46 CHAPTER 2 Exploring Data with Tables and Graphs 14.5 20.5 0 14 89.5 104.5 STEP 1: List the class limits from Table 2-2. STEP 2: Split the difference as shown. STEP 3: Find the first and last values of 20.5 and 104.5 by projecting the same pattern from Step 2. 15 29 29.5 30 44 44.5 45 59 59.5 60 74 74.5 75 89 90 104 FIGURE 2-1 Finding Class Boundaries from Class Limits in Table 2-2 Procedure for Constructing a Frequency Distribution We construct frequency distributions to (1) summarize large data sets, (2) see the distribution of the data, (3) identify outliers, and (4) have a basis for constructing graphs (such as histograms, introduced in Section 2-2). Technology can generate frequency distributions, but here are the steps for manually constructing them: 1. Select the number of classes, usually between 5 and 20. The number of classes might be affected by the convenience of using round numbers. (According to “Sturges’ guideline,” the ideal number of classes for a frequency distribution can be approximated by 1 + 1logn2>1log22 where n is the number of data values. We don’t use this guideline in this book.) 2. Calculate the class width. Class width ≈ 1maximum data value2 - 1minimum data value2 number of classes Round this result to get a convenient number. (It’s usually best to round up.) Using a specific number of classes is not too important, and it’s usually wise to change the number of classes so that they use convenient values for the class limits. 3. Choose the value for the first lower class limit by using either the minimum value or a convenient value below the minimum. 4. Using the first lower class limit and the class width, list the other lower class limits. (Do this by adding the class width to the first lower class limit to get the second lower class limit. Add the class width to the second lower class limit to get the third lower class limit, and so on.) CAUTION Finding the correct class width can be tricky. For class width, don’t make the most common mistake of using the difference between a lower class limit and an upper class limit. See Table 2-2 and note that the class width is 15, not 14. CAUTION For class boundaries, remember that they split the difference between the end of one class and the beginning of the next class, as shown in Figure 2-1. No Phones or Bathtubs Many statistical analyses must consider changing characteristics of populations over time. Here are some observations of life in the United States from 100 years ago: • 8% of homes had a telephone. • 14% of homes had a bathtub. • The mean life expectancy was 47 years. • The mean hourly wage was 22 cents. • There were approximately 230 annual murders in the entire United States. Although these observations from 100 years ago are in stark contrast to the United States of today, statistical analyses should always consider changing population characteristics that might have more subtle effects.
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