88 Chapter 3 Numerical Summaries of Data New 90.0 92.2 94.9 92.7 91.6 88.2 92.0 98.2 96.0 91.1 89.8 91.5 91.5 90.6 93.1 88.9 92.5 92.4 96.7 93.7 93.9 87.9 90.4 92.0 90.5 95.2 94.3 92.0 94.6 93.7 94.0 89.3 90.1 91.3 92.7 94.5 Recycled 91.8 94.5 93.9 77.3∗ 92.0 89.9 87.9 92.8 93.3 92.6 90.3 92.8 91.6 92.7 91.7 89.3 95.5 93.6 92.4 91.7 91.6 91.1 88.0 92.4 88.7 92.9 92.6 91.7 97.4 95.1 96.7 77.5∗ 91.4 90.5 95.2 93.1 ∗Measurement is in error due to a defective gauge. It is difficult to determine by looking at the tables whether the thicknesses tend to differ between new and recycled wafers. To interpret these data sets, we need to summarize them inways that will reveal the important features. Histograms, stem-and-leaf plots, and dotplots are graphical summaries of data sets. While graphs are excellent tools for visualizing the important features of a data set, they have limitations. In particular, graphs often cannot measure a feature precisely; for precise descriptions, we need to use numbers. In this chapter, we will learn about several of the most commonly used numerical summaries of data. Some of these describe the center of the data; these are called measures of center. Others describe how spread out the data values are; these are called measures of spread. Still others, called measures of position, specify the proportion of the data that is less than a given value. In the case study at the end of the chapter, you will be asked to use some of the summaries introduced in the chapter to help determine which type of wafer will produce better results. SECTION 3.1 Measures of Center Objectives 1. Compute the mean of a data set 2. Compute the median of a data set 3. Compare the properties of the mean and median 4. Find the mode of a data set 5. Approximate the mean with grouped data Objective 1 Compute the mean of a data set The Mean How do instructors determine your final grade? It’s the end of the semester, and you have just finished your statistics class. During the semester, you took five exams, and your scores were 78, 83, 92, 68, and 85. Your instructor must find a single number to give a summary of your performance. The quantity he or she is most likely to use is the arithmetic mean, which is often simply called the mean. To find the mean of a list of numbers, add the numbers, then divide by how many numbers there are. EXAMPLE 3.1 Computing the mean Explain It Again The mean and the average: Some people refer to the mean as the ‘‘average.’’ In fact, there are many kinds of averages; the mean is just one of them. Find the mean of the exam scores 78, 83, 92, 68, and 85. Solution Step 1: Add the numbers. 78 + 83 + 92 + 68 + 85 = 406 Step 2: Divide the sum by the number of observations. There were five observations. Therefore, the mean is Mean = 406 5 = 81.2
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