42 CHAPTER 2 Descriptive Statistics TRY IT YOURSELF 1 Construct a frequency distribution using the points scored by the 55 winning teams listed on page 39. Use six classes. Answer: Page A35 Note in Example 1 that the classes do not overlap, so each of the original data entries belongs to exactly one class. Also, the classes are of equal width. In general, all classes in a frequency distribution have the same width. However, this may not always be possible because a class can be open-ended. For instance, the frequency distribution for the population of Iowa shown at the left has an open-ended class, “80 and older.” After constructing a standard frequency distribution such as the one in Example 1, you can include several additional features that will help provide a better understanding of the data. These features (the midpoint, relative frequency, and cumulative frequency of each class) can be included as additional columns in your table. The midpoint of a class is the sum of the lower and upper limits of the class divided by two. The midpoint is sometimes called the class mark. Midpoint = (Lower class limit) + (Upper class limit) 2 The relative frequency of a class is the portion, or percentage, of the data that falls in that class. To find the relative frequency of a class, divide the frequency f by the sample size n. Relative frequency = Class frequency Sample size = f n Note that n = Σf. The cumulative frequency of a class is the sum of the frequencies of that class and all previous classes. The cumulative frequency of the last class is equal to the sample size n. DEFINITION You can use the formula shown above to find the midpoint of each class, or after finding the first midpoint, you can find the remaining midpoints by adding the class width to the previous midpoint. For instance, the midpoint of the first class in Example 1 is Midpoint = 155 + 190 2 = 172.5. Midpoint of first class. Using the class width of 36, the remaining midpoints are 172.5 + 36 = 208.5 Midpoint of second class. 208.5 + 36 = 244.5 Midpoint of third class. 244.5 + 36 = 280.5 Midpoint of fourth class. and so on. You can write the relative frequency as a fraction, decimal, or percent. The sum of the relative frequencies of all the classes should be equal to 1, or 100%. Due to rounding, the sum may be slightly less than or greater than 1. So, values such as 0.99 and 1.01 are sufficient. Population of Iowa Ages Frequency 0 –9 396,037 10 –19 418,823 20 –29 422,774 30 –39 399,314 40 – 49 363,809 50 –59 392,536 60 – 69 385,737 70 –79 229,536 80 and older 146,504 The last class, 80 and older, is open-ended. (Source: U.S. Census Bureau) For help with evaluating formulas, order of operations, multiplication and division of integers, operations with fractions and decimals, and significant digits and rounding, see Integrated Review at MyLab® Statistics
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