Discuss each of the following topics in class or review the topics on your own. Then write a brief but complete essay in which you summarize the main points. Please include formulas and graphs as appropriate.
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An average is an attempt to summarize a collection of data into just one number. Discuss how the mean, median, and mode all represent averages in this context. Also discuss the differences among these averages. Why is the mean a balance point? Why is the median a midway point? Why is the mode the most common data point? List three areas of daily life in which you think the mean, median, or mode would be the best choice to describe an “average.”
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Why do we need to study the variation of a collection of data? Why isn’t the average by itself adequate? We have studied three ways to measure variation. The range, the standard deviation, and, to a large extent, a box-and-whisker plot all indicate the variation within a data collection. Discuss similarities and differences among these ways to measure data variation. Why would it seem reasonable to pair the median with a box-and-whisker plot and to pair the mean with the standard deviation? What are the advantages and disadvantages of each method of describing data spread? Comment on statements such as the following:
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The range is easy to compute, but it doesn’t give much information;
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the box-and-whisker plot is fairly easy to construct, and it gives a lot of information at a glance.
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Why is the coefficient of variation important? What do we mean when we say that the coefficient of variation has no units? What advantage can there be in having no units? Why is relative size important?
Consider robin eggs; the mean weight of a collection of robin eggs is ounce, and the standard deviation is ounce. Now consider elephants; the mean weight of elephants in the zoo is tons, with a standard deviation tons. The units of measurement are different and there is a great deal of difference between the weight of an elephant and that of a robin’s egg. Yet the coefficient of variation is about the same for both. Comment on this from the viewpoint of the size of the standard deviation relative to that of the mean.
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What is Chebyshev’s theorem? Suppose you have a friend who knows very little about statistics. Write a paragraph or two in which you describe Chebyshev’s theorem for your friend. Keep the discussion as simple as possible, but be sure to get the main ideas across to your friend. Suppose he or she asks, “What is this stuff good for?” and suppose you respond (a little sarcastically) that Chebyshev’s theorem applies to everything from butterflies to the orbits of the planets! Would you be correct? Explain.
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