112 Chapter 3 Numerical Summaries of Data Step 5: The sum of the frequencies is 50. Since we are considering the data to be a sample, we subtract 1 from this sum to obtain 49. The sample variance is 292,800/49 = 5975.51020. Step 6: The sample standard deviation is √ 5975.51020 = 77.3014. Objective 5 Use the Empirical Rule to summarize data that are unimodal and approximately symmetric The Empirical Rule Many histograms have a single mode near the center of the data, and are approximately symmetric. Such histograms are often referred to as bell-shaped. Other histograms are strongly skewed; these are not bell-shaped. When a data set has a bell-shaped histogram, it is often possible to use the standard deviation to provide an approximate description of the data using a rule known as the Empirical Rule. The Empirical Rule When a population has a histogram that is approximately bell-shaped, then • Approximately 68% of the data will be within one standard deviation of the mean. In other words, approximately 68% of the data will be between μ − σ and μ + σ. • Approximately 95% of the data will be within two standard deviations of the mean. In other words, approximately 95% of the data will be between μ − 2σ and μ + 2σ. • All, or almost all, of the data will be within three standard deviations of the mean. In other words, all, or almost all, of the data will be between μ−3σ and μ+3σ. CAUTION The Empirical Rule should not be used for data sets that are not approximately bell-shaped. The Empirical Rule holds for many bell-shaped data sets. Figure 3.6 illustrates the Empirical Rule. Almost all ≈ 95% ≈ 68% μ − 3σ μ − 2σ μ − σ μ μ + σ μ + 2σ μ + 3σ Figure 3.6 The Empirical Rule. Approximately 68% of the data values are between μ − σ and μ + σ, approximately 95% are between μ − 2σ and μ + 2σ, and almost all are between μ − 3σ and μ + 3σ. EXAMPLE 3.17 Using the Empirical Rule to describe a data set Table 3.8 presents the percentage of the population aged 65 and over in each state and the District of Columbia. Figure 3.7 presents a histogram of these data. Compute the mean and standard deviation, and use the Empirical Rule to describe the data. Table 3.8 Percentage of People Aged 65 and Over in Each of the 50 States and District of Columbia 14.1 8.1 13.9 14.3 11.5 10.7 14.4 14.1 11.5 17.8 14.0 10.2 14.3 12.0 12.4 12.7 14.9 13.4 13.1 12.6 15.6 12.2 13.7 12.8 12.4 12.8 13.9 15.0 13.8 12.3 12.6 13.7 14.1 13.6 12.4 15.3 13.7 13.8 13.0 15.5 14.1 13.6 14.6 13.3 10.5 9.0 14.3 12.4 12.2 16.0 13.5
navidi_monk_essential_statistics_1e_ch1_3
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