2-1 Frequency Distributions for Organizing and Summarizing Data 51 Gaps Example 5 illustrates this principle: The presence of gaps can suggest that the data are from two or more different populations. The converse of this principle is not true, because data from different populations do not necessarily result in gaps. Table 2-9 is a frequency distribution of the weights (grams) of randomly selected pennies. Examination of the frequencies reveals a large gap between the lightest pennies and the heaviest pennies. This suggests that we have two different populations: Pennies made before 1983 are 95% copper and 5% zinc, but pennies made after 1983 are 2.5% copper and 97.5% zinc, which explains the large gap between the lightest pennies and the heaviest pennies represented in Table 2-9. YOUR TURN. Do Exercise 21 “Analysis of Last Digits” and determine whether there is a gap. If so, what is a reasonable explanation for it? EXAMPLE 5 Exploring Data: What Does a Gap Tell Us? TABLE 2-9 Randomly Selected Pennies Weight (grams) of Penny Frequency 2.40–2.49 18 2.50–2.59 19 2.60–2.69 0 2.70–2.79 0 2.80–2.89 0 2.90–2.99 2 3.00–3.09 25 3.10–3.19 8 Statistical Literacy and Critical Thinking 1.Boston Commute Time The accompanying table summarizes daily commute times in Boston. How many commute times are included in the summary? Is it possible to identify the exact values of all of the original data amounts? 2. Boston Commute Time Refer to the accompanying frequency distribution. What problem would be created by using classes of 0–30, 30–60, . . . , 120–150? 3.Relative Frequency Distribution Use percentages to construct the relative frequency distribution corresponding to the accompanying frequency distribution for daily commute time in Boston. 2-1 Basic Skills and Concepts Table for Exercises 1, 2 & 3. Daily Commute Time in Boston (minutes) Frequency 0–29 468 30–59 422 60–89 92 90–119 10 120–149 8 Frequency Distributions Access tech supplements, videos, and data sets at www.TriolaStats.com Frequency distributions are often easy to obtain after generating a histogram, as described in Section 2-2. With Statdisk, for example, we can generate a histogram with a desired starting point and class width, then move the cursor over the histogram to see the frequency for each class. If histograms are not used, “sort” the data (arrange them in order) so that we can see the maximum data value and the minimum data value used for computing the class width. Once the class limits are established, it is easy to find the frequency for each class using sorted data. Every statistics software package includes a sort feature. TECH CENTER
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