1-2 Types of Data 19 DEFINITION Data are at the ratio level of measurement if they can be arranged in order, differences can be found and are meaningful, and there is a natural zero starting point (where zero indicates that none of the quantity is present). For data at this level, differences and ratios are both meaningful. Ratio Level EXAMPLE 7 The following are examples of data at the ratio level of measurement. Note the presence of the natural zero value, and also note the use of meaningful ratios of “twice” and “three times.” 1. Heights of Students: Heights of 180 cm and 90 cm for a high school student and a preschool student (0 cm represents no height, and 180 cm is twice as tall as 90 cm.) 2. Class Times: The times of 50 min and 100 min for a statistics class (0 min represents no class time, and 100 min is twice as long as 50 min.) YOUR TURN. Do Exercise 27 “Areas of States.” TABLE 1-2 Levels of Measurement Level of Measurement Brief Description Example Ratio There is a natural zero starting point and ratios make sense. Heights, lengths, distances, volumes Interval Differences are meaningful, but there is no natural zero starting point and ratios are meaningless. Body temperatures in degrees Fahrenheit or Celsius Ordinal Data can be arranged in order, but differences either can’t be found or are meaningless. Ranks of colleges in U.S. News & World Report Nominal Categories only. Data cannot be arranged in order. Eye colors HINT The distinction between the interval and ratio levels of measurement can be a bit tricky. Here are two tools to help with that distinction: 1. Ratio Test Focus on the term “ratio” and know that the term “twice” describes the ratio of one value to be double the other value. To distinguish between the interval and ratio levels of measurement, use a “ratio test” by asking this question: Does use of the term “twice” make sense? “Twice” makes sense for data at the ratio level of measurement, but it does not make sense for data at the interval level of measurement. 2. True Zero For ratios to make sense, there must be a value of “true zero,” where the value of zero indicates that none of the quantity is present, and zero is not simply an arbitrary value on a scale. The temperature of 0°F is arbitrary and does not indicate that there is no heat, so temperatures on the Fahrenheit scale are at the interval level of measurement, not the ratio level. Six Degrees of Separation Social psychologists, historians, political scientists, and communications specialists are interested in “The Small World Problem”: Given any two people in the world, how many intermediate links are necessary to connect the two original people? In the 1950s and 1960s, social psychologist Stanley Milgram conducted an experiment in which subjects tried to contact other target people by mailing an information folder to an acquaintance who they thought would be closer to the target. Among 160 such chains that were initiated, only 44 were completed, so the failure rate was 73%. Among the successes, the number of intermediate acquaintances varied from 2 to 10, with a median of 6 (hence “six degrees of separation”). The experiment has been criticized for its high failure rate and its disproportionate inclusion of subjects with above-average incomes. A more recent study conducted by Microsoft researcher Eric Horvitz and Stanford Assistant Professor Jure Leskovec involved 30 billion instant messages and 240 million people. This study found that for instant messages that used Microsoft, the mean length of a path between two individuals is 6.6, suggesting “seven degrees of separation.” Work continues in this important and interesting field.
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