772 CHAPTER 12 Statistics In addition, it is often suggested that a frequency distribution should be constructed with 5 to 12 classes. If there are too few or too many classes, the distribution may become difficult to interpret. For example, if you use fewer than 5 classes, you risk losing too much information. If you use more than 12 classes, you may gain more detail but you risk losing clarity. Let the spread of the data be a guide in deciding the number of classes to use. To understand these rules, let’s consider a set of observed integral values that go from a low of 0 to a high of 26. Let’s assume that the first class is arbitrarily selected to go from 0 through 4. Thus, any of the data with values of 0, 1, 2, 3, 4 would belong in this class. We say that the class width is 5, since there are five integral values that belong to the class. This first class ended with 4, so the second class must start with 5. If this class is to have a width of 5, at what value must it end? The answer is 9 (5, 6, 7, 8, 9). The second class is 5–9. Continuing in the same manner, we obtain the following set of classes. Classes Lower class limits 0–4 5–9 10–14 15–19 20–24 25–29 ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ Upper class limits We need not go beyond the 25–29 class because the largest value we are considering is 26. The classes meet our three criteria: They have the same width, there is no overlap among the classes, and each of the values from a low of 0 to a high of 26 belongs to one and only one class. The choice of the first class, 0 – 4, was arbitrary. If we wanted to have more classes or fewer classes, we would make the class widths smaller or larger, respectively. The numbers 0, 5, 10, 15, 20, 25 are called the lower class limits, and the numbers 4, 9, 14, 19, 24, 29 are called the upper class limits. Each class has a width of 5. Note that the class width, 5, can be obtained by subtracting the first lower class limit from the second lower class limit: 5 0 5. − = The difference between any two consecutive lower class or upper class limits is also 5. Table 12.1 Example 2 A Frequency Distribution of Novels Table 12.1 shows the number of copies sold for the 30 top-selling novels of all time. The number of copies sold is rounded to the nearest million. Construct a frequency distribution of the data, letting the first class be 50–94 million. Solution The 30 pieces of data are given in descending order from highest to lowest. We are given that the first class is to be 50–94. The second class must therefore start at 95. This is the lower class limit of the second class. To determine the class width, we subtract 50, the lower class limit of the first class, from 95, the lower class limit of the second class, to obtain a class width of 95 50 45. − = The upper class limit of the second class is determined by adding the class width, 45, to the upper class limit of the first class, 94. Therefore, the upper class limit of the second class is 94 45 139. + = Thus, 50–94 first class 95–139 second class The other classes are determined using a similar technique. They are − 140 184, − − − − − − − − 185 229, 230 274, 275 319, 320 364, 365 409, 410 454, 455 499, 500 544. Novel Copies Sold (millions) Don Quixote 500 A Tale of Two Cities 200 The Lord of the Rings 150 The Little Prince 140 Harry Potter and the Sorcerer’s Stone 120 And Then There Were None 100 The Hobbit 100 Dream of the Red Chamber 100 Alice’s Adventures in Wonderland 100 The Lion, the Witch, and the Wardrobe 85 She: A History of Adventure 83 The Adventures of Pinocchio 80 Vardi Wala 80 The Da Vinci Code 80 Harry Potter and the Chamber of Secrets 77 Harry Potter and the Prisoner of Azkaban 65 Harry Potter and the Goblet of Fire 65 Harry Potter and the Order of the Phoenix 65 Harry Potter and the Half-Blood Prince 65 Harry Potter and the Deathly Hallows 65 The Alchemist 65 The Catcher in the Rye 65 Think and Grow Rich 60 The Bridges of Madison County 60 The Adventures of Sherlock Holmes 60 20,000 Leagues Under the Sea 60 One Hundred Years of Solitude 50 Lolita 50 Heidi’s Years of Living and Travel 50 Anne of Green Gables 50 Source: The New York Times
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