STATISTICS IN SUMMARY
Chapter Specifics
• Stemplots, histograms, and boxplots all describe the distributions of quantitative variables.
• Density curves also describe distributions. A density curve is a curve with area exactly 1 underneath it whose shape describes the overall pattern of a distribution.
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• An area under the curve gives the proportion of the observations that fall in an interval of values.
• You can roughly locate the median (equal-areas point) and the mean (balance point) by eye on a density curve.
• Stemplots, histograms, and boxplots are created from samples. Density curves are intended to display the idealized shape of the distribution of the population from which the samples are taken.
• Normal curves are a special kind of density curve that describes the overall pattern of some sets of data. Normal curves are symmetric and bell-shaped. A specific Normal curve is completely described by its mean and standard deviation. You can locate the mean (center point) and the standard deviation (distance from the mean to the change-of-curvature points) on a Normal curve. All Normal distributions obey the 68–95–99.7 rule.
• Standard scores express observations in standard deviation units about the mean, which has standard score 0. A given standard score corresponds to the same percentile in any Normal distribution. Table B gives percentiles of Normal distributions.
Chapters 10, 11, and 12 provided us with a strategy for exploring data on a single quantitative variable.
• Make a graph, usually a histogram or stemplot.
• Look for the overall pattern (shape, center, variability) and striking deviations from the pattern.
• Choose the five-number summary or the mean and standard deviation to briefly describe the center and variability in numbers.
In this chapter, we added another step: sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth density curve, such as the Normal curve. This step also allows us to identify “a large number of observations” as a population and use density curves to describe the distribution of a population. We did precisely this when we used the Normal distribution to describe the distribution of the heights of all young women or the scores of all students on the SAT exam.
Using a density curve to describe the distribution of a population is a convenient summary, allowing us to determine percentiles of the distribution without having to see a list of all the values in the population. It also suggests the nature of the conclusions we might draw about a single quantitative variable. Use statistics that describe the distribution of the sample to draw conclusions about parameters that describe the distribution of a population. We will explore this in future chapters.
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CASE STUDY EVALUATED The Normal curve that best approximates the distribution of body temperatures in Figures 13.2 and 13.3 has mean 98.25°F and standard deviation 0.73°F. Use what you have learned in this chapter to answer the following questions.
1. According to the 68–95–99.7 rule, 68% of body temperatures fall between what two values? Between what two values do 95% of body temperatures fall?
2. There was a time when 98.6°F was considered the average body temperature. Given what you know about the distribution of body temperatures given in Figures 13.2 and 13.3, what percentage of individuals would you expect to have body temperatures greater than 98.6°F? What percentage would you expect to have body temperature less than 98.6°F?
Online Resources
• LearningCurve has good questions to check your understanding of the concepts.