50 CHAPTER 2 Exploring Data with Tables and Graphs Table 2-7 satisfies these two conditions. The frequencies start low, increase to the maximum of 18, and then decrease to a low frequency. Also, the frequencies of 2, 4, and 10 that precede the maximum are a mirror image of the frequencies 10, 4, and 2 that follow the maximum. Real data sets are usually not so perfect as Table 2-7, and judgment must be used to determine whether the distribution comes close enough to satisfying those two conditions. (There are more objective procedures described later.) If we consider the Los Angeles commute times from Table 2-1 and their frequency distribution summarized in Table 2-2, we see that the frequencies start low at 6, increase to a maximum of 18 and then decrease to a low frequency of 1. The distribution is far from symmetric, so the data do not satisfy the criteria for being a normal distribution. The Los Angeles commute times have some other distribution that is clearly not a normal distribution. TABLE 2-7 Frequency Distribution Showing a Normal Distribution Time Frequency Normal Distribution 0–14 2 dFrequencies start low, . . . 15–29 4 30–44 10 45–59 18 dIncrease to this maximum, . . . 60–74 10 75–89 4 90–104 2 dDecrease to become low again. Analysis of Last Digits Example 4 illustrates this principle: Frequencies of last digits sometimes reveal how the data were collected or measured. TABLE 2-8 Last Digits of Pulse Rates from the National Health and Examination Survey Last Digit of Pulse Rate Frequency 0 455 1 0 2 461 3 0 4 479 5 0 6 425 7 0 8 399 9 0 Upon examination of measured pulse rates from 2219 adults included in the National Health and Examination Survey, the last digits of the recorded pulse rates are identified and the frequency distribution for those last digits is as shown in Table 2-8. Here is an important observation of those last digits: All of the last digits are even numbers. If the pulse rates were counted for 1 full minute, there would surely be a large number of them ending with an odd digit. So what happened? One reasonable explanation is that even though the pulse rates are the number of heartbeats in 1 minute, they were likely counted for 30 seconds and the number of beats was doubled. (The original pulse rates are not all multiples of 4, so we can rule out a procedure of counting for 15 seconds and then multiplying by 4.) Analysis of these last digits reveals to us the method used to obtain these data. In many surveys, we can determine that surveyed subjects were asked to report some values, such as their heights or weights, because disproportionately many values end in 0 or 5. This is a strong clue that the respondent is rounding instead of being physically measured. Fascinating stuff! YOUR TURN. Do Exercise 21 “Analysis of Last Digits.” EXAMPLE 4 Exploring Data: How Were the Pulse Rates Measured?
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