UNIT 7:
Normal Curves
Summary of the Video
We open by comparing the batting averages of baseball players in different eras. After a brief outline of the statistical content of the video, a histogram of the distribution of salaries in a large company appears. To display only the overall shape of the distribution, we draw a smooth curve through the tops of the bars of the histogram. This is a density curve if we insist that the total area be exactly 1. The salary distribution was skewed to the right. The distribution of heights of young women, on the other hand, is quite symmetric.
The video presents the Boston Beanstalks Club, a social club for tall people—women must be at least 5 feet 10 inches, men at least 6 feet 2 inches—and asks how common people this tall are. Heights are approximately described by a normal curve . We see the form of normal curves and learn to locate the mean and standard deviation. Any normal curve is completely described by its mean µ and standard deviation
, and all normal curves are alike when we measure in standard deviations away from the mean. This leads to the 68—95—99.7 rule. Now we can locate a Boston Beanstalk with the minimum 70inch height for women on the normal curve of young women's heights.
Then Stephen Jay Gould of Harvard asks about the decline of .400 hitters in baseball. He notes that the mean batting average has remained about the same, but that the standard deviation has steadily decreased over time. So .400 is a higher average now in the sense of being more standard deviations above the mean. We can make that precise by standardizing, that is, by measuring in standard deviations away from the mean. It turns out that great hitters of all eras are about equally far above the average if we use standardized batting averages.
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