Variable Class Interval Bernoulli Trial
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