2-1 Frequency Distributions for Organizing and Summarizing Data 49 Cumulative Frequency Distribution Another variation of a frequency distribution is a cumulative frequency distribution in which the frequency for each class is the sum of the frequencies for that class and all previous classes. Let’s return to the Los Angeles commute times provided at the beginning of this chapter. Table 2-6 is the cumulative frequency distribution found from Table 2-2. Using the original frequencies of 6, 18, 14, 5, 5, 1, 1, we add 6 + 18 to get the second cumulative frequency of 24, then we add 6 + 8 + 14 to get the third, and so on. See Table 2-6, and note that in addition to the use of cumulative frequencies, the class limits are replaced by “less than” expressions that describe the new ranges of values. TABLE 2-5 Daily Commute Times in New York, NY and Boise, ID (minutes) Daily Commute Time (minutes) New York, NY Boise, ID 0–14 8.6% 30.3% 15–29 20.3% 45.5% 30–44 24.8% 17.0% 45–59 17.2% 3.5% 60–74 18.5% 2.2% 75–89 3.3% 0.3% 90–104 4.7% 0.3% 105–119 0.1% 0.0% 120–134 0.0% 0.0% 135–149 2.5% 0.9% YOUR TURN. Do Exercise 23 “Oscar Winners.” TABLE 2-6 Cumulative Frequency Distribution of Daily Commute Times in Los Angeles Daily Commute Times in Los Angeles (minutes) Cumulative Frequency Less than 15 6 Less than 30 24 Less than 45 38 Less than 60 43 Less than 75 48 Less than 90 49 Less than 105 50 Critical Thinking: Using Frequency Distributions to Understand Data At the beginning of this section we noted that a frequency distribution can help us understand the distribution of a data set, which is the nature or shape of the spread of the data over the range of values (such as bell-shaped). In statistics we are often interested in determining whether the data have a normal distribution. (Normal distributions are discussed extensively in Chapter 6.) Data that have an approximately normal distribution are characterized by a frequency distribution with the following features. Normal Distribution 1. The frequencies start low, then increase to one or two high frequencies, and then decrease to a low frequency. 2. The distribution is approximately symmetric: Frequencies preceding the maximum frequency should be roughly a mirror image of those that follow the maximum frequency. Growth Charts Updated Pediatricians typically use standardized growth charts to compare their patient’s weight and height to a sample of other children. Children are considered to be in the normal range if their weight and height fall between the 5th and 95th percentiles. If they fall outside that range, they are often given tests to ensure that there are no serious medical problems. Pediatricians became increasingly aware of a major problem with the charts: Because they were based on children living between 1929 and 1975, the growth charts had become inaccurate. To rectify this problem, the charts were updated in 2000 to reflect the current measurements of millions of children. The weights and heights of children are good examples of populations that change over time. This is the reason for including changing characteristics of data over time as an important consideration for a population.
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