806 CHAPTER 12 Statistics Suppose your mathematics teacher states that exam scores for the previous exam followed a bell-shaped distribution and that your score was 1.5 standard deviations above the mean. How does your exam grade compare with the exam grades of your classmates? What percentage of students in your class had an exam grade below your exam grade? In this section, we will discuss sets of data whose histograms approximate bell-shaped distributions and learn how to determine the percentage of data that fall below a particular piece of data in the set of data. The Normal Curve SECTION 12.5 LEARNING GOALS Upon completion of this section, you will be able to: 7 Understand properties of a normal distribution. 7 Calculate a z-score and use it to determine the area under a normal curve. 7 Calculate the percentage of data between any two values in a normal distribution. Why This Is Important There are many real-life applications, such as IQ scores, heights and weights of males, heights and weights of females, and wearout mileage of automobile tires, whose histograms approximate a bell-shaped distribution. Understanding the properties of bell-shaped distributions will help us determine the percentage of data that fall in certain intervals of the distributions of these many real-life applications. When examining data using a histogram, we can refer to the overall appearance of the histogram as the shape of the distribution of the data. Certain shapes of distributions of data are more common than others. In this section, we will illustrate and discuss a few of the more common ones. In each case, the vertical scale is the frequency and the horizontal scale is the observed values. In a rectangular distribution (Fig. 12.17), all the observed values occur with about the same frequency. If a die is rolled many times, we would expect the numbers 1– 6 to occur with about the same frequency. The distribution representing the outcomes of the die is rectangular. Rectangular distribution Frequency Values Figure 12.17 In J-shaped distributions, the frequency is either constantly increasing (Fig. 12.18(a)) or constantly decreasing (Fig. 12.18(b)). The number of hours studied J-shaped Distributions (a) (b) Figure 12.18 Shutterstock
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