124 Chapter 3 Numerical Summaries of Data SECTION 3.3 Measures of Position Objectives 1. Compute and interpret z-scores 2. Compute the percentiles of a data set 3. Compute the quartiles of a data set 4. Compute the five-number summary for a data set 5. Understand the effects of outliers 6. Construct boxplots to visualize the five-number summary and outliers Objective 1 Compute and interpret z-scores The z-Score Who is taller, a man 73 inches tall or a woman 68 inches tall? The obvious answer is that the man is taller. However, men are taller than women on the average. Let’s ask the question this way: Who is taller relative to their gender, a man 73 inches tall or a woman 68 inches tall? One way to answer this question is with a z-score. The z-score of an individual data value tells how many standard deviations that value is from its population mean. So, for example, a value one standard deviation above the mean has a z-score of 1. A value two standard deviations below the mean has a z-score of −2. DEFINITION Let x be a value from a population with mean μ and standard deviation σ. The z-score for x is z = x − μ σ EXAMPLE 3.22 Computing and interpreting z-scores A National Center for Health Statistics study states that the mean height for adult men in the United States is μ = 69.4 inches, with a standard deviation of σ = 3.1 inches. The mean height for adult women is μ = 63.8 inches, with a standard deviation of σ = 2.8 inches. Who is taller relative to their gender, a man 73 inches tall, or a woman 68 inches tall? Solution We compute the z-scores for the two heights: z-score for man’s height = x − μ σ = 73 − 69.4 3.1 = 1.16 z-score for woman’s height = x − μ σ = 68 − 63.8 2.8 = 1.50 The height of the 73-inch man is 1.16 standard deviations above the mean height for men. The height of the 68-inch woman is 1.50 standard deviations above the mean height for women. Therefore, the woman is taller, relative to the population of women, than the man is, relative to the population of men. z-scores and the Empirical Rule z-scores work best for populations whose histograms are approximately bell-shaped—that is, for populations for which we can use the Empirical Rule. Recall that the Empirical Rule says that for a bell-shaped population, approximately 68% of the data will be within one standard deviation of the mean, approximately 95% will be within two standard deviations, and almost all will be within three standard deviations. Since the z-score is the number
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